The Existence of Curves of Genus Two with Elliptic Differentials

Ernst Kani^{∗}

Introduction

The main aim of this paper, which is the sequel to [Ka], is to prove the existence of curves C of genus 2 admitting morphisms to two given elliptic curves E and E′. More precisely, we are interested in the following problem.

Question. Given two elliptic curves E and E′ over an algebraically closed field K and an integer N ≥ 2, does there exist a curve C of genus 2 which admits two morphisms

f : C → E, f′ : C → E′,

of degree N such that the induced maps f^{∗} and f′_{∗} on the associated Jacobian varieties fit into an exact sequence

0 → J_{E} →^{f^{∗}} J_{C} →^{f′_{∗}} J_{E′} → 0 ? (1)

To put this question into its proper perspective, it may be useful to recall the following facts (cf. [Ka] for more details and historical remarks). If a curve C of genus 2 admits any non-constant morphism f_{1} : C → E_{1} to an elliptic curve E_{1} at all — in which case we say (mainly for historical reasons) that C admits an elliptic differential — then we have in fact the situation as described above, for f_{1} factors over a morphism f : C → E, and there is a complementary morphism f′ : C → E′ (with deg(f′) = deg(f) =: N) such that the induced morphisms on the Jacobians fit into an exact sequence (1). Since f and f′ are uniquely determined by f_{1} up to isomorphism, we say that (E, E′, N) is the type of the elliptic differential (or of the covering).

If char(K) |̸ N, then it is easy to see that there are only finitely many curves C of genus 2 admitting an elliptic differential of type (E, E′, N); in fact, if we let n(E, E′, N) denote the number of such curves C, each counted with multiplicity according to its automorphisms, then we have the estimate

*NSERC University Research Fellow

n(E, E′, N) ≤ sl(N) := #Sl_{2}(ℤ/Nℤ) = N^{3} ∏_{p|N} (1 − \frac{1}{p^{2}} ) ,

and equality holds if E and E′ are not isogenous. If, however, E and E′ are isogenous, then the exact value of n(E, E′, N) is much more complicated and was determined in [Ka]. Nevertheless, the mere knowledge of a formula for the number of curves still doesn’t prove their existence, for it is difficult to see from the fomula that the number in question is positive, particularly if N is not prime (and/or E and E′ are supersingular). To this end, we therefore establish the following simple lower bound, which generalizes the bound obtained in [Ka] (for N prime) to all composite N’s prime to the characteristic of K.

Theorem 1 (“Existence Theorem”). If char(K) |̸ N then we have

\frac{1}{6}sl(N) < n(E, E′, N) ≤ sl(N), (2)

except when j(E) = j(E′) = 0 and E ≃ E′ is supersingular. Thus, aside from this exception, there always exist a curve C of genus 2 of type (E, E′, N).

In particular, we see that the above question has a positive answer whenever p := char(K) = 0 or p ≡ 1 (3). Nevertheless, it can happen in the above exceptional case that no curve of type (E, E′, N) exists, as the following “non-existence theorem” shows.

Theorem 2. If char(K) = 2 or 3 and E/K is supersingular (hence j(E) = 0), then n(E, E, N) = 0 for all N ≥ 2 with char(K) |̸ N. Thus, there is no curve of genus 2 of type (E, E, N).

It seems likely that the case treated in Theorem 2 is the only type for which no curve exist:

Conjecture. There is no curve of type (E, E′, N) (with p = char(K) |̸ N) if and only if p = 2 or 3 and j(E) = j(E′) = 0.

As was already remarked, Theorem 1 shows that the conjecture is true if p = 0 or if p ≡ 1 (3). Moreover, if p ≡ 2 (3), then it can be shown that for each p there are at most finitely many exceptions. This follows from the following result which determines the precise order of magnitude of the function n(E, E, N) when E is supersingular and which includes Theorem 2 as a special case.

Theorem 3. If E is a supersingular curve over an algebraically closed field of characteristic p, then for each N ≥ 2 with p |̸ N we have

n(E, E, N) = \frac{(p − 2)(p − 3)}{p^{2} + 1}sl(N) + R(N) (3)

where the error term satisfies the estimate

R(N) = O(N^{2+ε}). (4)

Moreover, R(N) = 0 if p = 2 or 3. In particular, n(E, E, N) > 0 if p ≥ 5 and N ≥ N_{0}(p) is sufficiently large.

Aside from the results of [Ka], the proof of the above theorem requires the theory of modular forms and uses the Petersson-Ramanujan Conjecture (which was proved by Eichler and Deligne).

Remark: In the case that E = E′ is an (ordinary) elliptic curve with complex multiplication, I. Kiming [Ki] has determined the asymptotic behaviour of the function r(E, E, N) := sl(N) − n(E, E, N) when restricted to prime numbers N; this partly complements Theorem 3.

Even though the proof of Theorem 3 shows that the above constant N_{0}(p) is effectively computable (in principle) for each given p, this does not lead directly to any practical bounds on N_{0}(p). Nevertheless, if p is small, then the method of proof of Theorem 3 can be refined so as to yield useful lower bounds, and hence the above conjecture can be verified in these cases.

Theorem 4. If p < 23 then the above conjecture is true; i.e. if p ≠ 2, 3 then we have n(E, E′, N) > 0 for all N ≥ 2 with p |̸ N.

We now discuss the contents of this paper in more detail.

