Modular Diagonal Quotient Surfaces

E. Kani and W. Schanz

Abstract. The main objective of this paper is to analyze the geometry of the modular diagonal quotient surface Z_{N,ε} = Δ_{ε}\(X(N) × X(N)) which classifies pairs of elliptic curves E_{1} and E_{2} together with an isomorphism of “determinant ε” between their associated modular representations mod N. In particular, we calculate some of the numerical invariants of its minimal desingularization Z̃_{N,ε} such as its Betti and Chern numbers and determine its place in the Enriques-Kodaira classification table.

Introduction

Let X(N) = Γ(N)\ℌ^{∗} denote the modular curve of level N which admits the group G_{N} = Sl_{2}(ℤ/Nℤ)/{±1} as a subgroup of its automorphism group. Then G_{N} × G_{N} acts on the product surface Y_{N} = X(N) × X(N), and hence so does the “graph subgroup” Δ_{ε} = {(g, α_{ε}(g)) : g ∈ G_{N}} associated to the automorphism α_{ε} ∈ Aut(G) which is defined by conjugation by the element Q_{ε} = \binom{ε 0}{0 1} ∈ Gl_{2}(ℤ/Nℤ). We propose to call the resulting quotient surface Z_{N,ε} = Δ_{α}\Y a (twisted) modular diagonal quotient surface; it is a special case of the general diagonal quotient surfaces studied in [10].

The modular diagonal quotient surfaces occur naturally as the compactifications of the (coarse) moduli spaces associated to certain moduli problems, as will be explained below. Furthermore, there is a close analogy between these surfaces and the Hilbert modular surfaces studied by Hirzebruch and others (cf. van der Geer[3]).

Since Δ_{ε} has finitely many fixed points on Y_{N}, the quotient surfaces Z_{N,ε} have finitely many isolated quotient singularities which may be described explicitly (cf. Theorem 2.1 and Corollary 2.4 below); in particular, we have:

Theorem 1 If N ≥ 5, the modular diagonal quotient surface Z_{N,ε} has precisely r_{0}+r_{1}+r_{∞} singularities, where r_{0} = h(−4N^{2}) and r_{1} = h(−3N^{2}) are certain class numbers of binary quadratic forms, and r_{∞} = ∑_{1<d|N} ϕ(d)ϕ(\frac{N}{d}) denotes the number of cusps of X_{1}(N). All are cyclic quotient singularities.

The above theorem is an easy consequence of the general results of [10], once certain local invariants have been calculated; this is done in section 1. In the same way we can determine some of the numerical invariants of its minimal desingularization Z̃_{ε} (cf. Theorem 2.6):

Theorem 2 The desingularization Z̃_{N,ε} of the modular diagonal quotient surface Z_{N,ε} is a regular surface whose geometric genus p_{g,ε} and Chern numbers K_{ε}^{2} = c_{1}^{2}(Z̃_{N,ε}) and χ_{top,ε} = c_{2}(Z̃_{N,ε}) are given by

p_{g,ε} = \frac{m(N − 12)}{144N} + \frac{ϕ(N)}{8} + \frac{r_{0}}{8} + \frac{2r_{1}}{9} − \frac{s_{1,1,ε}}{9} + \frac{r_{∞}}{3} + \frac{𝕃_{∞,ε}}{12} − \frac{R_{∞,ε}}{12} − 1,
K_{ε}^{2} = \frac{m(N − 12)}{18N} + ϕ(N) − \frac{s_{1,1,ε}}{3} + 3r_{∞} − R_{∞,ε},
χ_{top,ε} = \frac{m(N − 12)}{36N} + \frac{ϕ(N)}{2} + \frac{3r_{0}}{2} + \frac{8r_{1}}{3} − s_{1,1,ε} + r_{∞} + 𝕃_{∞,ε},

provided that N ≥ 5. Here, m = |G_{N}|, and r_{0}, r_{1} and r_{∞} are as in Theorem 1. In addition, s_{1,1,ε} = \frac{1}{2}r_{1} if 3 |̸N, and s_{1,1,ε} = \frac{1}{2}(1+(\frac{−3}{ε}))r_{1}, if 3|N, and 𝕃_{∞,ε} and R_{∞,ε} are certain sums involving the lengths of the finite continued fraction expansions of \frac{N}{q}, where 0 < q < N.

It is, however, much more difficult to determine the Kodaira dimension κ of Z̃_{N,ε} since there does not seem to be an easy method for computing it (cf. the discussion of [10], section 4.) Nevertheless, it turns out rather surprisingly that κ and even the type of the surface Z̃_{N,ε} is completely characterized by its geometric genus (cf. Theorem 2.11):

Theorem 3 The Kodaira dimension of the modular diagonal quotient surface is given by

κ(Z̃_{N,ε}) = min(2, p_{g,ε} − 1).

Despite the simplicity of the result, its proof is not so direct, for we succeded in proving it only as a consequence of the following much more precise result. To state it, we adopt the convention that we view ε as an element of the quotient group (ℤ/Nℤ)^{×}/((ℤ/Nℤ)^{×})^{2}: this is permissible because if ε′ ≡ εd^{2} (mod N), then α_{ε′} and α_{ε} differ by an inner automorphism of G, and so the resulting surfaces are isomorphic.

Theorem 4 a) Z̃_{N,ε} is a rational surface if and only if p_{g}(Z̃_{N,ε}) = 0, and this is the case precisely for N ≤ 5 or for (N, ε) = (6, 1), (7, 1) or (8, 1).

b) Z̃_{N,ε} is a (blown-up) elliptic K3-surface if and only if p_{g}(Z̃_{N,ε}) = 1, i.e. if and only if (N, ε) = (6, 5), (7, 3), (8, 3), (8, 5), (9, 1) or (12, 1).

c) Z̃_{N,ε} is a (blown-up) elliptic surface with κ = 1 if and only if p_{g}(Z̃_{N,ε}) = 2. This is the case for (N, ε) = (8, 7), (9, 2), (10, 1), (10, 3) or (11, 1).

d) Z̃_{N,ε} is a surface of general type if and only if p_{g}(Z̃_{N,ε}) ≥ 3, or equivalently, if N ≥ 13 or if (N, ε) = (11, 2), (12, 5), (12, 7) or (12, 11).

It is interesting to observe that almost all of Theorem 4 follows relatively easily from the general criteria of [10], with the exception of the two cases (N, ε) = (10, 3) and (11, 1) which require a more detailed analysis (cf. Proposition 2.14).

As was already mentioned above, the modular diagonal quotient surfaces Z_{N,ε} have a natural modular interpretation. Indeed, by using the well-known modular interpretation of X′(N) = Γ\ℌ = X(N)\ {cusps}, it is easy to see that the points of Z′_{N,ε} := Δ_{ε}\(X′(N) × X′(N)) ⊂ Z_{N,ε} can naturally be identified with triplets (E_{1}, E_{2}, ψ), where E_{1} and E_{2} are elliptic curves and ψ : E_{1}[N] →^{∼} E_{2}[N] is an isomorphism of the N-torsion subgroups E_{i}[N] of determinant ε; the latter means that the Weil pairings are related by the formula

e_{E_{1},N} ∘ (ψ × ψ) = e_{E_{2},N}^{ε}.

Furthermore, by using Galois descent it is not difficult to see that the surface Z_{N,ε} has a “canonical model” over ℚ (and even over Spec(ℤ[\frac{1}{N}]) which serves as a coarse moduli scheme for the moduli functor Z_{N,ε} defined by

Z_{N,ε}(S) = {(E_{1}, E_{2}, ψ) : E_{1}, E_{2} are elliptic curves over S and ψ : E_{1}[N] →^{∼} E_{1}[N] is an S-isomorphism with det(ψ) = ε}/(isomorphisms).

Note that if S = Spec(K), K a number field, then this set may be identified with the set of isomorphism classes of pairs of elliptic curves together with an isomorphism ψ of the associated modular Galois representations ρ̄_{E_{i},N} : Gal(K̄/K) → Aut(E_{i}[N]).

As a result of this modular interpretation, the above theorem has several interesting applications which are worth mentioning.

First of all, the above theorem has an interesting connection with a question of Mazur which was studied by Kraus and Oesterlé [12]. In this question Mazur [15] poses the problem of finding examples of pairs (E, E′) of non-isogenous elliptic curves over ℚ such that for some integer N ≥ 7 their associated Galois representations on their N-torsion points are symplectically isomorphic, and in [12] such an example is exhibited for N = 7. In view of the above modular interpretation, Mazur’s question can be rephrased as asking for the existence of ℚ-rational points on modular diagonal quotient surface Z_{N,1} which do not lie on any “obvious” curves on Z_{N,1} (such as the curves at infinity or those coming from the relation of isogeny, i.e. on “graphs of Hecke correspondences”). Viewed in this light, the above theorem therefore shows that the Kraus-Oesterlé example should come as no surprise: indeed, since Z_{7,1} is rational (over ℚ̄) by Theorem 4, there should exist infinitely many such examples, at least over a sufficiently large number field K. More generally, one might expect the following conjecture to be true:

Conjecture 5 For any number field K and any N ≤ 12, there are infinitely many pairs (E_{1}, E_{2}) of non-isogenous elliptic curves defined over K whose associated Galois representations mod N are isomorphic.

Theorem 4 gives partial evidence for this: since for N ≤ 12, at least one of the surfaces Z̃_{N,ε} is either rational or an elliptic fibring, the above conjecture is true if we replace “any number field” by “any sufficiently large number field” and “non-isogeneous” by “not isogenous by an isogeny of degree ≤ d” (for any fixed d).

It is an intriguing and important question to determine to what extent the converse of Conjecture 5 is valid, i.e.

Question 6 For which integers N is it true that there are only finitely many pairs (E, E′) (up to ℚ̄-isomorphism) of non-isogenous elliptic curves defined over a number field K whose associated Galois representations mod N over K are isomorphic?

Indeed, H. Darmon and others have conjectured that this question has a positive answer for all sufficiently large N, and this would have far-reaching diophantine consequences such as the Asymtotic Fermat Conjecture (cf. Frey[2]).

Again, Theorem 4 gives some partial evidence for this, particularly if we admit the conjecture of S. Lang [14] that almost all of the K-rational points of an algebraic surface Z of general type are contained in the “exceptional locus” which by definition is the union of the (conjecturally finitely many) curves of genus ≤ 1 lying on Z. What remains to be done, however, is to identify these curves as Hecke correspondences (at least when N is large), but this seems to be a rather formidable problem!

Yet another application concerns the moduli space M_{2}^{ell}(N) of curves of genus 2 with an elliptic differential of degree N which was defined and studied in [8] (see also [7]). By definition, M_{2}^{ell}(N) classifies isomorphism classes of pairs ⟨C, f⟩ consisting of a curve C of genus 2 and a (minimal) covering f : C → E of degree N to an elliptic curve E, and it can be shown (cf. [9]) that this space is a double cover of the Humbert surface with invariant Δ = N^{2} (as defined in van der Geer [3], p. 210ff; cf. also [9]). More precisely, one can prove (cf. [8]) that M_{2}^{ell}(N) is an open subvariety of the twisted diagonal quotient surface Z_{N,−1}, and so we obtain from Theorem 4:

Theorem 7 The (desingularization of the) compactification \overline{M}_{2}^{ell}(N) = Z_{N,−1} of the moduli space M_{2}^{ell}(N) is rational if N ≤ 5, is a (blown-up) elliptic K3-surface if N = 6 or 7, is an elliptic surface (with κ = 1) if 8 ≤ N ≤ 10, and is of general type if N ≥ 11.

After we had proved a preliminary version of Theorems 4 and 7, R. Weissauer drew our attention to the article of C. F. Hermann [5] who had also determined the classification type of the surfaces Z_{N,ε}. Unfortunately, the interesting connection between the geometric genus p_{g} of Z̃_{N,ε} and the Enriques-Kodaira classification type is not mentioned there; in addition, the (more subtle) case (N, ε) = (10, 3) is missing in Hermann’s list of elliptic surfaces.^{1} However, it should be emphasized that our method is substantially different from that of Hermann in that most of Theorem 6 is deduced from the general classification Theorems 1 – 5, and only in two cases do we need to analyze the surfaces in more detail. This should be compared to the 14 cases or so which would have to be studied in detail in Hermann’s article (but were not since virtually all details of his proof were suppressed). In particular, we do not need Hermann’s curves F_{n} (which admittedly are of interest in connection with Mazur’s problem) for the proof of the above theorems, with the possible exception of the case (N, ε) = (11, 1). Furthermore, there are a number of typographical errors and other inaccuracies in his paper; cf. Remarks 1.7, 2.2 and 2.12 below.

Acknowledgements: The authors are very thankful for helpful and stimulating discussions with G. Frey, A. Geramita, E. Viehweg, R. Weissauer, and J. Zarhin. The second author wants to thank Queen’s University for its hospitality during a visit supported by NSERC and ARC research grants held by the first author.

This research was supported in part by the Natural Science and Engineering Research Council of Canada (NSERC) and also by the Advisory Research Council (ARC) of Queen’s University.

1After this manuscript was completed, Hermann showed the first author a copy of his (unpublished)

Habilitationsschrift in which both these lacunae were correctly addressed.

1 Invariants associated to the modular curve X(N)

1.1 The genus of X_{k}

For each N ≥ 1, the modular curve X(N) is the compactification of the quotient Γ(N)\ℌ of the upper half plane ℌ by the action of the principal congruence subgroup Γ(N) = {A ∈ SL_{2}(ℤ) : A ≡ 1 mod N\}, where A = \binom{a b}{c d} ∈ Γ(1) acts on z ∈ ℌ by the linear transformation z ↦ \frac{az+b}{cz+d}. For the basic properties of X(N) we refer to Schoeneberg [18] or Miyake[16]. The action of Γ(1) on ℌ induces an action of G = G_{N} = Γ(1)/ ± Γ(N) ≃ SL_{2}(ℤ/Nℤ)/{±1} on X(N), and the quotient X̄ = G_{N}\X(N) = X(1) ≃ ℙ^{1} is rational.

The quotient map π : X(N) → X̄ is ramified precisely at the points P̄_{0}, P̄_{1} and P̄_{∞} ∈ X̄ which are the images of the points i, ρ = e^{\frac{2πi}{3}} and ∞ ∈ ℌ^{∗} = ℌ ∪ ℚ ∪ \{∞\}, respectively. The corresponding ramification indices e_{k} = eP̄_{k} are e_{0} = 2, e_{1} = 3 and e_{∞} = N. More precisely, if P_{k} = P_{k,N} (k = 0, 1, ∞) denote (respectively) the images of i, ρ and ∞ ∈ ℌ^{∗} in X(N), then the stabilizers G_{k} = G_{P_{k}} = ⟨σ_{k,N}⟩ of P_{k,N} are cyclic groups of order e_{k} with generators

σ_{0} = σ_{0,N} := ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) , σ_{1} = σ_{1,N} := ( \begin{matrix}0 & −1 \\ 1 & 1 \end{matrix} ) , σ_{∞} = σ_{∞,N} := ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) .

Since the level N will be fixed from now on , we shall suppress the dependence on N in our notation, as long as this does not lead to any confusion.

For ε ∈ (ℤ/Nℤ)^{×} we let α_{ε} denote the automorphism of G_{N} induced by conjugation with the matrix Q_{ε} = \binom{ε 0}{0 1} ∈ Gl_{2}(ℤ/Nℤ); i.e.

α_{ε}(g) = Q_{ε}gQ_{ε}^{−1} = ( \begin{matrix}a & εb \\ ε^{−1}c & d \end{matrix} ) , if g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) .

As was explained in the introduction, we want to study the geometry of the diagonal quotient surface Z = (X(N) × X(N))/G_{N}, as well as that of its twists Z_{ε} = Z_{α_{ε}}. From [10] we know that the fundamental invariants (Betti, Hodge and Chern numbers) of the desingularization Z̃_{ε} of Z_{ε} are determined by three basic invariants 𝔾_{ε} = 𝔾_{α_{ε}}, 𝕊_{ε} = 𝕊_{α_{ε}} and 𝕃_{ε} = 𝕃_{α_{ε}} (together with the genera g = g(X) and ḡ = g(X̄) = 0), which therefore have to be computed here. Since all three are defined by invariants involving the quotient curves X_{k,ε} = ⟨α_{ε}^{−1}(σ_{k,N})⟩\X(N), we begin by calculating their genera g_{k,ε} = g(X_{k,ε}). For this, let r_{k,ε} denote the number of ramification points on X_{k,ε} of the covering π_{k,ε} : X → X_{k,ε} and t_{k,ε} the number of fixed points of the group G_{k,ε} = ⟨α_{ε}^{−1}(σ_{k}⟩) on X. Then by the Riemann-Hurwitz formula (together with Schoeneberg [10], p. 76) we have:

Proposition 1.1 If N ≥ 3, the genus g = g(X(N)) of X(N) and the genera g_{k,ε} of its quotients X_{k,ε} = G_{k,ε}\X(N) are given by

g = 1 + m\frac{N − 6}{12N},  where m = |G_{N}| = \frac{N^{3}}{2} ∏_{p|N} (1 − \frac{1}{p^{2}} ) ,
g_{0,ε} = \frac{1}{2} (g + 1 − \frac{1}{2}r_{0,ε}) ,  g_{1,ε} = \frac{1}{3} (g + 2 − r_{1,ε}) ,
g_{∞,ε} = \frac{g − 1}{N} + 1 − \frac{1}{2}r_{∞,ε} + \frac{1}{2N} t_{∞,ε}.

We next want to compute the numbers r_{k,ε} and t_{k,ε}, as well as the related invariant r_{n,ε}(P̄_{k}), which denotes the number of points x ∈ X_{k,ε} which have ramification degree e_{x}(π_{k,ε}) = n. It turns out that these are closely related to the following expressions which may be interpreted as class numbers h(d) of binary quadratic forms of discriminant d (cf. Hua [6], pp. 321-2):

Notation 1.2 For a positive integer N > 1 put

h(−4N^{2}) = \frac{1}{2}N ∏_{p|N} (1 − \frac{1}{p} (\frac{−4}{p})) ,

h(−3N^{2}) = \frac{1}{3}N ∏_{p|N} (1 − \frac{1}{p} (\frac{−3}{p})) ,

where (\frac{d}{p}) denotes the Legendre/Kronecker symbol; in particular, (\frac{−4}{2}) = 0 and (\frac{−3}{2}) = −1.

The relation between these expressions and the invariants r_{k,ε}, t_{k,ε} and r_{n,ε}(P̄_{k}) is given by the following result which will be proved in the next section as a corollary of the computation of the finer “local invariants” s_{ν,ε}(P̄_{l}, P̄_{k}).

Proposition 1.3 For N ≥ 5, the isomorphism class of the curve X_{k} = X_{k,ε} does not depend on ε ∈ (ℤ/Nℤ)^{×} and hence neither does its genus g_{k} = g_{k,ε}. Furthermore, the same is true for the invariants r_{k} = r_{k,ε}, t_{k} = t_{k,ε} and r_{n}(P̄_{k}) = r_{n,ε}(P̄_{k}); these are given by

a) r_{0} = t_{0} = h(−4N^{2}),

b) r_{1} = t_{1} = h(−3N^{2}),

c) r_{n}(P̄_{∞}) = \frac{1}{2}ϕ(n)ϕ(\frac{N}{n}), for 1 < n | N,

d) r_{∞} = \frac{1}{2} ∑_{n|N} ϕ(n)ϕ(\frac{N}{n}) − \frac{1}{2}ϕ(N) and t_{∞} = \frac{m}{N} − \frac{1}{2}Nϕ(N).

Remark 1.4 Some of the above formulae can be found in the literature albeit in a different guise. For example, since the stabilizer subgroup of P_{∞} in G_{N} is G_{∞} = ±Γ_{1}(N)/ ± Γ(N), where Γ_{1}(N) = \{A ∈ Γ(1) : A ≡ \binom{1 ∗}{0 1} mod N\}, the quotient curve X_{∞} is the modular curve usually denoted by X_{1}(N) = ±Γ_{1}(N)\ℌ^{∗}. Thus, the above two propositions yield the well-known relation for the genus of X_{1}(N),

g_{∞} = g(X_{1}(N)) = 1 + \frac{m}{12N} − \frac{1}{2}ν_{∞},

where ν_{∞} = \frac{1}{2} ∑_{ν|N} ϕ(ν)ϕ(\frac{N}{ν}) for N > 4; cf. Miyake[16], Theorem 4.2.9.

Similarly, the genus of X_{0} and X_{1} was determined by Hecke [4] for the case of prime level N; however, the presentation of his result is somewhat more indirect and does not easily generalize to composite N’s.

1.2 The local invariants

As was explained in [10], much of the structure of the twisted diagonal quotient surface Z_{ε} associated to X = X(N) is known once we have determined the local invariants s_{ν,ε}(P̄_{l}, P̄_{k}) := s_{ν,α_{ε}}(P̄_{l}, P̄_{k}) which count the number of points x ∈ X_{k,ε} lying above P̄_{l} whose “ramification characters” λ_{x,ε} are of a certain type (cf. [10], Notation 1.4 and Remark 1.5). Here we want to compute these numbers and hence derive in particular the values for r_{k}, t_{k}, etc., which were asserted in Proposition 1.3. In order to state the result, we first introduce the following notation.

Notation 1.5 For any pair of natural numbers N, k ∈ ℕ with (k, N) = 1, let

ρ(k, N) = |{x ∈ ℤ/Nℤ : x^{2} = k}|

denote the number of square roots of k mod N. Thus, ρ(k, N) = 0 if k ∉ ((ℤ/Nℤ)^{×})^{2} and otherwise ρ(k, N) = ρ(N), where

ρ(N) := |(ℤ/Nℤ)_{2}^{×}| = [(ℤ/Nℤ)^{×} : ((ℤ/Nℤ)^{×})^{2}] = 2^{s+t}, (1)

where s denotes the number of odd primes dividing N, and t = 0 if 4 |̸N, t = 2 if 8|N and t = 1 otherwise (cf. Landau [13], Satz 88).

Theorem 1.6 The twisted local invariants s_{ν,ε} = s_{ν,α_{ε}} are related to the standard (untwisted) invariants s_{ν} = s_{ν,id} by the formula

s_{ν,ε}(P̄_{l}, P̄_{k}) = s_{(κν)}(P̄_{l}, P̄_{k}), for 1 ≤ ν ≤ e_{k}, (2)

where κ = κ_{k,ε} is defined by

(3) κ_{k,ε} = \{ \begin{matrix}1 & if k = 0 or k = 1 and 3 |̸N \\ ε & if k = ∞ or k = 1 and 3|N, \end{matrix}

and (κν) denotes the least positive residue of κν (mod e_{k}). In particular,

r_{n}(P̄_{l}, P̄_{k}) = r_{n,α_{ε}}(P̄_{l}, P̄_{k}), where n|e_{k}, (4)

is independent of ε. Furthermore, with the notation introduced in Notations 1.2 and 1.5 we have

a) s_{ν,ε}(P̄_{l}, P̄_{k}) = δ_{νe_{k}}r_{1,ε}(P̄_{l}, P̄_{k}) = δ_{νe_{k}} \frac{m}{e_{l}e_{k}} , if l ≠ k;
b) s_{1,ε}(P̄_{0}, P̄_{0}) = r_{2,ε}(P̄_{0}, P̄_{0})  = r_{0} = h(−4N^{2}),
s_{2,ε}(P̄_{0}, P̄_{0}) = r_{1,ε}(P̄_{0}, P̄_{0})  = \frac{m}{4} − \frac{1}{2}r_{0};
c) s_{ν,ε}(P̄_{1}, P̄_{1}) = \frac{1}{2}h(−3N^{2})  = \frac{1}{2}r_{1}, if ν = 1, 2 and 3 |̸N,
s_{ν,ε}(P̄_{1}, P̄_{1}) = h(−3N^{2})  = r_{1}, if ν = 1, 2, ν ≡ ε (mod 3) and 3|N,
s_{ν,ε}(P̄_{1}, P̄_{1}) = 0,  if ν = 1, 2, ν ≢ ε (mod 3) and 3|N,
s_{3,ε}(P̄_{1}, P̄_{1}) = r_{1}(P̄_{1}, P̄_{1})  = \frac{m}{9} − \frac{1}{3}r_{1} = \frac{m}{9} − \frac{1}{3}h(−3N^{2});
d) s_{ν,ε}(P̄_{∞}, P̄_{∞}) = \frac{1}{2}ρ(\frac{εν}{d} , \frac{N}{d})ϕ(d), where d = (ν, N),
r_{d,ε}(P̄_{∞}, P̄_{∞}) = \frac{1}{2}ϕ(d)ϕ(\frac{N}{d}), if d|N.

Remark 1.7 As the explicit formulae in Theorem 1.6 show, the local invariants s_{ν,ε}(P̄_{l}, P̄_{k}) depend only on the square class ε · ((ℤ/Nℤ)^{×})^{2} of ε modulo N. This may also be seen directly as follows. First note that if ε = e^{2} is a square modulo N, then conjugation by Q_{ε} is the same as conjugation by \binom{e 0}{0 e^{−1}} ∈ Sl_{2}(ℤ/nℤ), and hence the map ε ↦ α_{ε} induces a homomorphism

ᾱ : (ℤ/Nℤ)^{×} / ((ℤ/Nℤ)^{×})^{2} → Aut(G_{N})/Inn(G_{N}) = Out(G_{N}).

Thus, since the local invariants s_{ν,α} depend only on the image of α in Out(G) (cf. [10], Remark 1.5d), the assertion follows.

As regards to the map ᾱ, note that ᾱ is injective for N > 4 (as is easy to see), but that this is false for n = 4 since G_{4} is isomorphic to the symmetric group S_{4} which has no outer automorphisms whereas ρ(4) = |(ℤ/4ℤ)^{×}/((ℤ/4ℤ)^{×})^{2}| = 2.

In Hermann [5], p. 96, it is asserted that ᾱ is in fact an isomorphism (even for N = 4!). For a proof, Hermann refers to Praetorius [17] §2, who actually only considers the case N = p^{r} with a prime number p > 3. By showing that G_{p^{r}} is a characteristic subgroup of G_{N}, if p^{r} ∥ N and p > 3, one can deduce from Praetorius that this is true if (N, 6) = 1. In the general case, however, it is still unclear which automorphisms actually exist.

The proof of Theorem 1.6 is based on the group-theoretical method presented in [10], Proposition 1.7. There it was shown that the local invariants s_{ν,α}(x̄, ȳ) are determined by the orders of certain “normalizing subsets” N_{ν,i}^{∗}(σ, τ) of G attached to generators σ and τ of the ramification groups G_{x} and G_{y,α} = α^{−1}(G_{y}), where x ∈ π^{−1}(x̄) and y ∈ π^{−1}(ȳ). Recall from [10], Notation 1.6, that these normalizing sets are defined as follows. If r = (e_{x}, e_{y}) and k = (ν, r), σ̄ = σ^{e_{x}/r}, τ̄ = τ^{e_{y}/r}, then for any i ∈ ℤ:

N_{ν}(σ, τ) = N_{k}(σ, τ) := {g ∈ G : gσ̄^{ν}g^{−1} ∈ ⟨τ̄^{ν}⟩} = {g ∈ G : g⟨σ̄^{ν}⟩g^{−1} = ⟨τ̄^{ν}⟩},
N_{ν}^{i}(σ, τ) = N_{k}^{i}(σ, τ) := {g ∈ G : gσ̄^{k}g^{−1} = τ̄^{ki}},
N_{ν}^{∗}(σ, τ) = N_{k}^{∗}(σ, τ) := N_{k}(σ, τ) \ ⋃_{d|k,d≠k} N_{d}(σ, τ),
N_{ν,i}^{∗}(σ, τ) = N_{k,i}^{∗}(σ, τ) := N_{ν}^{∗}(σ, τ) ∩ N_{ν}^{i}(σ, τ).

In the case that σ = σ_{k} and τ = α_{ε}^{−1}(σ_{k}), these normalizing sets are closely related to the following sets, as we shall see in Lemma 1.9.

Notation 1.8 For k = 0, 1 and ε ∈ (ℤ/Nℤ)^{×} let

M̃_{k,ε}(N) = \{A ∈ Gl_{2}(ℤ/Nℤ) : A = ( \begin{matrix}x & −y \\ y & x + δ_{1k}y \end{matrix} ) and det(A) = ε\} ,
M_{k,ε}(N) = {(x, y) ∈ ℤ/Nℤ × ℤ/Nℤ : q_{k}(x, y) = ε},

where q_{k}(x, y) = x^{2} + δ_{k1}xy + y^{2}. These two sets are related by the map p : \binom{x y}{z w} ↦ (x, z) which induces a bijection

p_{k,ε} : M̃_{k,ε}(N) →^{∼} M_{k,ε}(N) .

Furthermore, for d|N and (n, \frac{N}{d}) = 1 we put

M̃_{∞,d,n}(N) = \{\binom{x y}{z w} ∈ Sl_{2}(ℤ/Nℤ) : x ≡ n (mod \frac{N}{d}), (z, N) = \frac{N}{d} \},
M_{∞,d,n}(N) = \{(x, y) ∈ ℤ/Nℤ × ℤ/Nℤ : x ≡ n (mod \frac{N}{d}) and (y, N) = \frac{N}{d} \},
M̃_{∞,d,n}^{(2)}(N) = ⋃_{ν^{2}=n}M̃_{∞,d,ν}(N) and M_{∞,d,n}^{(2)}(N) = ⋃_{ν^{2}=n}M_{∞,d,ν}(N).

Here the above map p : \binom{x y}{z w} ↦ (x, z) induces a bijection

p_{∞,d,n} : M̃_{∞,d,n}(N)/⟨σ_{∞,N}⟩ →^{∼} M_{∞,d,n}(N).

Lemma 1.9 If d|N ≥ 5 and (n, \frac{N}{d}) = (ε, N) = 1, then

a) N_{1,1}^{∗}(σ_{0,N}, α_{ε}^{−1}(σ_{0,N})) = Q_{ε}^{−1} (M̃_{0,ε}(N) ∪ Q_{−1}M̃_{0,−ε}(N)) /{±1};
b) N_{1,1}^{∗}(σ_{1,N}, α_{ε}^{−1}(σ_{1,N})) = Q_{ε}^{−1}M̃_{1,ε}(N)/{±1},
N_{1,2}^{∗}(σ_{1,N}, α_{ε}^{−1}(σ_{1,N})) = Q_{ε}^{−1} (M̃_{1,−ε}(N)\binom{1 1}{0−1}) /\{±1\};
c) N_{d,n}^{∗}(σ_{∞,N}, α_{ε}^{−1}(σ_{∞,N})) = Q_{ε}^{−1}M̃_{∞,d,εn}^{(2)}(N)/{±1}.