As was already mentioned, although the results of [Ka] yield an explicit formula for n(E, E′, N), the task of extracting from this the general lower bound asserted in Theorem 1 still requires considerable work, particularly if Hom(E, E′) is large (e.g. if E and E′ are supersingular). Indeed, for N prime, this formula has the form

n(E, E′, N) = sl(N) − r(E, E′, N) = sl(N) − \frac{1}{2} ∑_{k=1}^{N−1} h(E, E′, k(N − k)), (5)

where h(E, E′, m) denotes the number of homomorphisms h : E → E′ of degree m. If N is composite, then there is a similar but much more complicated formula for r(E, E′, N) := sl(N) − n(E, E′, N) (cf. section 2); as a result, naive estimates of the right hand side of (5) tend to be negative. To circumvent this problem, a certain “mass formula” was proved in [Ka] which shows (in principle, at least) that the term r(E, E′, N) is “on average” much smaller than sl(N). While this is by no means immediately evident from the version proved in [Ka], it will become clear once we have verified the following remarkable identity in elementary number theory which, by the way, was discovered for this purpose with the help of a computer.

Theorem 5. Let σ(m, N) denote the arithmetical function defined by

σ(m, N) = ∑_{{d|N \atop d^{2}|m}} μ(d)σ(m/d^{2}),

where μ(d) denotes the Moebius μ-function and σ(n) = ∑_{d|n} d the sum of divisors function. Then

∑_{k=1}^{N−1} σ(k(N − k), N) = (\frac{5}{12} − \frac{1}{2N} ) sl(N). (6)

This identity, which depends on a classical identity of Glaisher[Gl], is derived in section 1. There we also show how this leads to the following version of the “Mass Formula” which was proved in another form in [Ka].

Theorem 6. Let E be an elliptic curve over K. Then

∑_{E′} \frac{r(E, E′, N)}{#Aut(E′)} ≤ (\frac{5}{24} − \frac{1}{4N} ) sl(N), (7)

where the sum on the left extends over a system of representatives of the isomorphism classes of elliptic curves E′/K. Moreover, equality holds in (7) if and only if char(K) = 0 or if N ≤ char(K).

Remark. As the proof below shows, the above inequality also holds if char(K) | N provided that we define, as in [Ka], r(E, E′, N) as the number of reducible antiisometries ψ : E[N] → E′[N]. (This agrees with the above definition in the case that char(K) |̸ N.) Thus r(E, E′, N) is always finite, whereas n(E, E′N) is infinite if (and only if) E and E′ are supersingular and char(K) | N (cf. [Ka], Theorem 3.4).

From Theorem 6 (and the results of [Ka]) it is easy to deduce the main existence theorem (Theorem 1); this will be done in section 2.

Finally, in section 3 we use the theory of modular forms to derive the order of magnitude of the function n(E, E, N) in the case that E is a supersingular elliptic curve and thereby prove Theorems 3 and 4.

This paper and its predecessor [Ka] developed out of the joint work [FK] with G. Frey, whom I would like to thank very much for the many stimulating and fruitful discussions as well as for his continued interest in this research. In addition, I have greatly benefitted from discussions with A. Brumer, I. Kiming, B. Mazur, F. Oort and J.-P. Serre on this topic, as well as from the comments of the referee. Finally, I would like to gratefully acknowledge support from the Natural Sciences and Engineering Research Council of Canada (NSERC).

1 An arithmetical identity

The purpose of this section is to prove the following remarkable identity (Theorem 1.2) concerning the arithmetical function σ(n) (= sum of divisors function) and to use it to prove the “mass formula” (Theorem 6) of the introduction.

Notation 1.1 If f : ℕ → ℂ is any arithmetical function, then we define the twovariable function f(m, n) by

f(m, n) = ∑_{{d|n \atop d^{2}|m}} μ(d)f(m/d^{2}), (1.1)

where μ(n) denotes the Moebius μ-function. Note that if f is multiplicative, i.e. f(nm) = f(n)f(m) if (n, m) = 1, then so is f( · , n), and hence we have

f(m, n) = ∏_{{p |̸ n \atop p^{r}||m}} f(p^{r}) ∏_{{p|n \atop p^{r}||m}} f(p^{r}, p) = f(m/m_{n})f(m_{n}, n), where m_{n} = ∏_{{p|n \atop p^{r}||m}}p^{r} (1.2)

denotes the n-primary part of m.

In the sequel we shall be particularly interested in the case that f(n) = σ(n) is the sum of divisors function, i.e., σ(n) = ∑_{d|n} d. In this case (1.2) shows that

σ(m, n) = ψ(m_{n})σ(m/m_{n}), where ψ(n) = n ∏_{p|n} (1 + \frac{1}{p}) (1.3)

denotes as usual the Dedekind ψ-function.

The function σ(n, m) satisfies the following curious identity.

Theorem 1.2 For every n ≥ 2 we have

∑_{k=1}^{n−1} σ(k(n − k), n) = \frac{1}{12}(5n − 6)ϕ(n)ψ(n) = (\frac{5}{12} − \frac{1}{2n}) sl(n), (1.4)

where ϕ denotes the Euler ϕ-function and sl(n) = #Sl_{2}(ℤ/nℤ).

Remark 1.3 Although the identity (1.4) seems to be new, the essential ingredient of its proof is, as we shall see presently, the following identity due to Glaisher (1884) which was generalized by Ramanujan in 1915 (cf. [Gl], [Ra], and also [Di], p. 300):

s(n) := ∑_{k=1}^{n−1} σ(k)σ(n − k) = \frac{1}{12}[5σ_{3}(n) − 6nσ(n) + σ(n)]. (1.5)

Note that this identity follows immediately from the identity

∂_{2}P = −Q − P^{2}

of Lang [La], ch. X, Th. 5.3 (p. 161) by comparing coefficients of the q-expansions of Q, P and ∂_{2}P as given on pp. 156, 160 of [La].