Proof. Let f : G̃ := Sl_{2}(ℤ/Nℤ) → G̃/{±1} = G_{N} denote the projection map and put

Ñ_{ε}^{i}(σ_{k}) := f^{−1}(N_{1}^{i}(σ_{k}, α_{ε}^{−1}(σ_{k}))), for k ∈ {0, 1, ∞};

thus we have N_{1}^{i}(σ_{k}, α_{ε}^{−1}(σ_{k})) = Ñ_{ε}^{i}(σ_{k})/{±1}.

a) Since σ_{0} has prime order (in G_{N}), we have N_{1,1}^{∗}(σ_{0}, α_{ε}^{−1}(σ_{0})) = N_{1}^{1}(σ_{0}, α_{ε}^{−1}(σ_{0})) = Ñ_{ε}^{1}(σ_{0})/{±1}, and so it is enough to calculate the “twisted centralizer” Ñ_{ε}^{1}(σ_{0}). For this, let A = \binom{x y}{z w} ∈ Sl_{2}(ℤ/Nℤ). Then

A ∈ Ñ_{ε}^{1}(σ_{0}) ⇔ ( \begin{matrix}x & y \\ z & w \end{matrix} ) ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) = ± ( \begin{matrix}0 & −ε^{−1} \\ ε & 0 \end{matrix} ) ( \begin{matrix}x & y \\ z & w \end{matrix} ) and xw − yz = 1

⇔ y = −ε^{−1}z, w = εx, εx^{2} + ε^{−1}z^{2} = 1 or y = ε^{−1}z, w = −εx, −εx^{2} − ε^{−1}z^{2} = 1
⇔ A ∈ Q_{ε}^{−1}M̃_{0,ε}(N) ∪ Q_{−ε}^{−1}M̃_{0,−ε}(N).

b) To determine the twisted centralizer of σ_{1} we observe that

A ∈ Ñ_{ε}^{1}(σ_{1}) ⇔ ( \begin{matrix}x & y \\ z & w \end{matrix} ) ( \begin{matrix}0 & −1 \\ 1 & 1 \end{matrix} ) = ± ( \begin{matrix}0 & −ε^{−1} \\ ε & 1 \end{matrix} ) ( \begin{matrix}x & y \\ z & w \end{matrix} ) and xw − yz = 1
⇔ ( \begin{matrix}y & −x + y \\ w & −z + w \end{matrix} ) = ± ( \begin{matrix}−ε^{−1}z & −ε^{−1}w \\ εx + z & εy + w \end{matrix} ) and xw − yz = 1
⇔ y = −ε^{−1}z, w = εx + z, xw − yz = 1 ⇔ A ∈ Q_{ε}^{−1}M̃_{1,ε}(N).

Here, the second last equivalence is valid because the case of the minus sign implies 2w = 2y = 0, so 2 = 2(xw − yz) = 0, which is false for N > 2, and the last because Q_{ε}^{−1}M̃_{1,ε} = \{\binom{ε^{−1}x −ε^{−1}z}{z x+z} : x^{2} + xz + z^{2} = ε\} = \{\binom{x −ε^{−1}z}{z εx+z} ∈ Sl_{2}(ℤ/Nℤ)\}.

In the same way one can determine Ñ_{ε}^{2}(σ_{1,N}) = \{A ∈ G̃ : Aσ_{1} = ±\binom{−1 −ε^{−1}}{ε 0}A\}.

c) For a divisor μ of N we have the following equivalences which will be justified below:

A ∈ NG̃(⟨±σ_{∞}^{μ}⟩)

⇔ there is a k: ( \begin{matrix}x & y \\ z & w \end{matrix} ) ( \begin{matrix}1 & μ \\ 0 & 1 \end{matrix} ) = ± ( \begin{matrix}1 & kμ \\ 0 & 1 \end{matrix} ) ( \begin{matrix}x & y \\ z & w \end{matrix} ) and xw − yz = 1
⇔ there is a k : ( \begin{matrix}x & xμ + y \\ z & zμ + w \end{matrix} ) = ( \begin{matrix}x + kzμ & y + kwμ \\ z & w \end{matrix} ) and xw − yz = 1
⇔ z ≡ 0 mod \frac{N}{μ} and xw − yz = 1.

To justify the second equivalence, note that Aσ_{∞}^{μ} = −σ_{∞}^{μ}A leads to z = −z and 2w = zμ = −zμ = −2w, hence 4 = 4(xw − yz) = 0 which is a contradiction since N > 4.

The last equivalence may be derived as follows. First of all, if zμ+w = w (in ℤ/Nℤ), then z ≡ 0( mod \frac{N}{μ}), which proves one implication. Conversely, assume that z ≡ 0 (mod \frac{N}{μ}) and xw − yz = 1. Thus 1 ≡ xw − yz ≡ xw (mod \frac{N}{μ}), i.e. (w, \frac{N}{μ}) = 1 and consequently there are k′, l such that 1 = k′w+l\frac{N}{μ}. By multiplying with xμ, it follows that xμ = xk′μ+lxN ≡ wkμ (mod N) (with k = xk′), which is the condition of the third line. Therefore

N_{d}^{∗}(σ_{∞,N}) = \{A ∈ G_{N} : z ≡ 0 mod \frac{N}{d} , z ≢ 0 mod \frac{N}{μ} , μ|d, μ < d\}\} = \{A : (z, N) = \frac{N}{d} \} .

Moreover, if A ∈ N_{d}^{∗}(σ_{∞}, σ_{∞}), then the above computation shows that in fact A ∈ N_{d}^{k}(σ_{∞}, σ_{∞}) for wkd ≡ xd (mod N) or, equivalently, for k ≡ x^{2} (mod \frac{N}{d}), and so the assertion follows for ε = 1.

From this the general case follows easily. Indeed, if ε^{∗}ε ≡ 1 (mod N), then α_{ε}^{−1}(σ_{∞}) = σ_{∞}^{ε^{∗}} and M̃_{∞,d,ε^{∗}n}^{(2)}(N) = Q_{ε}^{−1}M̃_{∞,d,εn}^{(2)}(N), and so we see that

N_{d,n}^{∗}(σ_{∞}, α_{ε}^{−1}(σ_{∞})) = N_{d,n}^{∗}(σ_{∞}, σ_{∞}^{ε^{∗}})) = N_{d,ε^{∗}n}^{∗}(σ_{∞}, σ_{∞}) = M̃_{∞,d,ε^{∗}n}^{(2)}(N)=Q_{ε}^{−1}M̃_{∞,d,εn}^{(2)}(N).

We now count the number of elements in each of the above sets.

Lemma 1.10 For any N, d, ε and n ∈ ℕ with d | N and (ε, N) = (n, \frac{N}{d}) = 1 we have

a) |M̃_{0,ε}(N)|  = |M_{0,ε}(N)|  = 2h(−4N^{2}), if 4 |̸N,
|M̃_{0,ε}(N)|  = |M_{0,ε}(N)|  = 2h(−4N^{2}) (1 + (\frac{−1}{ε})) , if 4|N;
b) |M̃_{1,ε}(N)|  = |M_{1,ε}(N)|  = 3h(−3N^{2}), if 3 |̸N,
|M̃_{1,ε}(N)|  = |M_{1,ε}(N)|  = 3h(−3N^{2}) (1 + (\frac{−3}{ε})) , if 3|N;
c) |M̃_{∞,d,n}(N)| = |M_{∞,d,n}(N)| · N = dϕ(d)N,
|M̃_{∞,d,n}^{(2)}(N)| = |M_{∞,d,n}^{(2)}(N)| · N = ρ (n, \frac{N}{d}) dϕ(d)N.

Proof. a) Keller [11] (The proof is similar to b) below.)

b) As was already remarked in Notation 1.8, we have a bijection M̃_{1,ε} →^{∼} M_{1,ε}, so the the first equality in each case is clear. Since M̃ := ∪_{ε}M̃_{1,ε} is a subgroup of Gl_{2}(ℤ/Nℤ), and since M̃_{1,1} = Ker(det_{|M̃}), we see that M̃_{1,ε} is either empty or an M̃_{1,1}-coset, and that the latter occurs precisely when ε ∈ D := det(M̃) ⊂ (ℤ/Nℤ)^{×}. Thus, the assertion follows once we have determined D and have shown that |M̃_{1,1}(N)| = 3h(−3N^{2}); for this we shall use the quadratic form interpretation of M̃_{1,ε}(N) →^{∼} M_{1,ε}.

Since by Chinese remainder theorem the number of solutions of the given equation is multiplicative (as a function of N), as is 3h(−3N^{2}), it suffices to consider powers of primes N = p^{r}. If p ≠ 2, we can transform the quadratic form a^{2} + ac + b^{2} to x^{2} + 3y^{2} by putting a = x + y and c = −2y. Thus, if p > 3 then we see that D = (ℤ/Nℤ)^{×} (cf. [6], p. 310), and so by using the same arguments as in Keller[11] we get the result in this case.

If p = 3, then clearly −1 ∉ D, so D = ((ℤ/Nℤ)^{×})^{2} since D contains the squares which have index 2 in (ℤ/Nℤ)^{×}. Furthermore, since (\frac{1−3y^{2}}{3^{r}} ) = (\frac{1}{3}) = 1, the equation x^{2} = 1−3y^{2} has exactly two solutions for each y ∈ ℤ/3^{r}ℤ, which means that |M_{1,1}| = 2 · 3^{r}. Thus, the formula is true in this case.

Finally, if p = 2 then D = (ℤ/Nℤ)^{×}. Indeed, D certainly contains all squares, i.e. all x ≡ 1 mod 8. Choosing a ≡ c ≡ 1 (mod 2) we have a^{2} + ac + c^{2} ≡ 2 + ac (mod 8) and thus all other residue classes modulo 8 appear in D, i.e. D = (ℤ/Nℤ)^{×}. Since a^{2} + ac + c^{2} ≡ 0 (mod 2) ⇔ a ≡ b ≡ 0 (mod 2) we obtain |M̃_{1,1}(N)| = |M̃|/|D| = (2^{2r} − 2^{2(r−1)})/2^{(r−1)} = 2^{r−1} · 3 = 3h(−3 · 2^{2r}).

c) It is clear that the second formula follows from the first. To prove the latter, we may assume by the Chinese remainder theorem that N = p^{r} and d = p^{s} are prime powers. We now distinguish two cases: Case 1: r = s. Here we have only the condition (z, N) = 1. The number of choices for z is therefore ϕ(N), and for each choice of x, w (mod N), there is a unique y (mod N) such that xw − yz ≡ 1 (mod N). Thus, we have in total ϕ(N)N^{2} matrices in M̃_{∞,d,n}(N) and ϕ(N)N vectors in M_{∞,d,n}(N). Case 2: r > s. The number of x (mod N) with x ≡ n (mod \frac{N}{d}) is clearly d. Note that for each such x we have (x, p) = 1 because (n, p^{r−s}) = 1 by hypothesis. Thus, for each y, z there is a unique w (mod N) such that xw − yz ≡ 1 (mod N). Of these, the number of z satisfying (z, N) = \frac{N}{d} is ϕ(d), so in total we have dϕ(d)N matrices and dϕ(d) vectors.

For the proof of Theorem 1.6 we also need the following fact.

Lemma 1.11 If N ≥ 5, ε ∈ (ℤ/Nℤ)^{×} and x̄ ≠ ȳ ∈ X̄, then G_{x} ∩ G_{y,ε} = {1}, for all x ∈ π^{−1}(x̄) and y ∈ π^{−1}(ȳ), and hence s_{ν,ε}(x̄, ȳ) = 0, for 1 ≤ ν < eȳ.

Proof. Since ±Γ_{1}(N) has no elliptic elements for N > 4 (cf. [16], Th. 4.2.9), we have G_{x} ∩ G_{y} = {1} for y = P_{∞} and for all x ∈ π^{−1}(P̄_{k}), k = 0, 1. Now since α_{ε}(σ_{∞}) = σ_{∞}^{ε}, we see that G_{y,ε} = α_{ε}^{−1}(G_{y}) = G_{y}, and hence also G_{x} ∩ G_{y,ε} = {1}. Furthermore, if y′ = gy ∈ π^{−1}(ȳ), then G_{x} ∩ G_{y′,ε} is a conjugate of G_{x′} ∩ G_{y,ε} for a suitable x′ = g′x, and hence is also trivial. Thus, the assertion holds for x̄ = P̄_{k}, k = 0, 1 and ȳ = P̄_{∞}, and also for x̄ = P̄_{∞} and ȳ = P̄_{k}, k = 0, 1, since G_{y} ∩ G_{x,ε} = α_{ε}^{−1}(G_{x} ∩ G_{y,ε^{−1}}). This proves the first statement since the assertion is trivial for all other pairs of distinct points, and the second follows immediately from the first from the definition of s_{ν,ε}(x̄, ȳ); cf. [10], Notation 1.4.

Proof of Theorem 1.6. It is enough to verify the formulae in a) – d), for (2) clearly follows from these, and (4) follows from (2) since by Remark 1.5 of [10] we have (with e = eȳ)

r_{n,α}(x̄, ȳ) = ∑_{{ν=1 \atop (ν,e)= \frac{e}{n}}}^{e} s_{ν,α}(x̄, ȳ), (5)

and since multiplication by κ permutes the ν with (ν, e_{k}) = \frac{e_{k}}{n}.

a) For ν ≠ e_{k}, this follows directly from Lemma 1.11, and hence, in view of (5), the result also holds for ν = e_{k} because we always have (cf. [10], Remark 1.5):

∑_{ν=1}^{e_{k}} s_{ν,ε}(P̄_{l}, P̄_{k})(ν, e_{k}) = ∑_{d|e_{k}} r_{d,ε}(P̄_{l}, P̄_{k})\frac{e_{k}}{d} = \frac{m}{e_{l}}. (6)

b) Fix ε and put σ′_{k} := α_{ε}^{−1}(σ_{k}). Then we have trivially λ_{P_{k},α_{ε}}(σ′_{k}) = λ_{P_{k}}(σ_{k}), where λ_{P_{k},α_{ε}} = λ_{P_{k}} ∘α_{ε} is the twisted local ramification character (cf. [10], Notation 1.2, 1.3), and hence it follows from [10], Proposition 1.7c) that

s_{ν,ε}(P̄_{k}, P̄_{k}) = \frac{1}{de_{k}} |N_{ν,ν̄^{∗}}^{∗}(σ_{k}, σ′_{k})| = \frac{1}{de_{k}} |N_{d,ν̄^{∗}}^{∗}(σ_{k}, σ′_{k})|, (7)

where d = (ν, e_{k}) and ν̄^{∗} \frac{ν}{d} ≡ 1 (mod \frac{e_{k}}{d}). Thus, since the combination of Lemmas 1.9a) and 1.10a) yields |N_{1,1}^{∗}(σ_{0}, σ′_{0})| = 2h(−4N^{2}), we obtain from (7) that s_{1,ε}(P̄_{0}, P̄_{0}) = \frac{1}{2}|N_{1,1}^{∗}(σ_{0}, σ′_{0})| = h(−4N^{2}), which does not depend on ε. Furthermore, since e_{0} = 2 we have by (5) and a) that s_{1,ε}(P̄_{0}, P̄_{0}) = r_{2,ε}(P̄_{0}, P̄_{0}) = ∑_{x̄∈X̄} r_{2,ε}(x̄, P̄_{0}) =: r_{0,ε} = r_{0}. The second equation of b) follows from the first by using (5) and (6).

c) From Lemmas 1.9b) and 1.10b) we have |N_{1,1}^{∗}(σ_{1}, σ′_{1})| = \frac{3}{2}h(−3N^{2}), if 3 |̸N (and |N_{1,1}^{∗}(σ_{1}, σ′_{1})| = 3h(−3N^{2}), if 3|N and ε ≡ 1 (mod 3), and |N_{1,1}^{∗}(σ_{1}, σ′_{1})| = 0, if 3|N and ε ≢ 1 (mod 3)), and so by the same argument as in a) we obtain the indicated values for s_{1,ε}(P̄_{1}, P̄_{1}) = r_{1,ε}(P̄_{1}, P̄_{1}). The formula for s_{2,ε}(P̄_{1}, P̄_{1}) is proven similarly, and the formula for s_{3,ε}(P̄_{1}, P̄_{1}) follows from (5) and (6).

d) Let ν̄^{∗} be such that ν̄^{∗} \frac{ν}{d} ≡ 1 (mod \frac{N}{d}), where d = (ν, N). Then by Lemmas 1.9c) and 1.10c) we have |N_{d,ν̄^{∗}}^{∗}(σ_{∞}, σ′_{∞})| = \frac{1}{2}ρ(εν̄^{∗}, \frac{N}{d})dϕ(d)N = \frac{1}{2}ρ(\frac{εν}{d} , \frac{N}{d})dϕ(d)N, the latter since (\frac{ν}{d})/ν̄^{∗} is a square mod \frac{N}{d}. Thus, by (7) we obtain s_{ν,ε}(P̄_{∞}, P̄_{∞}) = \frac{1}{2}ρ(\frac{εν}{d} , \frac{N}{d})ϕ(d), as asserted. The last formula of d) follows from the previous one by using (5) and the fact that

∑_{{ν=1 \atop (ν,n)=1}}^{n} ρ(εν, n) = ϕ(n).