Proof of Theorem 1.2. Since sl(n) = n^{3} ∏_{p|n} (1 − \frac{1}{p^{2}} ) = nϕ(n)ψ(n), the second equality in (1.4) is clear. To prove the first, let

s^{∗}(n) = ∑_{k=1}^{n−1} σ(k(n − k), n) and s′(n) = ∑_{k=1}^{n−1} σ(k(n − k)).

These two functions are related to s(n) by the formulae

s^{∗}(n) = ∑_{d|n} μ(d)s′(n/d) and s′(n) = ∑_{d|n} μ(d)ds(n/d). (1.6)

To see this, note first that we have μ(d) ≠ 0, d|n, d^{2}|k(n − k) ⟺ μ(d) ≠ 0, d|n, d|k because if n = p_{1} ⋯ p_{r}, d|n and d |̸ k, then p_{i} |̸ k for at least one i, so p_{i} |̸ n − k, and then p_{i} |̸ k(n − k) and d^{2} |̸ k(n − k). Thus we have

s^{∗}(n) = ∑_{k=1}^{n−1} σ(k(n − k), n) = ∑_{k=1}^{n−1} ∑_{{d|n \atop d^{2}|k(n−k)}} μ(d)σ (\frac{k(n − k)}{d^{2}})

= ∑_{k=1}^{n−1} ∑_{{d|n \atop d|k}} μ(d)σ (\frac{k(n − k)}{d^{2}}) = ∑_{d|n} μ(d) ∑_{{k=1 \atop d|k}}^{n−1} σ (\frac{k}{d} (\frac{n}{d} − \frac{k}{d} ))

= ∑_{d|n} μ(d) ∑_{k=1}^{n/d−1} σ (k (\frac{n}{d} − k)) = ∑_{d|n} μ(d)s′(n/d),

which establishes the first formula of (1.6). To prove the second formula, we shall use the identity

σ(nm) = ∑_{{d|n \atop d|m}} σ(n/d)σ(m/d)μ(d)d, ∀n, mϵℕ, (1.7)

which is well-known in the theory of modular forms and Hecke-Operators. An elementary proof of this may be found in McCarthy [Mc], p. 24. By (1.7) we obtain

s′(n) = ∑_{k=1}^{n−1} σ(k(n − k)) = ∑_{k=1}^{n−1} ∑_{{d|k \atop d|n−k}} σ(k/d)σ (\frac{n − k}{d} ) μ(d)d

= ∑_{d|n} μ(d)d ∑_{{k=1 \atop d|k}}^{n−1} σ(k/d)σ(n/d − k/d) = ∑_{d|n} μ(d)d ∑_{k=1}^{n/d−1} σ(k)σ(n/d − k)

which proves the second formula of (1.6).

Substituting the second formula of (1.6) into the first yields the relation

s^{∗}(n) = ∑_{d|n} u(d)s(n/d), (1.8)

where u(n) is the arithmetical function defined by

u(n) = ∑_{d|n} dμ(d)μ(n/d).

We shall now deduce (1.4) from (1.8) by substituting Glaisher’s identity (1.5) and applying the identities

∑_{d|n} σ_{k}(d)u(n/d) = nJ_{k−1}(n), and ∑_{d|n} dσ_{k}(d)u(n/d) = J_{k+1}(n), (1.9)

where, as in McCarthy [Mc], p. 13, J_{k}(n) = ∑_{d|n} d^{k}μ(n/d). Specifically we obtain

s^{∗}(n) = \frac{1}{12} ∑_{d|n} u(n/d)[5σ_{3}(d) − 6dσ(d) + σ(d)] = \frac{1}{12}[5nJ_{2}(n) − 6J_{2}(n) + nJ_{0}(n)],

which is (1.4) because J_{2}(n) = ϕ(n)ψ(n) and J_{0}(n) = 0 for n > 1.

It thus remains to verify the identities (1.9). For this we shall use the formalism of the Dirichlet product (cf. [Mc], p. 2). As in [Mc], let ζ_{k}(n) = n^{k}. Then by definition u = (ζ_{1}μ) ∗ μ, σ_{k} = ζ_{0} ∗ ζ_{k} and so the left hand side of the first identity of (1.9) is (ζ_{0} ∗ ζ_{k}) ∗ (ζ_{1}μ ∗ μ) = ζ_{k} ∗ (ζ_{1}μ) = ζ_{1}(ζ_{k−1} ∗ μ) = ζ_{1}J_{k−1}. This proves the first identity. Similarly, the left hand side of the second one is ζ_{1}(ζ_{0} ∗ ζ_{k}) ∗ (ζ_{1}μ) ∗ μ = ζ_{1}(ζ_{0} ∗ζ_{k} ∗μ)∗μ = (ζ_{1}ζ_{k})∗μ = ζ_{k+1} ∗μ = J_{k+1}. This proves (1.9) and hence Theorem 1.2.

For later reference, let us note here that the identity (1.4) can also be written in the following form (1.10) which accentuates its connection to the formula for r(E, E′, N) in [Ka], Theorem 3.1 (cf. also Theorem 2.1 below).

Corollary 1.4 For every n ≥ 2 we have

∑_{{d|n \atop d≠n}} ∑_{{k=1 \atop (k,n/d)=1}}^{n/d} \frac{ψ(dk(n − dk))}{ψ(k(\frac{n}{d} − k))}σ(k(\frac{n}{d} − k), n) = (\frac{5}{12} − \frac{1}{2n}) sl(n). (1.10)

Proof. Clearly, the right hand sides of (1.4) and (1.10) are equal. Moreover, the left hand sides of (1.4) and (1.10) are equal term-by-term because we have the identity

(1.11) σ(m, n) = \frac{ψ(m)}{ψ(m/d)}σ (\frac{m}{d}, n) , if d|(m, n^{r}), for some r ≥ 1,

which follows easily from (1.3). Indeed, if we put \overline{m} = m/d, then its n-component is \overline{m}_{n} = m_{n}/d and so we have m/m_{n} = \overline{m}/\overline{m}_{n}. Thus, by (1.3) we obtain

σ(m, n) = ψ(m_{n})σ(m/m_{n}) = ψ(m_{n})σ(\overline{m}/\overline{m}_{n}) = ψ(m_{n})/ψ(\overline{m}_{n})σ(\overline{m}, n),

from which (1.11) is immediate.

We also note here the following corollary which will be used later in section 3.

Corollary 1.5 If n = p^{r} is a prime power then

s′′(n) := ∑_{{k=1 \atop (k,n)=1}}^{n−1} σ(k)σ(n − k) = α(p)n^{3} + β(p)n (1.12)

where α(p) = \frac{5}{12} \frac{(p−1)(p^{2}−1)}{p(p^{2}+1)} and β(p) = \frac{1}{12} \frac{(p^{2}−1)(p−2)(p−3)}{p(p^{2}+1)}.

Proof. Since p |̸ k(p^{r} − k)/d^{2} where d = (k, p^{r}) = p^{i}, we have that σ(k(p^{r} −k), p^{r}) = σ(k(p^{r} − k)/d^{2}) = σ(k/d)σ(p^{r} − k)/d). Thus, the identity (1.10) reduces in the prime power case to the formula

∑_{d|p^{r}} ψ(d^{2})s′′(p^{r}/d) = \frac{1}{12}(5p^{r} − 6)p^{2r} (1 − \frac{1}{p^{2}} ) ,

from which the identity (1.12) follows by (a somewhat tedious) induction on r.

By combining the results of [Ka] with the above theorem, we can now easily prove Theorem 6 of the introduction.

Proof of Theorem 6. Combining equations (4.1) and (4.2) of the Mass Formula (Theorem 4.1) of [Ka], we obtain

∑_{E′} \frac{r(E, E′, N)}{#Aut(E′)} ≤ \frac{1}{2} ∑_{k=1}^{N−1} σ(k(n − k), N),

with equality holding if and only if char(K) = 0 or if char(K) ≥ N. From this (7) follows immediately in view of the fundamental identity (1.4).

2 Bounding n(E, E′, N) : E ordinary or j(E) ≠ 0

The purpose of this section is to prove the existence of curves of genus 2 with elliptic differentials by establishing the lower bound for the number n(E, E′, N) asserted in Theorem 1 of the introduction. For this we shall use both the mass formula (Theorem 6) established in the previous section, as well as the following special case of Theorem 3.4 of [Ka].

Theorem 2.1 If char(K) |̸ N, then the weighted number of curves of genus 2 of type (E, E′, N) is given by the formula

n(E, E′, N) = sl(N) − \frac{1}{2} ∑_{k=1}^{N−1} w(k, N)h (E, E′, \frac{k(N − k)}{(k, N)^{2}}, N) , (2.1)

where the weighting factor w(k, N) is defined by

w(k, N) = \frac{ψ(k(N − k))}{ψ(k(N − k)/(k, N)^{2}))} = (k, N)^{2} ∏_{{p|(k,N) \atop p |̸ \frac{k(N−k)}{(k,N)^{2}}}} (1 + \frac{1}{p}) , (2.2)

and h(E, E′, m, N) denotes the number of N-primitive homomorphisms h : E → E′ of degree m, which is related to the number h(E, E′, m) of all homomorphisms of degree m by the formula

h(E, E′, m, N) = ∑_{{k|N \atop k^{2}|m}} μ(k)h(E, E′, \frac{m}{k^{2}} ). (2.3)

Proof of Theorem 1. The upper bound in (2) follows directly from the fact that r(E, E′, N) := sl(N) − n(E, E′, N) ≥ 0; cf. Theorem 2.1. For the lower bound we shall establish the equivalent inequality

r(E, E′, N) < \frac{5}{6}sl(N). (2.4)

For this, suppose first that Min(a(E), a(E′)) ≤ 4 where, for brevity, a(E) := #Aut(E). Since r(E, E′, N) is symmetric in E and E′ (as follows easily from the description given in [Ka]), we obtain from the mass formula (7) the estimate

r(E, E′, N) ≤ 4 (\frac{5}{24} − \frac{1}{4N} ) sl(N) < \frac{5}{6}sl(N),

which yields (2.4) in the case that Min(a(E), a(E′)) ≤ 4.

Now suppose that Min(a(E), a(E′)) > 4. Looking at the table of groups of automorphisms of elliptic curves, we see that we then must have j(E) = j(E′) = 0 (cf. Silverman [Si], p. 103). Thus, to finish the proof it remains to consider the case that j(E) = j(E′) = 0 and E ≃ E′ is ordinary. Here we shall prove the following slightly better result:

Proposition 2.2 Let E/K be the elliptic curve with j(E) = 0 (so E is defined by y^{2} = x^{3} − 1) and suppose that E is ordinary (i.e. p := char(K) = 0 or p ≡ 1(3)). Then for all N ≥ 2 prime to p we have

n(E, E, N) ≥ \frac{1}{2}sl(N). (2.5)

Proof. We shall first establish the inequality

h(E, E, m, N) ≤ \frac{3}{2}σ(m, N), if m ≥ 2, N ≥ 1, (2.6)

where h(E, E, m, N) is as in (2.3) and σ(m, N) as in (1.3).