Proof of Proposition 1.3. The first assertion follows directly from Lemma 1.12 below, and the independence assertions follow from (4). It thus remains to prove the asserted formulae.

a), b) If ex̄ = p is prime, then rx̄ = tx̄ = r_{p}(x̄) = ∑ s_{ν}(x̄), where the sum is over all ν with 1 ≤ ν < ex̄ and (ν, ex̄) = 1, and so both a) and b) follow directly from Theorem 1.6.

c) Since r_{n,ε}(P̄_{∞}) = ∑_{x̄∈X̄} r_{n,ε}(x̄, P̄_{∞}) by definition, we have by Theorem 1.6a), d) that r_{n,ε}(P̄_{∞}) = r_{n,ε}(P̄_{∞}, P̄_{∞}) = \frac{1}{2}ϕ(n)ϕ(\frac{N}{d}).

d) The formula for r_{∞} follows directly from part c), and that for t_{∞} follows from c) together with the following (well-known) identity which, in view of Theorem 1.6d), follows from formula (6) above:

\frac{1}{2} ∑_{ν|N} νϕ(ν)ϕ (\frac{N}{ν} ) = \frac{m}{N} = \frac{1}{2}N^{2} ∏_{p|N} (1 − \frac{1}{p^{2}} ) . (8)

In the above proof we had used the following result.

Lemma 1.12 For each k = 0, 1, ∞ the stabilizer G_{k,ε} = α_{ε}^{−1}(G_{k}) is conjugate to G_{k}; more precisely, we have α_{ε}(σ_{k}) = τ_{k}^{−1}σ_{k}^{κ_{ε}}τ_{k}, where κ_{ε} = κ_{k,ε} is as in (3) and τ_{k} = τ_{k,ε} ∈ Sl_{2}(ℤ/Nℤ)/{±1}; in fact, we can take τ_{k} = 1 if k = ∞ and τ_{k} = g_{k,ε}^{±}Q_{±ε}^{−1} with g_{k,ε}^{±} ∈ M̃_{k,±ε}, if k = 0, 1.

Proof. Since α_{ε}(σ_{∞}) = σ_{∞}^{ε}, the assertions are clear for k = ∞.

For k = 0, 1,we first observe that either M̃_{k,ε}(N) or M̃_{k,−ε}(N) is non-empty (cf. Lemma 1.10), so there exists an element g_{k,ε}^{±} ∈ M̃_{k,±ε}. Clearly τ_{k} := g_{k,ε}^{±}Q_{±ε}^{−1} ∈ Sl_{2}(ℤ/Nℤ) and τ_{k}Q_{±ε} = g_{k,ε}^{±} ∈ ⋃_{μ} M̃_{k,μ}(N) = C_{Gl_{2}(ℤ/Nℤ)}(σ_{k}). Thus Q_{±ε}^{−1}τ_{k}^{−1}σ_{k}τ_{k}Q_{±ε} = σ_{k}, or τ_{k}^{−1}σ_{k}τ_{k} = Q_{±ε}σ_{k}Q_{±ε}^{−1} = α_{ε}(σ_{k}^{±1}). This proves the assertions, for the (−1)-case is only plays a role if k = 1 and in this case is only necessary if 3|N and ε ≡ 2 (mod 3) (cf. Lemma 1.10).

Corollary 1.13 The character h_{α_{ε}}^{1} afforded by the G-module H^{1}(X(N), ℂ) (endowed with the twisted G-action) is independent of ε, i.e. h_{α_{ε}}^{1} = h^{1}.

Proof. The above Lemma 1.12 implies that (1_{α^{−1}(G_{k})})^{G} = (1_{G_{k}})^{G}, for k = 0, 1, ∞, and so the assertion follows directly from [10], Proposition 1.8, Equation (16).

Although h_{ε}^{1} = ω_{ε} + \overline{ω_{ε}}, where ω_{ε} = ω ∘ α_{ε} denotes the character afforded by the Gmodule H^{0}(X, ω_{X}) of holomorphic differentials (with the twisted G-action), it is not true in general that the character ω_{ε} is independent of ε. However, we have:

Proposition 1.14 For any character χ of G and any ε ∈ (ℤ/Nℤ)^{×} we have χ_{−ε} = \overline{χ_{ε}}, where χ_{ε} := χ ∘ α_{ε}, and hence also ω_{−ε} = \overline{ω_{ε}}. In particular, if −1 is a square mod N, then ω_{ε} = \frac{1}{2}h^{1} is independent of ε.

Proof. In view of the above corollary and the remarks after it (together with Remark 1.7), the last assertions clearly follow from the first. Furthermore, since α_{−ε} = α_{ε} ∘ α_{−1}, it is enough to verify the first assertion for ε = 1, and for this it suffices to prove that α_{−1}(A) = \binom{a −b}{−c d} is conjugate to A^{−1} for all A = \binom{a b}{c d} ∈ Sl_{2}(ℤ/Nℤ), where N = p^{r} is a prime power. For the p-adic valuation we have w.l.o.g. v_{p}(a − d) ≥ min{v_{p}(b), v_{p}(c)} =: s (otherwise consider A′ = \binom{1 0}{1 1}A^{−1} = \binom{a−b b}{a−d+b+c b+d}). If s = v_{p}(c) then \frac{c}{p^{s}} ∈ (ℤ/p^{r}ℤ)^{∗} and

hence S := \binom{1 \frac{a−d}{p^{s}} ( \frac{c}{p^{s}})^{−1}}{0 1} ∈ Sl_{2}(ℤ/p^{r}ℤ). Observing that c\frac{a−d}{p^{s}}(\frac{c}{p^{s}})^{−1} = a − d (in ℤ/p^{r}ℤ) we find Sα_{−1}(A)S^{−1} = A^{−1}. In the case that s = v_{p}(b) we use S = \binom{1 0}{\frac{d−a}{p^{s}} ( \frac{b}{p^{s}})^{−1} 1}.

2 The diagonal quotient surfaces Z_{N,ε} and Z̃_{N,ε}

2.1 The singularities of Z_{N,ε}

Now that we have calculated the local invariants s_{ν,α_{ε}} of the modular curve X = X(N), we can apply the general results of [10] to determine the geometry of the modular diagonal quotient surface Z_{ε} = Z_{N,ε} = Δ_{ε}\(X(N) × X(N)), where Δ_{ε} = {(g, α_{ε}(g)) : g ∈ G_{N}}.

We begin by summarizing the nature of the singularities of Z_{N,ε}. Recall from [10] that every diagonal quotient surface has only cyclic quotient singularities and that these may be classified according to their type (n, q); this means that the singularity in question has the form A_{n,q} = C_{n}\ℂ^{2}, where the cyclic group C_{n} acts on ℂ^{2} via σ(z_{1}, z_{2}) = (χ_{1}(σ)z_{1}, χ_{2}(σ)z_{2}) where the “weights” χ_{i} : C_{n} → ℂ^{×} have order n and satisfy χ_{1} = χ_{2}^{q}.

To state the result, we let φ and ψ denote the canonical quotient maps induced by the inclusion of subgroups {1} ≤ Δ_{ε} ≤ G_{N} × G_{N}:

Y = X(N) × X(N) ⟶^{φ} Z_{ε} ⟶^{ψ} Ȳ = X(1) × X(1) ≅ ℙ^{1} × ℙ^{1}.

Theorem 2.1 a) For N ≥ 5 the set S_{ε} of singularities of Z_{ε} decomposes into the disjoint union of the sets S_{k,ε} = S_{ε} ∩ ψ^{−1}(P̄_{k}, P̄_{k}) of singularities lying over (P̄_{k}, P̄_{k}) ∈ Ȳ, where k = 0, 1, ∞.

b) The total number of singularities over (P̄_{k}, P̄_{k}) is given by |S_{k,ε}| = r_{k}, where r_{k} is as in Proposition 1.3; in particular, this number does not depend on ε. Moreover:

1. All r_{0} singularities in S_{0,ε} are of type (2, 1).

2. We have r_{1} = s_{1,1,ε} + s_{1,2,ε}, where s_{1,q,ε} denotes the number of singularities in S_{1,ε} of type (3, q), q = 1, 2. Moreover, if 3 |̸N, then both types occur equally often (so s_{1,q,ε} = \frac{r_{1}}{2} ), whereas if 3|N, then all r_{1} singularities are of type (3, q), where q ≡ ε (mod 3).

3. For each d|N, d ≠ N, and each q ∈ ((ℤ/\frac{N}{d} ℤ)^{×})^{2}, there are \frac{1}{2}ρ(\frac{N}{d})ϕ(d) singularities of type (\frac{N}{d} , εq) in S_{∞,ε}, and other types do not occur. In particular, if N is prime, Z_{ε} has exactly one singularity of type (N, q) for every 1 ≤ q < N with (\frac{q}{N} ) = (\frac{ε}{N} ) .

Remark 2.2 In [5], Hilfssatz 1, Hermann claims that |S_{0,ε}| = \frac{N}{2} ∏_{p|N} (1 − (\frac{−1}{p})) and

|S_{1,ε}| = \frac{N}{3} ∏_{p|N} (1 − (\frac{−3}{p})), rather than the above values |S_{0,ε}| = \frac{N}{2} ∏_{p|N} (1 − \frac{1}{p} (\frac{−1}{p}))

and |S_{1,ε}| = r_{1} = \frac{N}{3} ∏_{p|N} (1 − \frac{1}{p} (\frac{−3}{p})). Presumably this is a typographical error in [5].

Proof of Theorem 2.1. a) In view of [10], Theorem 2.3b), this is equivalent to the assertion that s_{ν,α_{ε}}(x̄, ȳ) = 0 if x̄ ≠ ȳ and 1 ≤ ν < eȳ, which is true by Theorem 1.6a).

b) By [10], Theorem 2.3c), we have |S_{k,ε}| = r_{k,α_{ε}}, which by (4) does not depend on ε (and hence is given by Proposition 1.3). The last assertions follow immediately from the fact that if d = (ν, e_{k}), then the number of singularities of type (\frac{e_{k}}{d} , \frac{ν}{d}) in S_{k,ε} is s_{ν,α_{ε}}(P̄_{k}, P̄_{k}) (cf. [10], Theorem 2.3) which by Theorem 1.6 has the indicated values.

The above theorem describes mainly the number of singularities of a given type. However, by analyzing the above proof, we see that we have also determined the singularities themselves as well in the process. To make this more precise, let us first introduce the following notation.

Notation 2.3 For ε ∈ (ℤ/Nℤ)^{×}, let

M_{k,ε}^{∗}(N) = {(x, y) ∈ ℤ/Nℤ × ℤ/Nℤ : q_{k}(x, y) ≡ ±ε (mod N)}, if k = 0, 1,
M_{∞,ε}^{∗}(N) = {(x, y) ∈ ℤ/Nℤ × ℤ/Nℤ : gcd(x, y, N) = 1, gcd(y, N) ≠ 1},

where q_{0}(x, y) = x^{2} + y^{2} and q_{1}(x, y) = x^{2} + xy + y^{2}; cf. Notation 1.8. Furthermore, if d|e_{k}, d ≠ e_{k} and 1 ≤ n < e_{k}, (n, \frac{e_{k}}{d}) = 1, then let M_{k,ε,d,n}^{∗} := M_{k,ε,d,n}^{∗}(N) ⊂ M_{k,ε}^{∗} := M_{k,ε}^{∗}(N) be defined as follows:

M_{0,ε,1,1}^{∗} = M_{0,ε}^{∗} = M_{0,ε} ∪ M_{0,−ε},
M_{1,ε,1,1}^{∗} = M_{1,ε},  M_{1,ε,1,2}^{∗} = M_{1,−ε},
M_{∞,ε,d,n}^{∗} = M_{∞,d,εn}^{(2)}.

It thus follows from the definitions that

M_{k,ε}^{∗} = · ; ⋃ ; d,n M_{k,ε,d,n}^{∗},

and from Lemma 1.9 that

p_{ε}(N_{d,n}^{∗}(σ_{k}, α_{ε}^{−1}(σ_{k}))) = M_{k,d,n,ε}^{∗}/{±1},

where p_{ε}(\binom{x y}{z w}) = (εx, z). In particular, for each (x, y) ∈ M_{k,ε,d,n}^{∗} there is an element σ_{k,ε,x,y} ∈ N_{d,n}^{∗}(σ_{k}, α_{ε}^{−1}(σ_{k})) ⊂ Sl_{2}(ℤ/Nℤ)/{±1} such that p_{ε}(σ_{k,ε,x,y}) = (x, y). (For example, σ_{0,ε,x,y} = Q_{ε}^{−1}\binom{x −y}{y x} = \binom{ε^{−1}x −ε^{−1}y}{y x}, if (x, y) ∈ M_{0,ε}, and σ_{0,ε,x,y} = Q_{−ε}^{−1}\binom{−x y}{y −x} =

\binom{ε^{−1}x ε^{−1}y}{y −x}, if (x, y) ∈ M_{0,−ε}, etc.)

Note that if we view the elements of M_{k,ε}^{∗} as column vectors, then ±σ_{k} acts on M_{k,ε,d,n}^{∗} and hence also on M_{k,ε}^{∗}; we shall denote the quotient space of this action by \overline{M}_{k,ε,d,n}^{∗} = ⟨±σ_{k}⟩\M_{k,ε,d,n}^{∗}, and by \overline{M}_{k,ε}^{∗} = ⟨±σ_{k}⟩\M_{k,ε}^{∗}.

Corollary 2.4 If N ≥ 5, then the map (a, c) ↦ z_{k,ε,a,c} := φ(σ_{k,ε,a,c}P_{k}, P_{k}) induces a natural bijection

ζ_{k,ε} : \overline{M}_{k,ε}^{∗} →^{∼} S_{k,ε}

between the set \overline{M}_{k,ε}^{∗} := ⟨±σ_{k}⟩\M_{k,ε}^{∗} of ±σ_{k}-orbits of M_{k,ε}^{∗} and the set S_{k,ε} of singularities of Z_{ε} which lie above (P̄_{k}, P̄_{k}). Furthermore, if (a, c) ∈ M_{k,ε,d,n}^{∗}, then z_{k,ε,a,c} = ζ_{k,ε}(a, c) is a singularity of type (\frac{e_{k}}{d} , q), where qn ≡ 1 (mod \frac{e_{k}}{d}). In particular, if a, c ∈ ℤ satisfy the conditions d := (c, N) > 1 and (a, d) = 1, then z_{∞,ε,εa,c} = (π̃(\frac{a}{c}), P_{∞}) is a singularity of type (d, q), where qεa^{2} ≡ 1 (d) and π̃ : ℌ^{∗} → X(N) = Γ(N)\ℌ^{∗} denotes the quotient map.

Proof. We first note that it follows from Lemma 1.9 that the above map p_{ε} : \binom{a b}{c d} ↦ (εa, c) induces a bijection

p̄_{ε} : G_{k,ε}\N_{d,n}^{∗}(σ_{k}, σ′_{k})/G_{k} →^{∼} \overline{M}_{k,ε,d,n}^{∗},

where σ′_{k} = α_{ε}^{−1}(σ_{k}). On the other hand, since λ_{k}(σ_{k}) = λ_{k,ε}(σ′_{k}) (cf. the proof of Theorem 1.6), it follows from [10], Corollary 2.5c), that the map g ↦ φ(gP_{k}, P_{k}) induces a bijection

G_{k,ε}\N_{d,n}^{∗}(σ_{k}, σ′_{k})/G_{k} →^{∼} S_{dq,α_{ε}}(P̄_{k}, P̄_{k}),

where S_{dq,α_{ε}}(P̄_{k}, P̄_{k}) denotes the set of singularities lying above (P̄_{k}, P̄_{k}) which are of type (\frac{e_{k}}{d} , q), where nq ≡ 1 (mod \frac{e_{k}}{d}). Combining these two bijections, we see that the assignment ζ_{k,ε} : (a, c) ↦ φ(p̄_{ε}^{−1}(a, c)P_{k}, P_{k}) = φ(σ_{k,ε,a,c}P_{k}, P_{k}) yields the desired bijection.