To prove this, recall that End(E) = ℤ[ρ] where ρ = \frac{−1+\sqrt{−3}}{2}. Since this has class number 1 and #(ℤ[ρ]^{×}) = 6, it follows that h(E, E, m) = 6ν(m), where ν(m) denotes the number of ideals of O = ℤ[ρ] of norm m, and hence we obtain

h(E, E, m, N) = 6ν(m, N), (2.7)

where ν(m, N) denotes the number of N-primitive ideals of norm m. From the decomposition of primes in O we obtain for a prime q and integer r ≥ 1:

ν(q^{r}, N) = 0  if q ≡ 2(3) and q|N or r = 1
ν(q^{r}, N) ≤ ν(q^{r}) ≤ 1 if q ≡ 2(3) and q |̸ N,
ν(q^{r}, N) = 2  if q ≡ 1(3) and q|N,
ν(q^{r}, N) = r + 1  if q ≡ 1(3) and q |̸ N,
ν(3^{r}, N) ≤ 1.

Thus, if we put \overline{ν}(m, N) = ν(m, N)/σ(n, N) then we obtain

\overline{ν}(q^{r}, N) = 0 if q ≡ 2(3) and q|N or r = 1,
\overline{ν}(q^{r}, N) ≤ \frac{1}{7} if q ≡ 2(3) and q |̸ N, r ≥ 2,

because \frac{1}{σ(p^{r})} ≤ \frac{1}{p^{2}+p+1} ≤ \frac{1}{7} (for r ≥ 2). On the other hand, if q ≡ 1(3) then q ≥ 7 so \overline{ν}(q^{r}, N) ≤ \frac{r+1}{σ(q^{r})} ≤ \frac{2}{p+1} ≤ \frac{1}{4}; in addition we have \overline{ν}(3^{r}, N) ≤ \frac{1}{σ(3^{r})} ≤ \frac{1}{4}, for r ≥ 1 Gathering these inequalities together, we thus obtain

ν(q^{r}, N) ≤ \frac{1}{4}σ(q^{r}, N) (2.8)

for all primes q and r ≥ 1, and so (2.6) follows by combining (2.7) and (2.8).

We now show how to deduce (2.5) from (2.6). Indeed, if d = (k, N), then from (2.2) and (2.6) we obtain the estimate

w(k, N)h(E, E, k(N − k)/d^{2}, N) ≤ \frac{3}{2} \frac{ψ(k(N − k))}{ψ(k(N − k)/d^{2})}σ (\frac{k(N − k)}{d^{2}}, N) , (2.9)

provided that k ≠ \frac{N}{2} (i.e. \frac{k(N−k)}{d^{2}} ≠ 1). Now by (1.11) the right hand side of (2.9) equals \frac{3}{2}σ(k(N − k), N), and so, if N is odd, then we obtain from (2.1), (2.9) and the fundamental identity (1.4) the inequality

r(E, E, N) ≤ \frac{1}{2} ∑_{k=1}^{N−1} \frac{3}{2}σ(k(N − k), N) = \frac{3}{4} (\frac{5}{12} − \frac{1}{2N} ) sl(N),

which yields the bounds

r(E, E, N) < \frac{5}{16}sl(N) and n(E, E, N) > \frac{11}{16}sl(N) > \frac{1}{2}sl(N), if N is odd.

Suppose next that N is even. Since by definition

w(N/2, N)h(E, E, \frac{(N/2)^{2}}{d^{2}} , N) = ψ((N/2)^{2})a(E) = 6ψ((N/2)^{2}),

and

σ(\frac{N}{2} , N) = ψ((\frac{N}{2} )^{2}) = \frac{N}{2} ψ(\frac{N}{2} ) ≤ \frac{N}{4} ψ(N) = \frac{sl(N)}{4ϕ(N)}, (2.10)

we obtain the estimate

w(N/2, N)h(E, E, (N/2)^{2}/d^{2}, N) ≤ \frac{3}{2}σ(N/2, N) + \frac{9}{2} · \frac{sl(N)}{4ϕ(N)}.

This leads to the bounds

2 · r(E, E, N) ≤ ∑_{k=1}^{N−1} \frac{3}{2}σ(k(N − k), N) + \frac{9sl(N)}{8ϕ(N)} = (\frac{5}{8} + \frac{9}{8ϕ(N)} − \frac{3}{4N} ) sl(N).

Now for N ≠ 2, 6 we have the estimate g(N) := \frac{3}{2ϕ(N)} − \frac{1}{N} ≤ \frac{1}{2} because g(N) ≤ \frac{3}{2·2} − \frac{1}{4} = \frac{1}{2} if 2 < N ≤ 4 and g(N) ≤ \frac{3}{2·4} < \frac{1}{2} if N ≥ 5, N ≠ 6. We thus obtain r(E, E, N) ≤ (\frac{5}{16} + \frac{3}{8} − \frac{1}{2}) sl(N) = \frac{1}{2}sl(N), and so (2.5) follows provided that N ≠ 2, 6. To include these two exceptional cases we note that if 2 |̸ (\frac{N}{2} ), then

the estimate (2.10) can be improved to the equality σ (\frac{N}{2} , N) = ψ ((\frac{N}{2} )^{2}) = \frac{sl(N)}{6ϕ(N)} which leads to the bound

r(E, E, N) ≤ (\frac{5}{16} + \frac{3}{8ϕ(N)} − \frac{3}{8N} ) sl(N) ≤ \frac{1}{2}sl(N),

and so (2.5) holds in all cases.

3 Bounding n(E_{1}, E_{2}, N): supersingular case

We now turn to the case not covered by the Existence Theorem 1, which is the case that E = E′ is supersingular with j-invariant j = 0. The main result here is Theorem 3 of the introduction which gives the precise order of magnitude of n(E, E, N) for any supersingular curve E/K.

This theorem will be proved below in two steps. The first step consists of relating r(E, E, N) to the sum

s_{p}^{∗}(N) = ∑_{k=1}^{N−1} d_{p}(k(N − k), N); (3.1)

the second consists of deriving an asymptotic formula for s_{p}^{∗}(N). Here, d_{p}(m, N) is defined in terms of the function d_{p}(m) by the rule (1.1), and d_{p}(m) is defined as in Hecke [He], p. 817:

d_{p}(m) = ∑_{{t|m \atop p |̸ t}} t = σ(m) − pσ(m/p) = σ(m/m_{p}).

Remark 3.1 Since E is assumed to be supersingular, it follows that

d_{p}(m) = σ(E, m)

is the number of subgroup schemes of E of order m (cf. [Ka], Proposition 4.3a)), and similarly, d_{p}(m, N) is the number σ(E, m, N) of N-primitive subgroup schemes of order m (cf. [Ka], Proposition 4.3b)). Thus, by the Mass Formula (in the version of [Ka], Theorem 4.1) we therefore see that s_{p}^{∗}(N) is twice the left hand side of (7):

∑_{E′} \frac{r(E, E′, N)}{#Aut(E′)} = \frac{1}{2}s_{p}^{∗}(N). (3.2)

We now turn to the first step in the proof of Theorem 3 which consists of establishing the following result.

Proposition 3.2 For p |̸ N we have

r(E, E, N) = \frac{12}{p − 1}s_{p}^{∗}(N) + R_{0}(N), (3.3)

where R_{0}(N) = 0 for p ≤ 13, p ≠ 11, and otherwise satisfies the estimate

|R_{0}(N)| ≤ c(ε)N^{2+ε}, ∀ε > 0. (3.4)

Proof. Since E is supersingular, End(E) is a maximal order of the quaternion algebra ramified at p and ∞, and so it follows that h(n) := h(E, E, N) is a quaternary quadratic form of discriminant p^{2} (cf. Deuring [De]). Thus, by Hecke’s theory (Hecke [He]), the function

H(τ) = ∑_{n≥0} h(n)q^{n} , q = e^{2πiτ},

is a modular form of weight 2 on Γ_{0}(p), i.e. H(τ) ∈ M_{2}(Γ_{0}(p)). We shall compare H(τ) to the modular form

E(p, τ) := \frac{p − 1}{24} + ∑_{n≥1} d_{p}(n)q^{n} ∈ M_{2}(Γ_{0}(p)); (3.5)

note that E(p, τ) = \frac{1}{8π^{2}} E(τ; p) in the notation of Schoeneberg [Sch], p. 177. (Note also the sign error in the q-expansion formula for E(τ; N) on p. 177 of [Sch]; it is stated correctly in [He], Satz 11).

Let F_{1}, ..., F_{g} ∈ S_{2}(Γ_{0}(p)) denote a basis for the cusp forms of weight 2 on Γ_{0}(p), which we can take to be a basis of normalized eigenforms (newforms) under the Hecke algebra. Since dim M_{2}(Γ_{0}(p))/S_{2}(Γ_{0}(p)) = 1 (cf. [Sch], pp. 171-2), it follows by comparing constant coefficients that H(τ) − \frac{24}{p−1}E(p, τ) ∈ S_{2}(Γ_{0}(p)), so

H(τ) = \frac{24}{p − 1}E(p, τ) + ∑_{j=1}^{g} c_{j}F_{j}(τ),

for certain c_{j} ∈ ℂ. We thus have the relations

h(n) = \frac{24}{p − 1}d_{p}(n) + ∑_{i=1}^{g} c_{j}f_{j}(n), ∀n ≥ 1, (3.6)

where F_{j}(τ) = ∑_{n≥1} f_{j}(n)q^{n}; in particular, for n = 1 we obtain the relation

a(E) = \frac{24}{p − 1} + ∑_{j=1}^{g} c_{j}. (3.7)

Thus, if h(m, n) and f_{j}(m, n) are defined by rule (1.1), then (3.6) yields the relations

h(m, n) = \frac{24}{p − 1}d_{p}(m, n) + ∑_{j=1}^{g} c_{j}f_{j}(m, n). (3.8)

We shall now substitute these in the formula (2.1) for r(E, E, N) = sl(N) − n(E, E, N), which we can write in the form

r(E, E, N) = \frac{1}{2} ∑_{t|N} ∑_{{k=1 \atop (k,N)=t}} w(k, N)h (k(N − k)/t^{2}, N) . (3.9)

Now it follows from (1.11), together with the fact that d_{p}(m, n) = σ(m/m_{p}, n) if p |̸ n, that

w(k, N)d_{p} (k(N − k)/t^{2}, N) = d_{p}(k(N − k), N), if p |̸ N. (3.10)

Thus, substituting (3.8) in (3.9) and using (3.10) yields

r(E, E, N) = \frac{12}{p − 1} ∑_{k=1}^{N−1} d_{p}(k(N − k), N) + R_{0}(N) = \frac{12}{p − 1}s_{p}^{∗}(N) + R_{0}(N), (3.11)

where R_{0}(N) = \frac{1}{2} ∑_{j=1}^{g} c_{j}f̃_{j}(N) and

f̃_{j}(N) = ∑_{t|N} ∑_{{k=1 \atop (k,N)=t}}^{N−1} w(k, N)f_{j} (k(N − k)/t^{2}, N) . (3.12)

Of course, if g := dim S_{2}(Γ_{0}(p)) = 0, then trivially R_{0}(N) = 0; this happens for p ≤ 13, p ≠ 11 (cf. [Sch], p. 103).