2.2 The configuration of exceptional curves on Z̃_{N,ε}

We now turn our attention to the minimal desingularization σ : Z̃_{ε} → Z_{ε} of Z_{ε}, which is obtained from Z_{ε} by replacing each singularity s ∈ S_{ε} by an “exceptional curve” E_{s}. Here E_{s} = C_{s,1} + … + C_{s,r_{s}} is a chain of ℙ^{1}’s whose length r_{s} and self-intersection numbers C_{s,j}^{2} = −c_{s,j} are determined by the continued fraction expansion \frac{n_{s}}{q_{s}} = [[c_{s,1}, …, c_{s,r_{s}}]] of \frac{n_{s}}{q_{s}}, where (n_{s}, q_{s}) denotes the type of the singularity s ∈ S_{ε}; cf. [1], III.2 or [10], section 3.2.

We can view the surface Z̃_{ε} as an X(N)-fibration over X̄ ≃ ℙ^{1} in two ways. Indeed, if ψ̃_{i} : Z̃_{ε} → X̄ denotes the composition of ψ̃ = ψ ∘ σ : Z̃_{ε} → Ȳ with the i-th projection map pr_{i} : Ȳ = X̄ × X̄ → X̄, where i = 1, 2, then all except three of the fibres ψ̃_{i}^{−1}(x̄) of ψ̃_{i} are isomorphic to X(N) (cf. [10], Proposition 2.1) whereas those over P̄_{k}, k = 0, 1, ∞, are reducible: their components consist of the above exceptional curves C_{s,j} together with the curves C̃_{k,i} which are the proper transforms under σ of the curves C_{k,1} = φ(P̄_{k} × X) and C_{k,2} = φ(X × P̄_{k}) on Z_{ε}. More precisely, we have:

Proposition 2.5 The divisor E = ∑_{s∈S_{ε}} E_{s} of exceptional curves on Z̃_{ε} is the sum of the three disjoint curves E_{k} = ∑_{s∈S_{k,ε}} E_{s} = ∑_{s∈S_{k,ε}} ∑_{j=1}^{r_{s}} C_{s,j}, where k = 0, 1, ∞. If we put

D_{k} = C̃_{k,1} + C̃_{k,2} + E_{k}, D_{k}^{∗} = ψ̃_{1}^{∗}(P̄_{k}) + ψ̃_{2}^{∗}(P̄_{k}), and D′_{k} = D_{k}^{∗} − D_{k},

then we have:

a) E_{0} is the sum of r_{0} disjoint (−2)-curves joining C̃_{0,1} to C̃_{0,2}, and we have

(9)  ψ̃_{i}^{∗}(P̄_{0}) = 2C̃_{0,i} + E_{0}, for i = 1, 2, and hence D_{0}^{∗} = 2D_{0}, D′_{0} = D_{0}.

b) E_{1} = E_{11} + E_{12} decomposes into two disjoint curves E_{11} and E_{12}, where E_{11} consists of s_{1,1,ε} disjoint (−3)-curves joining C̃_{1,1} to C̃_{1,2}, whereas E_{12} consists of s_{1,2,ε} disjoint (−2)-chains of length 2 joining C̃_{1,1} to C̃_{1,2}. Moreover, if E_{12i} denotes the sum of the components of E_{12} which meet C̃_{1,i}, then

(10) ψ̃_{i}^{∗}(P̄_{1}) = 3C̃_{1,i} +E_{1} +E_{12i}, for i = 1, 2, and thus D_{1}^{∗} = 3D_{1} −E_{11}, D′_{1} = 2D_{1} −E_{11}.

c) E_{∞} = ∑_{s∈S_{∞}} E_{s} decomposes into r_{∞} disjoint ℙ^{1}-chains E_{s} = C_{s,1} + … + C_{s,r_{s}} indexed by s ∈ S_{∞,ε} = ζ_{∞,ε}(M_{∞,ε}^{∗}). Moreover, if s = z_{∞,ε,a,c}, where n_{s} := (c, N) ≠ 1, and (a, n_{s}) = 1, then s is a singularity of type (n_{s}, q_{s}), where a^{2}q_{s} ≡ ε (mod n_{s}) and hence C_{s,j}^{2} = −c_{s,j}, where \frac{n_{s}}{q_{s}} = [[c_{s,1}, …, c_{s,r_{s}}]] is the continued fraction expansion of \frac{n_{s}}{q_{s}}. Furthermore,

ψ̃_{i}^{∗}(P̄_{∞}) = NC̃_{∞,i} + ∑_{s∈S_{∞,ε}} ∑_{j=1}^{r_{s}} a_{s,i,j}C_{s,j}, (11)

where i = 1, 2 and the coefficients a_{s,i,j} are determined from the continued fraction expansion of \frac{n_{s}}{q_{s}} by the recursion relations a_{s,i,j+1} = c_{s,j}a_{s,i,j} − a_{s,i,j−1}, 1 ≤ j ≤ r_{s}, together with the boundary conditions a_{s,1,0} = a_{s,2,r_{s}+1} = N, a_{s,1,r_{s}+1} = a_{s,2,0} = 0.

d) Each C̃_{k,i} is a smooth curve of genus g_{k} whose self-intersection number is given by:

C̃_{0,i}^{2} = −r_{0}/2, C̃_{1,i}^{2} = −(2r_{1} − s_{1,1,ε})/3, and C̃_{∞,i}^{2} = − ∑_{ν=1}^{N−1} ϕ((ν, N)) ⟨\frac{ν^{2}ε}{N(ν, N)}⟩ ,

where ⟨x⟩ = x − [x] denotes the fractional part. Furthermore, we have (C̃_{l,1}.C̃_{k,2}) = r_{1}(P̄_{l}, P̄_{k}) > 0, where r_{1}(P̄_{l}, P̄_{k}) is as in Theorem 1.6.

e) The canonical divisor K_{ε} of Z̃_{ε} is given by

K_{ε} ∼ −3D_{0} + D′_{1} + D′_{∞} ∼ D_{0} − D_{1} − D_{∞}. (12)

Proof. The first assertion follows from Theorem 2.1a). Moreover, since e_{0} = 2, e_{1} = 3, and e_{∞} = N, a), b) and c) follow easily from Propositions 3.5, 3.13 and 3.12 of [10], together with Theorem 2.1b). (For c), use also Corollary 2.4.) Moreover, d) follows from Corollary 3.15 of [10] by using Theorem 1.6. Finally, for e) we note that the different divisor D(ψ̃) (as defined in [10], Theorem 3.10) is by definition D(ψ̃) = D′_{0} +D′_{1} +D′_{∞} and that ψ̃^{∗}(KȲ) ∼ −2D_{k}^{∗}, for any k = 0, 1, ∞, and so the first equivalence in (12) follows from [10], Theorem 3.10, and (9). The second follows from the first by noting that D′_{1}+D′_{∞} ∼ 2D_{0}^{∗}−D_{1}−D_{∞} ∼ 4D_{0}−D_{1}−D_{∞} by (9) again.

2.3 The invariants of the surface Z̃_{ε}

We now turn to study the numerical invariants of Z̃_{ε}. Recall that by Theorem 3.7 of [10], the Betti, Hodge and Chern numbers of Z̃_{ε} can all be expressed in terms of the three fundamental invariants 𝔾_{ε}, 𝕊_{ε} and 𝕃_{ε}. Here these invariants have the following explicit values.

Theorem 2.6 The invariants 𝔾_{ε}, 𝕊_{ε} and 𝕃_{ε} are given by

𝔾_{ε} = \frac{m(N − 12)}{144N} − 1 + \frac{1}{8}ϕ(N) + \frac{1}{8}r_{0} + \frac{1}{6}r_{1} + \frac{1}{4}r_{∞},
𝕊_{ε} = \frac{1}{18} (2s_{1,1,ε} − r_{1}) + 𝕊_{∞,ε}
𝕃_{ε} = r_{0} + 2r_{1} − s_{1,1,ε} + 𝕃_{∞,ε},

where r_{0}, r_{1}, r_{∞} and s_{1,1,ε} are as in Theorem 2.1 and

𝕊_{∞,ε} = ∑_{{d|N \atop d≠N}} \frac{ϕ(d)}{2} ∑_{{n=1 \atop (n, \frac{N}{d})=1}}^{\frac{N}{d}} 𝕊 (εn^{2}, \frac{N}{d} ) and 𝕃_{∞,ε} = ∑_{{d|N \atop d≠N}} \frac{ϕ(d)}{2} ∑_{{n=1 \atop (n, \frac{N}{d})=1}}^{\frac{N}{d}} 𝕃 (\frac{N}{εn^{2}d}) ,

where 𝕊(q, n) = ∑_{k=1}^{n−1} ((\frac{k}{n})) ((\frac{kq}{n} )) denotes the usual Dedekind sum and 𝕃(\frac{n}{q}) the length of the continued fraction expansion of \frac{n}{(q)}, where (q) is the least positive residue of q mod n. Thus, if we put

R_{∞,ε} := 12𝕊_{∞,ε} + 𝕃_{∞,ε} + r_{∞} = ∑_{{d|N \atop d≠N}} \frac{ϕ(d)}{2} ∑_{{n=1 \atop (n, \frac{N}{d})=1}}^{\frac{N}{d}} (⟨\frac{εn^{2} + ε^{∗}(n^{∗})^{2}}{N/d} ⟩ + 𝕃 (\frac{N}{N − n^{2}ε})) ,

where n^{∗}n ≡ 1 (mod \frac{N}{d}) and 0 < n^{∗} < \frac{N}{d}, then the geometric genus p_{g,ε}, and the Chern numbers c_{1}^{2} = K_{ε}^{2} and c_{2} = χ_{top,ε} of Z̃_{ε} are given by

p_{g,ε} = 𝔾_{ε} − 𝕊_{ε} = \frac{m(N − 12)}{144N} + \frac{ϕ(N)}{8} + \frac{r_{0}}{8} + \frac{2r_{1}}{9} − \frac{s_{1,1,ε}}{9} + \frac{r_{∞}}{3} + \frac{𝕃_{∞,ε}}{12} − \frac{R_{∞,ε}}{12} − 1,
K_{ε}^{2} = 8 + 8𝔾_{ε} − 𝕃_{ε} − 12𝕊_{ε} = \frac{m(N − 12)}{18N} + ϕ(N) − \frac{s_{1,1,ε}}{3} + 3r_{∞} − R_{∞,ε},
χ_{top,ε} = 4 + 4𝔾_{ε} + 𝕃_{ε} = \frac{m(N − 12)}{36N} + \frac{ϕ(N)}{2} + \frac{3r_{0}}{2} + \frac{8r_{1}}{3} − s_{1,1,ε} + r_{∞} + 𝕃_{∞,ε}.

Proof. The formula for 𝔾_{ε} := \frac{1}{2}(g − g_{0,ε} − g_{1,ε} − g_{∞,ε}) (cf. [10], Remark 1.5), follows immediately from Propositions 1.1 and 1.3. Furthermore, by definition (cf. [10], Theorem 3.7) we have 𝕊_{ε} = 𝕊_{0,ε} + 𝕊_{1,ε} + 𝕊_{ε} where 𝕊_{k,ε} = ∑_{ν=1}^{e_{k}−1} s_{ν,ε}(P̄_{k})𝕊(\frac{ν}{(ν,e_{k})}, \frac{e_{k}}{(ν,e_{k})}) denotes the contribution from P̄_{k}. Since 𝕊(1, 2) = 0 and 𝕊(1, 3) = −𝕊(2, 3) = \frac{1}{18}, it follows that 𝕊_{0,ε} = 0 and 𝕊_{1,ε} = \frac{1}{18}s_{1,1,ε} − \frac{1}{18}s_{1,2,ε} = \frac{1}{18}(2s_{1,1,ε} − r_{1}) (cf. Theorem 2.1b)). Moreover, by Theorem 2.1b) we have

𝕊_{∞,ε} = ∑_{{d|N \atop d≠N}} \frac{ϕ(d)}{2} ∑_{{n=1 \atop (n, \frac{N}{d})=1}}^{\frac{N}{d}} ρ (εn, \frac{N}{d} ) 𝕊 (n, \frac{N}{d} ) .

This equals the asserted expression because the inner sum runs only over the n satisfying εn ≡ x^{2} (mod \frac{N}{d}) for some x, and the number of x satisfying εx^{2} ≡ n (mod \frac{N}{d}) is precisely ρ(εn, \frac{N}{d}) = ρ(\frac{N}{d}).

The proof for 𝕃_{ε} is similar, using the fact that 𝕃(\frac{2}{1}) = 𝕃(\frac{3}{1}) = 1 and 𝕃(\frac{3}{2}) = 2.

The indicated expression for R_{∞,ε} follows directly from the above expressions for 𝕊_{∞,ε} and 𝕃_{∞,ε}, together with the identity

12𝕊(q, n) + 𝕃 (\frac{n}{q}) + 1 = 𝕃 (\frac{n}{n − q} ) + \frac{q}{n} + \frac{q^{∗}}{n} ≤ n, (13)

which was established in [10], Proposition 1.15. Finally, the first expression for p_{g,ε} (and similarly, those for K_{ε}^{2} and χ_{top,ε}) is that given in Theorem 3.7 of [10], and the second follows from the first by substituting the above values of 𝔾_{ε}, 𝕊_{ε} and 𝕃_{ε}.

Remark 2.7 Some of above invariants have a natural character–theoretic interpretation: we have by Theorem 3.7 of [10] that

𝔾_{ε} = \frac{1}{4}(h^{1}, h_{ε}^{1}), 𝕊_{ε} = \frac{1}{4}(h^{1}, h_{ε}^{1}) − (ω, \overline{ω_{ε}}), and p_{g,ε} = (ω, \overline{ω_{ε}}),

where h_{ε}^{1} and ω_{ε} are as in Corollary 1.13 and Proposition 1.14. It thus follows from [10], Corollary 3.3, together with Proposition 1.14, that p_{g}(Z̃_{ε}) + p_{g}(Z̃_{−ε}) = 2𝔾_{ε} is independent of ε, and hence, if −1 is a square modulo N, then the same is true for p_{g}(Z̃_{ε}) = 𝔾_{ε} itself.

In order to be able to classify the surfaces according to the Enriques–Kodaira classification scheme, we require some good lower bounds for K_{ε}^{2} and p_{g}.

Proposition 2.8 The following inequalities hold for all N ≥ 5 and all ε:

K_{ε}^{2} > \frac{m}{18N^{2}} (N − 1)(N − 30) > \frac{1}{60}N(N − 1)(N − 30). (14)

Moreover, we have K_{ε}^{2} > 0 whenever N ≥ 17 or N ≥ 13 and ε ≢ 1 mod ((ℤ/Nℤ)^{×})^{2}.

Proof. We shall bound each of the terms appearing in the formula for K_{ε}^{2}. For s_{1,1,ε} we have the estimate:

s_{1,1,ε} ≤ r_{1} = \frac{N}{3} ∏_{p|N} (1 − \frac{1}{p} (\frac{−3}{p})) ≤ \frac{N}{3} ∏_{p|N} (1 + \frac{1}{p}) = \frac{2m}{3Nϕ(N)} ≤ \frac{m}{6N} ,

since ϕ(N) ≥ 4 if N ≥ 5. Moreover, by Proposition 1.3d) and identity (8) we have

r_{∞} = \frac{1}{2} ∑_{d|N,d≠N} ϕ(d)ϕ(\frac{N}{d}) ≥ \frac{1}{2N} ∑_{d|N,d≠N} dϕ(d)ϕ(\frac{N}{d}) = \frac{m}{N^{2}} − \frac{ϕ(N)}{2}. (15)

Furthermore, using the inequality (13) and Theorem 1.6d) we obtain

R_{∞,ε} ≤ ∑_{1<d|N} r_{d,ε}(P̄_{∞}, P̄_{∞})d = \frac{1}{2} ∑_{d|N,d≠N} \frac{N}{d} ϕ(d)ϕ(\frac{N}{d}) = \frac{m}{N} − \frac{ϕ(N)}{2}, (16)

the latter by (8) again. Substituting these estimates in the formula for K_{ε}^{2} in Theorem 2.6 yields

K_{ε}^{2} > \frac{m(N − 12)}{18N} + ϕ(N) − \frac{m}{18N} + 3 (\frac{m}{N^{2}} − \frac{ϕ(N)}{2}) − \frac{m}{N} + \frac{ϕ(N)}{2}
= \frac{m}{18N^{2}} (N^{2} − 31N + 54)
> \frac{m}{18N^{2}} (N^{2} − 31N + 30) = \frac{m}{18N^{2}} (N − 1)(N − 30),

which proves the first of the desired inequalities (14). The second follows from this because

\frac{m}{N^{3}} = \frac{1}{2} ∏_{p|N} (1 − \frac{1}{p^{2}} ) ≥ \frac{1}{2} ∏_{p} (1 − \frac{1}{p^{2}} ) = \frac{1}{2ζ(2)} = \frac{3}{π^{2}} > \frac{3}{10}. (17)

From (14) we see that K_{ε}^{2} > 0 if N ≥ 30. By computing the values K_{ε}^{2} for N ≤ 29 (cf. Tables 1 and 2 below), one verifies that this is also true if N ≥ 16 or if N ≥ 13 and ε ≢ 1 (mod ((ℤ/Nℤ)^{×})^{2}).