Since (3.11) is identical to (3.3), the proof of Proposition 3.2 will be complete once we have shown that R_{0}(N) satisfies (3.4). For this, we shall first estimate f̃_{j}(N) by using the Ramanujan-Petersson conjecture (for weight 2) which was proved by Eichler [Ei] (see also Shimura [Sh], Theorem 7.12) and was complemented by Igusa:

|f_{j}(n)| ≤ d(n)n^{1/2} , if p|̸ n. (3.13)

Moreover, since for p^{r}∥n we have f_{j}(n) = f_{j}(p)^{r}f_{j}(n/p^{r}) (cf. Lang [La], p. 110) and since f_{j}(p) = ±1 (cf. [SB], Th. 3 or [Mi], Th. 4.6.17), we see that (3.13) can be improved to |f_{j}(n)| ≤ d(n/p^{r})(n/p^{r})^{1/2}, if p^{r}∥n; (3.14)

in particular, (3.13) is valid for all n ≥ 1.

From (3.13) we obtain the following estimate for f(m, n):

|f(m, n)| ≤ d(m)\frac{ψ((m, n))}{(m, n)}m^{1/2}. (3.15)

Indeed, since both sides of (3.15) are multiplicative in m (for n fixed), it is enough to consider the case that m = q^{r} is a prime power. Then both sides of (3.15) depend (for m fixed) only on q^{s}||n, and so we may assume n = q^{s}. Now if s = 0 or r ≤ 1 then f(q^{r}, q^{s}) = f(q^{r}), in which case (3.15) follows directly from (3.13). Thus, assume s ≥ 1 and r ≥ 2. Then f(q^{r}, q^{s}) = f(q^{r}) − f(q^{r−2}), so |f(q^{r}, q^{s})| ≤ |f(q^{r})|+|f(q^{r−2})| ≤ (r+1)q^{r/2}+(r−1)q^{r/2−1} ≤ (r+1)(1+ \frac{1}{q})q^{r/2} = d(m)\frac{ψ((m,n))}{(m,n)}m^{1/2}, which proves (3.15).

Next we note that with t = (k, N) and M = \frac{k(N−k)}{t^{2}}we have the identity

w(k, N)\frac{ψ((M, N))}{(M, N)} = tψ(t) (3.16)

because by (2.2) the left hand side is t^{2} ∏_{{q|t \atop q |̸ M}}(1 + \frac{1}{q}) ∏_{{q|M \atop q|N}} (1 + \frac{1}{q}) = t^{2} ∏_{q|t}(1 + \frac{1}{q}).

Thus, combining (3.15) and (3.16) yields

|w(k, N)f_{j}(k(N − k)/t^{2}, N)| ≤ w(k, N)d(M)\frac{ψ((M, N))}{(M, N)}M^{1/2} = tψ(t)d(M)M^{1/2},

and so, substituting this in (3.12), we obtain

|f̃_{j}(N)| ≤ ∑_{t|N} tψ(t) ∑_{{k=1 \atop (k,N)=t}}^{N−1} d (\frac{k(N − k)}{t^{2}}) (\frac{k(N − k)}{t^{2}})^{1/2} . (3.17)

Fix ε > 0 and let c_{ε} be such that d(n) ≤ c_{ε}n^{ε}. (3.18) Then from (3.17) we obtain

|f̃_{j}(N)| ≤ c_{ε} ∑_{{t|n \atop t≠N}} tψ(t)ϕ(N/t) (\frac{N}{2t} )^{1+2ε} = \frac{c_{ε}}{2^{1+2ε}} ∑_{{t|n \atop t≠N}} t^{−2ε}ψ(t)ϕ(N/t)N^{1+2ε} (3.19)

because \frac{k(N−k)}{t^{2}} ≤ (N/2t)^{2}, for all k. Finally, using the identity/inequality

∑_{t|N} ψ(t)ϕ (\frac{N}{t} ) = ∏_{q^{r}||N} (2q^{r} + (r − 1)q^{r}(1 − \frac{1}{q^{2}} )) ≤ d(N)N

we obtain, using (3.18) once more, that

|f̃_{j}(N)| ≤ (\frac{1}{2})^{1+2ε} c_{ε}^{2}N^{2+3ε}.