Proposition 2.9 For N ≥ 9 or for N ≥ 6 and ε ≢ 1 mod ((ℤ/Nℤ)^{×})^{2}, the geometric genus of Z̃_{ε} is positive: p_{g,ε} > 0. Moreover,

p_{g,ε} > \frac{1}{480}N(N − 1)(N − 23), (18)

and hence we have p_{g,ε} ≥ 3 if N ≥ 13.

Proof. We first derive the inequality (18). Since r_{0}, 𝕃_{∞,ε} and r_{1} ≥ s_{1,1,ε} are positive, substituting the estimates (15) and (16) in the formula for p_{g,ε} of Theorem 2.6 gives

p_{g}(Z̃_{ε}) + 1 > \frac{m(N − 12)}{144N} + \frac{1}{8}ϕ(N) + \frac{1}{3} (\frac{m}{N^{2}} − \frac{ϕ(N)}{2}) − \frac{1}{12} (\frac{m}{N} − \frac{ϕ(N)}{2})
= \frac{m}{144N^{2}} (N^{2} − 24N + 48).

Using the lower bound (17) we obtain

p_{g} > \frac{3N}{1440}(N^{2} − 24N + 48) − 1 = \frac{1}{480}N(N − 1)(N − 23) + \frac{5}{96}N − 1,

from which (18) follows readily.

If N ≥ 25, then (18) shows that p_{g} > \frac{1}{480}25·24·2 = \frac{5}{2} > 2, which proves the last assertion for N ≥ 25. For small N, the corresponding statement follows by explicit computations of the values (cf. Tables 1 and 2). Similarly, the first assertions may be verified in this way.

Remark 2.10 We can compute the geometric genus of Z̃_{ε} not only by the Noether formula as above but also with the help of the character ω. This yields a formula involving the class numbers h(−d) of the imaginary quadratic fields ℚ(\sqrt{−d}), where d|N. (As usual, we adopt the convention that h(−3) = \frac{1}{3} and h(−4) = \frac{1}{2}; cf. Notation 1.2.) To be precise, we have for N ≥ 4 the following formula:

p_{g}(Z̃_{ε}) = 𝔾_{ε}+\frac{1}{18}(s_{1,1,ε}−s_{1,2,ε})−\frac{1}{2} ∑_{ν|N} ϕ(ν)ϕ(\frac{N}{ν}) ∑_{μ|ν} \frac{1}{ϕ(μ)} ∑_{{d|μ \atop −d fund}}χ_{d}(ε) ∏_{{p|μ \atop p|/d}} (1−χ_{d}(p))^{2}h(−d)^{2},

where the above sum runs only over those d|N such that −d is a fundamental discriminant, and χ_{d} denotes the usual quadratic character on (ℤ/Nℤ)^{×} defined by χ_{d}(x) = (\frac{−d}{x}). Since we do not require this formula in the sequel, we shall not present its proof here.

2.4 Classification

We shall now classify the surfaces Z̃_{ε} according to their types in the Enriques–Kodaira classification table. For this, we may and shall view ε ∈ (ℤ/Nℤ)^{×}/((ℤ/Nℤ)^{×})^{2} since the isomorphism type of the diagonal quotient surfaces ℤ_{ε} depends only on the image of α_{ε} in the outer isomorphism group Out(G_{N}) (cf. Remark 1.7 and [10], Remark 3.9). It turns out rather surprisingly that the type of Z̃_{ε} is completely determined by its geometric genus p_{g}:

Theorem 2.11 a) Z̃_{ε} is a rational surface if and only if p_{g}(Z̃_{ε}) = 0, and this is the case precisely for N ≤ 5 or for (N, ε) = (6, 1), (7, 1) or (8, 1).

b) Z̃_{ε} is a (blown-up) elliptic K3-surface if and only if p_{g}(Z̃_{ε}) = 1, i.e. if and only if (N, ε) = (6, 5), (7, 3), (8, 3), (8, 5), (9, 1) or (12, 1).

c) Z̃_{ε} is a (blown-up) elliptic surface with κ = 1 if and only if p_{g}(Z̃_{ε}) = 2. This is the case for (N, ε) = (8, 7), (9, 2), (10, 1), (10, 3) or (11, 1).

d) Z̃_{ε} is a surface of general type if and only if p_{g}(Z̃_{ε}) ≥ 3, or equivalently, if N ≥ 13 or if (N, ε) = (11, 2), (12, 5), (12, 7) or (12, 11).

Remark 2.12 While this theorem is closely related to Satz 3 of Hermann[5], it is not identical with the latter. First of all, the interesting role of p_{g} in the classification of Z̃_{ε} is not mentioned by Hermann. Secondly, the somewhat tenacious case (N, ε) = (10, 3) seems to be missing in Satz 3c) of [5]. Thirdly, Hermann claims on p. 96 that the residue classes of (ℤ/Nℤ)^{×}/((ℤ/Nℤ)^{×})^{2} are represented by the elements ε ∈ (ℤ/Nℤ)^{×} with ε^{2} = 1, which is false e.g. for primes p ≡ 1 mod 4 (in which case the only solutions of ε^{2} = 1 are ±1 which are both squares). Thus the list of cases considered by Hermann seems to be incomplete.

The proof of this theorem is based on the classification criteria of [10], section 4. Here we apply these criteria to the curves C = C̃_{∞,i}, whose properties seem to control the type of Z̃_{ε}, as the following proposition shows. It is interesting to note that virtually all the cases of Theorem 2.11 are covered by this result.

Proposition 2.13 a) If g_{∞} = 0 and C̃_{∞,1}^{2} = −1, then Z̃_{ε} is rational.

b) If g_{∞} = 0 and C̃_{∞,1}^{2} = −2, then Z̃_{ε} is not of general type (i.e. κ(Z̃_{ε}) ≤ 1). If, in addition, p_{g}(Z̃_{ε}) = 1 and either ϕ(N) > 4 or ϕ(N) = 4 and −ε ∈ ((ℤ/dℤ)^{×})^{2} for some d|N, d ≠ 1, then Z̃_{ε} is a (blown-up) K3-surface; in the latter case, the minimal model of Z̃_{ε} is also an elliptic surface.

c) If g_{∞} = 0 and C̃_{∞,1}^{2} = −3 and

s(N, ε) := \frac{1}{2} ∑_{{1<d|N \atop −ε∈((ℤ/dℤ)^{×})^{2}}} ρ(d)ϕ (\frac{N}{d} ) > 5 − \frac{1}{2}ϕ(N),

then Z̃_{ε} is not of general type.

d) If g_{∞} ≥ 1 and d_{ε} := 2p_{g}(Z̃_{ε}) − 2g_{∞} + C̃_{∞,1}^{2} ≥ 1, then Z̃_{ε} is of general type.

Proof. Most of this proposition follows from Theorem 6 of [10], applied to x = P_{∞}. By Proposition 2.5d) we have, with the notation there:

c_{x,ε} = −C̃_{∞,1}^{2} = −C̃_{∞,2}^{2} and r_{1,x,ε} = (C̃_{∞,1}.C̃_{∞,2}) = \frac{1}{2}ϕ(N); (19)

in particular, the curves C̃_{∞,1} and C̃_{∞,2} always meet. Note that both are smooth of genus g_{∞} (cf. Proposition 2.5d). In addition we note that by Theorem 1.6d) the invariant s_{x,ε} of Theorem 6 is given by s_{x,ε} := ∑_{d|e_{x},d≠e_{x}} s_{e_{x}−d,ε}(x̄, x̄) = s(N, ε).

a) (cf. [10], Theorem 6a)) The hypothesis means that C̃_{∞,1} and C̃_{∞,2} are two (−1)-curves which meet, so Z̃_{ε} is rational (cf. van der Geer[3], VII.2.2).

b) If g ≤ 1, i.e. N ≤ 6, then κ(Z̃_{ε}) ≤ 0 by the classification Theorem 4.1c) of [10]. Thus, assume N ≥ 7, so ϕ(N) ≥ 4. But then C̃_{∞,1} and C̃_{∞,2} meet in at least 2 points (counting multiplicities), so {C̃_{∞,1}, C̃_{∞,2}} is a (−2)-configuration which cannot exist on a surface of general type (cf. vdG[3], VII.2.7). Thus κ(Z̃_{ε}) ≤ 1.

If, in addition, p_{g} = 1 etc., then we can apply Theorem 6b) of [10]. Here the hypotheses imply that k_{x,ε} := 4g_{∞} − 2C̃_{∞,1}^{2} − 4 = 0 and i_{x,ε} = 2r_{1,x,ε} + 2C̃_{∞,ε} = ϕ(N) − 4. Thus, the hypotheses on N and ε guarantee that s_{x,ε} > (k_{x,ε}^{2} −i_{x,ε})/2, and so it follows from Theorem 6b) of [10] that Z̃_{ε} is a K3-surface. Finally, to verify the last assertion we note that in the latter situation the hypotheses of [3], Proposition VII.2.9 apply (since {C̃_{∞,1}, C̃_{∞,2}} forms an elliptic configuration), and so the last assertion follows.

c) In this case k_{x,ε} = 2 and i_{x,ε} = ϕ(N) − 6, so again s_{x,ε} > (k_{x,ε}^{2} − i_{x,ε})/2, and the assertion follows from Theorem 6b) as before.

d) This follows directly from Theorem 6c) of [10].

Proof of Theorem 2.11. We first note that p_{g} has the asserted values. Indeed, if N ≥ 17, then p_{g} ≥ 3 by Proposition 2.9, whereas if N ≤ 16, then Table 1 below shows that p_{g} is as stated in the theorem. Thus, since the cases a) – d) cover all possibilities and are mutually exclusive, it is enough to show that Z̃_{ε} has the indicated type for each of the values (N, ε) listed.

a) If N ≤ 5, then g = 0, so Z̃_{ε} is rational since Z_{ε} is dominated by Y ≃ ℙ^{1} × ℙ^{1}; cf. also [10], Theorem 4.1a). For the other three cases the hypotheses of Proposition 2.13a) are satisfied (cf. Table 1), so Z̃_{ε} is rational in all these cases.

b) If (N, ε) = (6, 1), then Z̃_{ε} is an elliptic K3-surface by [10], Theorem 4.2c) because g = 1, ḡ = 0 and p_{g} = 1. Next, if (N, ε) = (8, 3), (8, 5) or (12, 1), then g_{∞} = 0 and C̃_{∞,i}^{2} = −2, and ϕ(N) = 4. Since 2|N, the last criterion of Proposition 2.13b) applies and so Z̃_{ε} is an elliptic K3-surface. Finally, if (N, ε) = (7, 3) or (9, 1), then we have g_{∞} = 0, C̃_{∞,i}^{2} = −2 and ϕ(N) = 6 > 4, so Z̃_{ε} is a (blown-up) K3-surface by Proposition 2.13b). It remains to show that the minimal models Z̄_{ε} of these two surfaces are elliptic.

Suppose first that (N, ε) = (7, 3). Since K_{ε}^{2} = −1 and Z̄_{ε} is a K3-surface, we see that K_{ε} = E, where E is the unique blow-down curve. Consider C := C̃_{1,1}, which is a smooth elliptic curve on Z̃_{ε}. Thus, by the adjunction formula (C.E) = (C.K_{ε}) = −C^{2} = 1, so the image C̄ of C on Z̄_{ε} is a smooth elliptic curve with C̄^{2} = 0. Thus, Z̄_{ε} has an elliptic configuration and is therefore an elliptic K3-surface (cf. [3], Proposition VII.2.9).

Finally, suppose that (N, ε) = (9, 1). Here K_{ε}^{2} = −6, so we have 6 blow-down curves E_{1}, …, E_{6}. Three of these are: E_{1} = Γ̃, the image of the diagonal (cf. [10], Remark 4.9), which meets a unique (−2)-curve E_{2} lying in the fibre (of ψ̃_{1}) over P̄_{0} and a unique (−3)-curve E_{3} in the fibre over P̄_{1}. Furthermore, E_{1} meets the fibre over P̄_{∞} only in the (−9)-component. Now the latter fibre contains four (−3)-curves: three meet C̃_{∞,1}, the fourth does not but is connected to C̃_{∞,1} by a (−2)-chain. Since the K3-surface Z̄_{ε} does not contain any curves with odd self-intersection number (by the adjunction formula), we see that each of these four (−3)-curves has to be met transversely by a blow-down curve. But we have only three such curves (since E_{1}, E_{2} and E_{3} cannot meet them), so one blowdown curve (say E_{4}) has to meet two of them. Thus, since C̃_{∞,1} is a (−2)-curve, we obtain, after blowing down E_{4}, an elliptic configuration consisting of a cycle of (−2)-curves, and hence Z̄_{ε} is again elliptic.

c) If (N, ε) = (10, 3) or (11, 1), then Z̃_{ε} is an elliptic surface by Proposition 2.14 below. For the other three cases we have s(N, ε) + \frac{1}{2}ϕ(N) = 6, so by Proposition 2.13c) we have κ(Z̃_{ε}) ≤ 1. Since p_{g} ≥ 2, we must have κ = 1, and so Z̃_{ε} is a (blown-up) elliptic surface, as asserted.

d) By Proposition 2.8 we have K_{ε}^{2} > 0 if N ≥ 17 or if N ≥ 13 and ε ≢ 1 mod ((ℤ/Nℤ)^{×})^{2}, so Z̃_{ε} is of general type in those cases (cf. [10], Corollary 4.6). We are thus left with the cases

(11, 2), (14, 1), (15, 1), (16, 1).

But for each of these cases we have g_{∞} ≥ 1 and d_{ε} := 2p_{g}(Z̃_{ε}) − 2g_{∞} + C_{∞,1}^{2} ≥ 1 (cf. Table 1), and so Proposition 2.13d) shows that these surfaces are all of general type as well.

To conclude the proof of Theorem 2.11, we still have to study the following two cases which seem to be much more subtle than the others in that they cannot be treated by general principles but instead require a detailed study of certain special curve configurations lying on them.

Proposition 2.14 If (N, ε) = (10, 3) or (11, 1), then Z̃_{ε} is a (blown-up) elliptic surface with κ = 1.

Proof. We first observe that it is enough to show that κ(Z̃_{ε}) ≤ 1 because p_{g}(Z̃_{ε}) = 2 and hence κ(Z̃_{ε}) ≥ 1 in both cases.

a) Let us first consider the case that (N, ε) = (10, 3). Then (with the notation of 2.5) we have D_{∞} = C̃_{∞,1} + C̃_{∞,2} + E_{∞,1} + … + E_{∞,6}, where E_{∞,1}, …, E_{∞,6} are 6 disjoint chains joining C̃_{∞,1} to C̃_{∞,2}. More explicitly, E_{∞,1} is a (−4, −2, −2)-chain, E_{∞,2} and E_{∞,3} are (−2)-curves, E_{∞,4} is a (−2, −3)-chain, E_{∞,5} a (−3, −2)-chain, and finally E_{∞,6} is a (−2, −2, −4)-chain; these correspond under the correspondence of Proposition 2.5c) to the points z_{∞,ε,1,0}, z_{∞,ε,1,2}, z_{∞,ε,1,4}, z_{∞,ε,1,5}, z_{∞,ε,2,5}, and z_{∞,ε,3,0}, respectively. (Here, ε = 3.) Thus, using the terminology of [10], Definition 4.16, we see that D = D(P̄_{∞}, P̄_{∞}) = C̃_{∞,1} + C̃_{∞,2} + E_{∞,2} + E_{∞,3} is a (−2)-join of breadth 2 of the two (−3)-curves C̃_{∞,1} and C̃_{∞,2} with intersection number (C̃_{∞,1}.C̃_{∞,2}) = \frac{1}{2}ϕ(10) = 2.

Claim 1: If Z̃_{ε} is of general type and C is a (−1)-curve on Z̃_{ε}, then C meets D_{∞} in each of its two (−4)-components transversely and in no other components, i.e. (C.D_{∞}) = 2.

First note that since none of the fibre components of ψ̃_{i} are (−1)-curves, C is not a component of a fibre and so (C.ψ̃_{i}(P̄_{k})) > 0; in particular, (C.D_{∞}) > 0.

Since Z̃_{ε} is of general type by hypothesis, we are in the situation of [10], Corollary 4.18 (with D = D(P̄_{∞}, P̄_{∞})), from which we conclude that C can only meet the (−4)- components of D_{∞}. Moreover, C has to meet any such component C′ transversely, for otherwise the image C̄′ of C′ on the blow-down surface Z̄ would be a singular curve with (KZ̄.C̄′) ≤ (KZ̃_{ε}.C′) − 2 = 0, which is impossible (cf. [1], VII.2.3).

We thus have (C.D_{∞}) = 1 or 2, depending on whether C meets one or both of the (−4)-components. Suppose the former were possible, i.e. that C meets only E_{∞,1} (or E_{∞,6}). Since the multiplicity of the (−4)-component of E_{∞,1} in ψ̃_{i}^{∗}(P̄_{∞}) is 3 respectively 1 for i = 1 respectively i = 2 (cf. 2.5), we have (C.ψ̃_{1}^{∗}(P̄_{∞})) = 3 and (C.ψ̃_{2}^{∗}(P̄_{∞})) = 1. Then also (C.ψ̃_{1}^{∗}(P̄_{1})) = 3 and (C.ψ̃_{2}^{∗}(P̄_{1})) = 1, which is impossible: on the one hand C has to meet a unique component C_{s,j} of D_{1} transversely, but then (C.ψ̃_{1}^{∗}(P̄_{1})) ≠ 3 (cf. 2.5). This, therefore, shows that the case (C.D_{∞}) = 1 is impossible, and so we must must have the situation of the claim. Claim 2: Let f : Z̃_{ε} → Z̄ denote the blow-down map with respect to a (−1)-curve C as in claim 1. Then Z̄ is minimal.

If not, then there is a (−1)-curve C̄ on Z̄. Since D̄ = f_{∗}(D_{∞}) consists entirely of (−2)- and (−3, −2)-chains (etc.) which join the (−3)-curves f(C̃_{∞,1}) and f(C̃_{∞,2}), we conclude from [10], Corollary 4.18 that C̄ does not meet D̄. Thus, if C̃ denotes the proper transform of C̄ with respect to f, then (C̃.(D_{∞} + 2C)) = (C̃.f^{∗}D̄) = (C̄.D̄) = 0, which means that C̃ meets neither C nor D_{∞}. But then C̃ is a (−1)-curve on Z̃_{ε} which doesn’t meet D_{∞}, which is impossible by claim 1. Thus, C̄ cannot exist, which verifies claim 2.

From claims 1 and 2 we can readily see that κ(Z̃_{ε}) ≤ 1. Indeed, since K_{ε}^{2} = −1 and p_{g} = 2 > 0, Z̃_{ε} is not minimal, so there exists a (−1)-curve C on Z̃_{ε}. Thus, if Z̃_{ε} were of general type, then by claims 1 and 2 the blow-down Z̄ is minimal. But K2 ; Z̄ = K2 ; Z̃ + 1 = 0, so Z̄ cannot be of general type, contradiction. Thus κ(Z̃_{ε}) ≤ 1 as desired.

b) We now turn to the much more difficult case that (N, ε) = (11, 1). Here we have (with the notation of 2.5) that E_{0}, resp. E_{11}, resp. E_{12} consists of 6 disjoint (−2)-curves, resp. of 2 disjoint (−3)-curves, resp. of 2 disjoint (−2)-chains of length 2. Moreover, E_{∞} = E_{∞,1} + …+E_{∞,5}, where E_{∞,i} is the ℙ^{1}-chain associated to the singularity z_{∞,1,i,0} ∈ S_{∞,1}; cf. Prop. 2.5. Thus, E_{∞,1} is a (−11)-chain, E_{∞,2} is a (−4, −3)-chain, E_{∞,3} is a (−3, −2, −2, −2, −2)- chain, E_{∞,4} is a (−2, −2, −2, −2, −3)-chain, and E_{∞,5} is a (−3, −4)-chain.

As in Remark 4.9 of [10], let Γ̃ denote the image of the diagonal of Y, and let Γ_{k} denote the unique component of E_{k} which meets Γ̃. From that remark we easily deduce: Claim 3: Γ̃ is a (−1)-curve which meets the (−2)-curve Γ_{0} and the (−3)-curve Γ_{1} transversely; in particular, there is a blow-down map f_{1} : Z̃ → Z̄_{1} which contracts each of Γ_{0}, Γ_{1} and Γ_{2} := Γ̃ to a point. Moreover, we have τ̃(Γ_{k}) = Γ_{k} for k = 0, 1, 2, where τ̃ denotes the involution of Z̃ induced by the one on Y = X × X which interchanges the two factors. Claim 4: There exist (−1)-curves Γ_{3} and Γ_{4} on Z̃ with the property that τ̃(Γ_{k}) = Γ_{k} and that Γ_{k} meets the (−k)-components of both E_{∞,2} and E_{∞,5} transversely for k = 3, 4.

To construct these Γ_{k} we shall use certain twists of the Hecke correspondences T_{n} = T′(1, n) on Y (cf. Shimura[19], §3.3, §7.3). Explicitly, for n, k ∈ ℕ with (nk, N) = 1, let T_{n,k} := φ((τ_{k}^{−1} × id_{X})T_{n}) ⊂ Z_{ε}, where τ_{k} = \binom{k^{−1} 0}{0 k} ∈ Sl_{2}(ℤ/Nℤ)/\{±1\}. Then T_{n,k} is an irreducible curve on Z_{ε} which is birationally isomorphic to the modular curve X_{0}(n) if and only if k^{2}nε ≡ 1 (mod N). If this is the case, then it follows from the definitions that its proper transform T̃_{n,k} on Z̃_{ε} is one of the curves F_{n}^{(i)} studied by Hermann[5], pp. 104-7. Moreover, it is easily verified that in that case we have

T_{n,k} ∩ S_{∞,ε} = {z_{∞,ε,εkn/d,0} : d|n}. (20)

In addition, if ε = 1, then we also have τ(T_{n,k}) = T_{n,k} and hence τ̃(T̃_{n,k}) = T̃_{n,k}.

Applying this to the case (N, ε) = (11, 1), we shall now show that Γ_{3} = T̃_{4,5} and Γ_{4} = T̃_{3,2} satisfy the properties of the claim. Indeed, Γ_{3} and Γ_{4} are (−1)-curves by Hermann[5], Hilfssatz 16. Moreover, (20) shows that T_{4,5} passes through z_{∞,1,2,0} and z_{∞,1,5,0}, so Γ_{3} meets E_{∞,2} and E_{∞,5} (and no other components of E_{∞}), and similarly, Γ_{4} meets E_{∞,1} = Γ_{∞}, E_{∞,2} and E_{∞,5}. Next we note that (Γ_{3}.ψ̃_{i}^{∗}(P̄_{∞})) = 6 and (Γ_{4}.ψ̃_{i}^{∗}(P̄_{∞})) = 4 because in general (T̃_{n,k}.ψ̃_{i}^{∗}(P̄)) = ψ(n) (by the projection formula). Thus, since the multiplicities of the components of E_{∞,2} in ψ̃_{1}^{∗}(P̄_{∞}) and in ψ̃_{2}^{∗}(P̄_{∞})) are (3, 1) and (1, 4), respectively, and those of E_{∞,5} are (4, 1) and (1, 3), respectively, one easily sees that Γ_{k} must meet the (−k)-components of E_{∞,2} and E_{∞,5} transversely, which verifies the claim.

Let us now put Γ_{5} = T̃_{5,3} and Γ_{6} = T̃_{9,4}. By exactly the same method as that of the proof of claim 4 one shows: Claim 5: There exist (−2)-curves Γ_{5} and Γ_{6} on Z̃ such that Γ_{5} (resp. Γ_{6}) meets the (−3)- components (resp. the extremal (−2)-components) of both E_{∞,3} and E_{∞,4} transversely. Claim 6: Let f : Z̃ → Z̄ denote the blow-down map, where Z̄ is the surface obtained by blowing down Z̄_{1} (cf. claim 3) via the images of Γ_{3} and Γ_{4} on Z̄_{1}. Then K2 ; Z̄ = 0 and the involution τ̃ on Z̃ induces an involution τ̄ on Z̄. Moreover, Ē_{0} := f_{∗}(E_{0}) and Ē_{1} := f_{∗}(E_{1}) consist entirely of (−2)-curves, whereas Ē_{∞} := f_{∗}(E_{∞}) consists of (−2) and (−3)-curves, together with the (−7)-curve Γ̄_{∞} = f_{∗}(Γ_{∞}).

The first assertion is clear since K2 ; Z̃ = −5 (cf. Table 1) and five curves are blown down, as is the second since τ̃(Γ_{k}) = Γ_{k} for 0 ≤ k ≤ 4. To prove the third, note first that Γ_{0}, Γ_{1} and Γ_{2} meet E_{0} (resp. E_{1}) only in Γ_{0} (resp. in Γ_{1}), which is blown down to a point by f. Moreover, by Hermann[5], Hilfssatz 11, Γ_{3} and Γ_{4} do not meet E_{0} (because ν_{2}(3) = ν_{2}(4) = 0),^{2} and so all the components of E_{0} other than Γ_{0} remain (−2)-curves. Similarly, Γ_{3} does not meet E_{1} (ν_{3}(4) = 0) whereas Γ_{4} meets a (−3)-component of E_{1} (because ν_{3}(3) = 1), which has to be different from Γ_{1}, for otherwise two (−1) curves would intersect on a suitable blow-down. Thus, this (−3)-component becomes a (−2)-component after blowing down Γ_{4}, and so the third assertion follows since all other components are (−2)-chains which do not meet Γ_{k}, 0 ≤ k ≤ 4 (cf. [5], Hilfssatz 11). Finally, the last assertion is immediate by claim 4 and the fact that Γ_{3} meets Γ_{∞} but Γ_{4} does not. Claim 7: For the minimal model Z_{min} of Z̃ (and of Z̄) we have K_{Z_{min}}^{2} ≤ 1.

We may assume that Z_{min} is of general type for otherwise K_{Z_{min}}^{2} = 0. The two curves C̄_{i} = f(C̃_{∞,i}), i = 1, 2, on Z̄ are joined by the (−2)-chain Ē_{1} arising from the images of the (−3)-components of E_{∞,2} and E_{∞,5}, and also by the (−2)-chain Ē_{2} obtained from the images of the extremal (−2)-curves of E_{∞,3} and E_{∞,4} together with Γ_{6} (cf. claim 5). Since no Γ_{k}, 0 ≤ k ≤ 4, meets C̃_{∞,1} +C̃_{∞,2}, we see that D̄ := C̄_{1} +C̄_{2} +Ē_{1} +Ē_{2} is a (−2)-join of breadth 2 with invariants k(C̄_{1}, C̄_{2}) := (K_{Z_{min}}.(C̄_{1} + C̄_{2})) = 4 and i(C̄_{1}, C̄_{2}) := (C̄_{1} + C̄_{2})^{2} = 6, and so we have K_{Z_{min}}^{2} ≤ 4^{2}/(2 · 2 + 6) < 2 by Proposition 4.17 of [10].

Claim 8: Z̄ is minimal and hence κ(Z̃) ≤ 1.

Suppose not; then there is a (−1)-curve C̄ on Z̄, and so by claims 7 and 6, the blowndown Z̄′ of Z̄ via C̄ is a minimal surface of general type. Thus C̄ is unique (so τ̄(C̄) = C̄) and C̄ cannot meet any (−2)-curve on Z̄. From claim 6 we therefore see that C̄ cannot meet Ē_{0} and Ē_{1}, and hence the total transform C of C̄ does not meet E_{0} + E_{1}. Thus, 2(C.C̃_{0,1}) = (C.ψ̃_{1}^{∗}(P̄_{0})) = (C.ψ̃_{1}^{∗}(P̄_{1})) = 3(C.C̃_{1,1}), and so it follows that (C.ψ̃_{1}^{∗}(P̄_{k})) ≡ 0 (mod 6), and hence (C.ψ̃_{1}^{∗}(P̄_{k})) ≥ 6 because C̄ cannot be a component of Ē_{k} by claim 6.

Next we note that (C.C̃_{∞,i}) = 0, for i = 1, 2. If not, then (C.C̃_{∞,i}) ≥ 1 for i = 1, 2 (because τ̃ fixes C and interchanges C̃_{∞,1} and C̃_{∞,2}), and so the image D̄′ = C̄′_{1} + C̄′_{2} + Ē′_{1} + Ē′_{2} on Z̄′ of the (−2)-join D̄ constructed in the proof of claim 7 would have invariants k(C̄′_{1}, C̄′_{2}) ≤ 2 and i(C̄′_{1}, C̄′_{2}) > 6, which violates the inequality of Proposition 4.17 of [10].

In addition, C̄ cannot meet any (−3)-components of Ē_{∞} = Ē_{∞,1} + … + Ē_{∞,5}, where Ē_{∞,j} := f_{∗}(E_{∞,j}). Indeed, if C̄ meets that of Ē_{∞,2} or of Ē_{∞,5}, then it also meets both because τ̄ interchanges these components, and then the images of Ē_{∞,2} and Ē_{∞,5} on Z̄′ would form an elliptic (−2)-configuration, which is impossible (cf. [3], VII.2.7). Similarly, if C̄ meets one of the (−3)-components of Ē_{∞,3} and Ē_{∞,4}, then it meets both, and the images of Ē_{∞,3}, Ē_{∞,4} and Γ_{6} on Z̄′ would form an elliptic (−2)-configuration, contradiction.

From claim 6 we thus see that C̄ can meet ψ̃_{1}^{∗}(P̄_{∞}) only in Γ_{∞} = E_{∞,1}, which has multiplicity 1 in ψ̃_{1}^{∗}(P̄_{∞}). Thus (C̄.Ē_{∞,1}) ≥ (C.E_{∞,1}) = (C.ψ̃_{1}^{∗}(P̄_{∞})) ≥ 6, and so the image Ē′ of Ē_{∞,1} on Z̄′ is a (singular) curve with (Ē′.KZ̄′) = (Ē_{∞,1}.KZ̄)−(C̄.Ē_{∞,1}) ≤ 5−6 = −1, which is impossible since Z̄′ is minimal. Thus, no such curve C̄ exists and so Z̄ is minimal.

2Note that the formulae for ν_{2}(n) and ν_{3}(n) on p. 104 of [5] are incorrect when 4|n and 9|n, and have to be modified as in Miyake [16], p. 108

\begin{matrix}N & ε & m & g & g_{0} & g_{1} & g_{∞} & C̃_{0}^{2} & C̃_{1}^{2} & C̃_{∞}^{2} & d & 2𝔾 & 𝕃 & 2𝕊 & p_{g} & c_{2} & K^{2} & κ \\ 5 & 1 & 60 & 0 & 0 & 0 & 0 & −1 & −1 & −1 & −1 & 0 & 10 & 0 & 0 & 14 & −2 & −1 \\ 5 & 2 & 60 & 0 & 0 & 0 & 0 & −1 & −1 & −1 & −1 & 0 & 9 & 0 & 0 & 13 & −1 & −1 \\ 6 & 1 & 72 & 1 & 0 & 0 & 0 & −2 & −1 & −1 & −1 & 1 & 10 & 1 & 0 & 16 & −4 & −1 \\ 6 & 5 & 72 & 1 & 0 & 0 & 0 & −2 & −2 & −2 & 0 & 1 & 18 & −1 & 1 & 24 & 0 & 0 \\ 7 & 1 & 168 & 3 & 1 & 1 & 0 & −2 & −1 & −1 & −1 & 1 & 12 & 1 & 0 & 18 & −6 & −1 \\ 7 & 3 & 168 & 3 & 1 & 1 & 0 & −2 & −1 & −2 & 0 & 1 & 19 & −1 & 1 & 25 & −1 & 0 \\ 8 & 1 & 192 & 5 & 2 & 1 & 0 & −2 & −2 & −1 & −1 & 2 & 14 & 2 & 0 & 22 & −10 & −1 \\ 8 & 3 & 192 & 5 & 2 & 1 & 0 & −2 & −2 & −2 & 0 & 2 & 18 & 0 & 1 & 26 & −2 & 0 \\ 8 & 5 & 192 & 5 & 2 & 1 & 0 & −2 & −2 & −2 & 0 & 2 & 18 & 0 & 1 & 26 & −2 & 0 \\ 8 & 7 & 192 & 5 & 2 & 1 & 0 & −2 & −2 & −3 & 1 & 2 & 28 & −2 & 2 & 36 & 0 & 1 \\ 9 & 1 & 324 & 10 & 4 & 3 & 0 & −3 & −1 & −2 & 0 & 3 & 20 & 1 & 1 & 30 & −6 & 0 \\ 9 & 2 & 324 & 10 & 4 & 3 & 0 & −3 & −2 & −3 & 1 & 3 & 28 & −1 & 2 & 38 & −2 & 1 \\ 10 & 1 & 360 & 13 & 6 & 3 & 0 & −2 & −3 & −3 & 1 & 4 & 30 & 0 & 2 & 42 & −6 & 1 \\ 10 & 3 & 360 & 13 & 6 & 3 & 0 & −2 & −3 & −3 & 1 & 4 & 25 & 0 & 2 & 37 & −1 & 1 \\ 11 & 1 & 660 & 26 & 12 & 8 & 1 & −3 & −2 & −2 & 0 & 5 & 27 & 1 & 2 & 41 & −5 & 1 \\ 11 & 2 & 660 & 26 & 12 & 8 & 1 & −3 & −2 & −3 & 1 & 5 & 34 & −1 & 3 & 48 & 0 & 2 \\ 12 & 1 & 576 & 25 & 11 & 7 & 0 & −4 & −2 & −2 & 0 & 7 & 22 & 5 & 1 & 40 & −16 & 0 \\ 12 & 5 & 576 & 25 & 11 & 7 & 0 & −4 & −4 & −4 & 4 & 7 & 38 & −1 & 4 & 56 & 4 & 2 \\ 12 & 7 & 576 & 25 & 11 & 7 & 0 & −4 & −2 & −4 & 2 & 7 & 30 & 1 & 3 & 48 & 0 & 2 \\ 12 & 11 & 576 & 25 & 11 & 7 & 0 & −4 & −4 & −6 & 6 & 7 & 58 & −5 & 6 & 76 & 8 & 2 \\ 13 & 1 & 1092 & 50 & 24 & 16 & 2 & −3 & −2 & −3 & 1 & 8 & 39 & 0 & 4 & 59 & 1 & 2 \\ 13 & 2 & 1092 & 50 & 24 & 16 & 2 & −3 & −2 & −3 & 1 & 8 & 34 & 0 & 4 & 54 & 6 & 2 \\ 14 & 1 & 1008 & 49 & 23 & 15 & 1 & −4 & −3 & −4 & 2 & 10 & 36 & 2 & 4 & 60 & 0 & 2 \\ 14 & 3 & 1008 & 49 & 23 & 15 & 1 & −4 & −3 & −5 & 5 & 10 & 49 & −2 & 6 & 73 & 11 & 2 \\ 15 & 1 & 1440 & 73 & 35 & 23 & 1 & −4 & −2 & −4 & 2 & 14 & 34 & 6 & 4 & 66 & −6 & 2 \\ 15 & 2 & 1440 & 73 & 35 & 23 & 1 & −4 & −4 & −6 & 4 & 14 & 44 & 2 & 6 & 76 & 8 & 2 \\ 15 & 7 & 1440 & 73 & 35 & 23 & 1 & −4 & −2 & −6 & 8 & 14 & 54 & −2 & 8 & 86 & 22 & 2 \\ 15 & 11 & 1440 & 73 & 35 & 23 & 1 & −4 & −4 & −8 & 10 & 14 & 76 & −6 & 10 & 108 & 24 & 2 \\ 16 & 1 & 1536 & 81 & 39 & 25 & 2 & −4 & −4 & −3 & 1 & 15 & 34 & 7 & 4 & 68 & −8 & 2 \\ 16 & 3 & 1536 & 81 & 39 & 25 & 2 & −4 & −4 & −5 & 5 & 15 & 44 & 1 & 7 & 78 & 18 & 2 \\ 16 & 5 & 1536 & 81 & 39 & 25 & 2 & −4 & −4 & −5 & 7 & 15 & 50 & −1 & 8 & 84 & 24 & 2 \\ 16 & 7 & 1536 & 81 & 39 & 25 & 2 & −4 & −4 & −7 & 11 & 15 & 80 & −7 & 11 & 114 & 30 & 2 \\ 17 & 1 & 2448 & 133 & 65 & 43 & 5 & −4 & −3 & −4 & 6 & 20 & 62 & 0 & 10 & 106 & 26 & 2 \\ 17 & 3 & 2448 & 133 & 65 & 43 & 5 & −4 & −3 & −4 & 6 & 20 & 47 & 0 & 10 & 91 & 41 & 2 \\ 18 & 1 & 1944 & 109 & 52 & 34 & 2 & −6 & −3 & −5 & 7 & 21 & 46 & 5 & 8 & 92 & 16 & 2 \\ 18 & 5 & 1944 & 109 & 52 & 34 & 2 & −6 & −6 & −8 & 14 & 21 & 82 & −5 & 13 & 128 & 40 & 2 \\ 19 & 1 & 3420 & 196 & 96 & 64 & 7 & −5 & −3 & −4 & 10 & 29 & 60 & 1 & 14 & 122 & 58 & 2 \\ 19 & 2 & 3420 & 196 & 96 & 64 & 7 & −5 & −3 & −5 & 11 & 29 & 67 & −1 & 15 & 129 & 63 & 2 \end{matrix}

\begin{matrix}N & ε & m & g & g_{0} & g_{1} & g_{∞} & C̃_{0}^{2} & C̃_{1}^{2} & C̃_{∞}^{2} & d & 2𝔾 & 𝕃 & 2𝕊 & p_{g} & c_{2} & K^{2} & κ \\ 20 & 1 & 2880 & 169 & 83 & 53 & 3 & −4 & −6 & −6 & 12 & 30 & 64 & 6 & 12 & 128 & 28 & 2 \\ 20 & 3 & 2880 & 169 & 83 & 53 & 3 & −4 & −6 & −8 & 14 & 30 & 62 & 2 & 14 & 126 & 54 & 2 \\ 20 & 11 & 2880 & 169 & 83 & 53 & 3 & −4 & −6 & −10 & 20 & 30 & 104 & −6 & 18 & 168 & 60 & 2 \\ 20 & 13 & 2880 & 169 & 83 & 53 & 3 & −4 & −6 & −8 & 18 & 30 & 74 & −2 & 16 & 138 & 66 & 2 \\ 21 & 1 & 4032 & 241 & 117 & 79 & 5 & −8 & −2 & −6 & 14 & 40 & 56 & 10 & 15 & 140 & 52 & 2 \\ 21 & 2 & 4032 & 241 & 117 & 79 & 5 & −8 & −4 & −8 & 16 & 40 & 64 & 6 & 17 & 148 & 68 & 2 \\ 21 & 5 & 4032 & 241 & 117 & 79 & 5 & −8 & −4 & −12 & 28 & 40 & 124 & −10 & 25 & 208 & 104 & 2 \\ 21 & 10 & 4032 & 241 & 117 & 79 & 5 & −8 & −2 & −10 & 26 & 40 & 100 & −6 & 23 & 184 & 104 & 2 \\ 22 & 1 & 3960 & 241 & 118 & 77 & 6 & −6 & −6 & −6 & 16 & 40 & 63 & 6 & 17 & 147 & 69 & 2 \\ 22 & 7 & 3960 & 241 & 118 & 77 & 6 & −6 & −6 & −9 & 25 & 40 & 102 & −6 & 23 & 186 & 102 & 2 \\ 23 & 1 & 6072 & 375 & 185 & 123 & 12 & −6 & −4 & −4 & 18 & 55 & 55 & 9 & 23 & 169 & 119 & 2 \\ 23 & 5 & 6072 & 375 & 185 & 123 & 12 & −6 & −4 & −7 & 33 & 55 & 112 & −9 & 32 & 226 & 170 & 2 \\ 24 & 1 & 4608 & 289 & 141 & 93 & 5 & −8 & −4 & −4 & 12 & 50 & 48 & 24 & 13 & 152 & 16 & 2 \\ 24 & 5 & 4608 & 289 & 141 & 93 & 5 & −8 & −8 & −10 & 30 & 50 & 88 & 0 & 25 & 192 & 120 & 2 \\ 24 & 7 & 4608 & 289 & 141 & 93 & 5 & −8 & −4 & −10 & 30 & 50 & 88 & 0 & 25 & 192 & 120 & 2 \\ 24 & 11 & 4608 & 289 & 141 & 93 & 5 & −8 & −8 & −12 & 36 & 50 & 120 & −8 & 29 & 224 & 136 & 2 \\ 24 & 13 & 4608 & 289 & 141 & 93 & 5 & −8 & −4 & −8 & 24 & 50 & 64 & 8 & 21 & 168 & 96 & 2 \\ 24 & 17 & 4608 & 289 & 141 & 93 & 5 & −8 & −8 & −10 & 30 & 50 & 92 & 0 & 25 & 196 & 116 & 2 \\ 24 & 19 & 4608 & 289 & 141 & 93 & 5 & −8 & −4 & −10 & 30 & 50 & 84 & 0 & 25 & 188 & 124 & 2 \\ 24 & 23 & 4608 & 289 & 141 & 93 & 5 & −8 & −8 & −16 & 48 & 50 & 208 & −24 & 37 & 312 & 144 & 2 \\ 25 & 1 & 7500 & 476 & 236 & 156 & 12 & −5 & −5 & −9 & 39 & 72 & 106 & 0 & 36 & 254 & 190 & 2 \\ 25 & 2 & 7500 & 476 & 236 & 156 & 12 & −5 & −5 & −9 & 39 & 72 & 99 & 0 & 36 & 247 & 197 & 2 \\ 26 & 1 & 6552 & 421 & 208 & 137 & 10 & −6 & −6 & −9 & 37 & 66 & 111 & 0 & 33 & 247 & 161 & 2 \\ 26 & 5 & 6552 & 421 & 208 & 137 & 10 & −6 & −6 & −9 & 37 & 66 & 84 & 0 & 33 & 220 & 188 & 2 \\ 27 & 1 & 8748 & 568 & 280 & 187 & 13 & −9 & −3 & −9 & 49 & 88 & 102 & 4 & 42 & 282 & 234 & 2 \\ 27 & 2 & 8748 & 568 & 280 & 187 & 13 & −9 & −6 & −12 & 54 & 88 & 132 & −4 & 46 & 312 & 252 & 2 \\ 28 & 1 & 8064 & 529 & 261 & 173 & 10 & −8 & −6 & −9 & 47 & 85 & 102 & 9 & 38 & 276 & 192 & 2 \\ 28 & 3 & 8064 & 529 & 261 & 173 & 10 & −8 & −6 & −15 & 59 & 85 & 162 & −9 & 47 & 336 & 240 & 2 \\ 28 & 5 & 8064 & 529 & 261 & 173 & 10 & −8 & −6 & −11 & 55 & 85 & 110 & −1 & 43 & 284 & 244 & 2 \\ 28 & 11 & 8064 & 529 & 261 & 173 & 10 & −8 & −6 & −13 & 51 & 85 & 106 & 1 & 42 & 280 & 236 & 2 \\ 29 & 1 & 12180 & 806 & 400 & 266 & 22 & −7 & −5 & −7 & 67 & 118 & 118 & 0 & 59 & 358 & 362 & 2 \\ 29 & 2 & 12180 & 806 & 400 & 266 & 22 & −7 & −5 & −7 & 67 & 118 & 109 & 0 & 59 & 349 & 371 & 2 \\ 30 & 1 & 8640 & 577 & 285 & 187 & 9 & −8 & −6 & −10 & 50 & 96 & 94 & 18 & 39 & 290 & 190 & 2 \\ 30 & 7 & 8640 & 577 & 285 & 187 & 9 & −8 & −6 & −12 & 64 & 96 & 110 & 2 & 47 & 306 & 270 & 2 \\ 30 & 11 & 8640 & 577 & 285 & 187 & 9 & −8 & −12 & −18 & 78 & 96 & 216 & −18 & 57 & 412 & 284 & 2 \\ 30 & 17 & 8640 & 577 & 285 & 187 & 9 & −8 & −12 & −16 & 64 & 96 & 132 & −2 & 49 & 328 & 272 & 2 \end{matrix}

References

[1] W. Barth, C. Peters, A. Van de Ven: Compact Complex Surfaces. Springer-Verlag, Berlin, 1984.

[2] G. Frey: On ternary equations of Fermat type and relations with elliptic curves. To appear.

[3] G. van der Geer: Hilbert Modular Surfaces. Springer-Verlag, Berlin, 1988.

[4] E. Hecke: Über ein Fundamentalproblem aus der Theorie der elliptischen Modulfunktionen. Abh. Math. Sem. Hamburg 6 (1928), 235-257 = Math. Werke, pp. 525-547.

[5] C.F. Hermann: Modulflächen quadratischer Diskriminante. Manuscr. math. 72 (1991), 95-110.

[6] Hua Loo Keng: Introduction to Number Theory. Springer-Verlag, Berlin, 1982.

[7] E. Kani: Curves of genus 2 with elliptic differentials and the height conjecture for elliptic curves. Proc. Conf. Number Theory and Arithmetic Geometry (G. Frey, ed.) Univ. Essen, Preprint No. 18 (1991), 30-39.

[8] E. Kani: Curves with elliptic differentials. Preprint.

[9] E. Kani: Elliptic curves on abelian surfaces. Manuscr. math. 84 (1994), 199-223.

[10] E. Kani, W. Schanz: Diagonal quotient surfaces. Manuscr. math. (to appear)

[11] O.-H. Keller: Darstellungen von Restklassen (mod n) als Summe von zwei Quadraten. Acta Sci. Math. (Szeged) 25 (1964), 191-192.

[12] A. Kraus, O. Oesterlé: Sur une question de B. Mazur. Math. Ann. 293 (1992), 259-275.

[13] E. Landau: Vorlesungen über Zahlentheorie. Chelsea Publ. Co., New York, 1950.

[14] S. Lang: Number Theory III. Ency. Math. Sci. vol. 60, Springer-Verlag, Berlin, 1991.

[15] B. Mazur: Rational isogenies of prime degree. Invent. math. 44 (1978), 129-162.

[16] T. Miyake: Modular Forms, Springer-Verlag, Berlin, 1989.

[17] H.W. Praetorius: Die Charaktere der Modulgruppe der Stufe q^{2}. Abh. Math. Sem. Univ. Hamburg 9 (1933), 365-394.

[18] B. Schoeneberg: Elliptic Modular Functions. Springer-Verlag, Berlin, 1974.

[19] G. Shimura: Introduction to the Arithmetic Theory of Automorphic Functions. Iwanami Shoten & Princeton University Press, 1971.

E. Kani

Department of Mathematics and Statistics Queen’s University Kingston, Ontario, Canada K7L 3N6 W. Schanz Schuppstr. 2 65191 Wiesbaden Germany