Thus, substituting this estimate in (3.11) yields the desired inequality (3.4) with

c(ε) = (\frac{1}{4})^{1+ε/3} (∑_{j=1}^{g} |c_{j}|) c_{ε/3}^{2}. (3.20)

Remark 3.3 a) In view of proving the existence of curves of type (E, E, N) (with E supersingular), Proposition 3.2 represents the main step, at least if p > 5. Indeed, since s_{p}^{∗}(N) ≤ s^{∗}(N) < \frac{5}{12}sl(N) by Theorem 5, it follows from Proposition (3.2) that

r(E, E, N) < \frac{5}{p − 1}sl(N) + R_{0}(N) < \frac{6}{p − 1}sl(N), if N^{1−ε} ≥ c(ε)\frac{π^{2}}{6} (p − 1), (3.21)

where c(ε) is as in (3.4) and/or (3.20). Indeed, by using (3.4) and the fact (cf. (3.48) below) that sl(N) > N^{3}/ζ(N) = (6/π^{2})N^{3}, we obtain

R_{0}(N) ≤ c(ε)N^{2+ε} < c(ε)(π^{2}/6)N^{ε−1}sl(N)

from which (3.21) follows readily. (See also Lemma 3.10 below.)

b) The above proof of Proposition 3.2 shows that the constant c(ε) appearing in (3.4) and (3.21) has the form

c(ε) = (\frac{1}{4})^{1+ε/3} c(E)c_{ε/3}^{2} (3.22)

where c_{ε} depends only on ε > 0 (and is defined by (3.18)) and c(E) = ∑_{j=1}^{g} |c_{j}| depends only on E. While for each ε > 0 the (best) constant c_{ε} can easily be determined explicitly, viz.

c_{ε} = ∏_{p^{ε}<2} Max (\frac{k(p)}{p^{(k(p)−1)ε}} , \frac{k(p) + 1}{p^{k(p)ε}}) , (3.23)

where k(p) = k(p, ε) = [\frac{1}{log(p^{ε})}], we have less information about c(E). A very crude estimate shows that we can always bound c(E) by c′p^{c′′p}, but it seems likely that much better bounds should be possible, for we have by (3.7)

c(E) ≥ ∑_{j=1}^{g} c_{j} = a(E) − \frac{24}{p − 1}, (3.24)

with equality holding if and only if c_{j} ≥ 0, ∀j. In particular, we have that equality holds in (3.24) if g := g(X_{0}(N)) := dim S_{2}(Γ_{0}(p)) ≤ 1 which, by [Sch] p. 103 (or [SB], Table 5) is true if p ≤ 19.

c) Note that H(τ) − \frac{24}{p−1}E(p, τ) ∈ S_{2}^{−}(Γ_{0}(p)), where S_{2}^{−} denotes the (−1)- eigenspace of the Fricke involution W_{p}. Indeed, since d_{p}(pm) = d_{p}(m) (by definition) and h(pm) = h(m) (because E is supersingular), for all m, we have that (H(τ) − \frac{24}{p−1}E(p, τ))_{|U_{p}} = H(τ) − \frac{24}{p−1}E(p, τ), from which the assertion follows since U_{p} = −W_{p} on S_{2}(Γ_{0}(p)) (cf. Atkin-Lehner [AL], Lemma 17(iii)).

We thus see that c_{j} = 0 if F_{j} ∉ S_{2}^{−}(Γ_{0}(p)). In particular, we see that equality holds in (3.24) as long as dim S_{2}^{−}(Γ_{0}(p)) ≤ 1, which, by Table 5 of the Antwerp Conference (see [SB]), is true if p ≤ 19 or if p = 37.

We now turn to determine the order of magnitude of the function s_{p}^{∗}(n). As a preparatory step, we first prove the following result which is interesting in itself and which generalizes another identity of Glaisher (cf. Remark 3.5 below).

Proposition 3.4 The order of magnitude of the function s_{p}(n) = ∑_{k=1}^{n−1} d_{p}(k)d_{p}(n−k) is given by the formula

s_{p}(n) = aσ_{3}(n) + bσ_{3}(n/p) − cd_{p}(n) + R_{1}(n) (3.25)

in which a = \frac{5}{12} \frac{(p−1)^{2}}{p^{2}+1} , b = p^{2}a and c = \frac{p−1}{12}, and the error term R_{1}(n) = R_{1}(p, n) satisfies the estimate |R_{1}(n)| ≤ c′d(n)n^{3/2} (3.26)

where c′ = c′(p) is a constant. Moreover, if p = 2 or 3 then there is no error term: R_{1}(2, n) = R_{1}(3, n) = 0.

Proof. The q-expansion of the square of the modular form E(p, τ) defined by (3.5) is given by

E(p, τ)^{2} = (\frac{p − 1}{24})^{2} + ∑_{n≥1} a_{n}q^{n} ∈ M_{4}(Γ_{0}(p)),

where a_{n} = ∑_{k=1}^{n−1} d_{p}(k)d_{p}(n − k) + 2 (\frac{p−1}{24}) d_{p}(n) = s_{p}(n) + \frac{p−1}{12}d_{p}(n).

Let F′_{1}, …, F′_{s} be a basis of normalized newforms of S_{4}(Γ_{0}(p)); note that s = \frac{p+1}{4}+\frac{1}{4} (1 + (\frac{−1}{p}))+(\frac{−3}{p}) (cf. Shimura[Sh], p. 25). Since dim M_{4}(Γ_{0}(p))/S_{4}(Γ_{0}(p)) = 2 (cf. [Sh], p. 46), it follows that E_{4}(τ), E_{4}(pτ), F′_{1}, ⋯, F′_{s} is a basis of M_{4}(Γ_{0}(p)) and so there exist a, b, c′_{1}, …, c′_{s} ∈ ℂ such that

E(p, τ)^{2} = aE_{4}(τ) + bE_{4}(pτ) + ∑_{i=1}^{s} c′_{i}F′_{i}(τ). (3.27)

Thus, if we write F′_{i}(τ) = ∑_{n≥1} f′_{i}(n)q^{n} then we have the relations

s_{p}(n) = aσ_{3}(n) + bσ_{3}(n/p) − \frac{p − 1}{12}d_{p}(n) + ∑_{i=1}^{s} c′_{i}f′_{i}(n), ∀n ≥ 1, (3.28)

because E_{4}(τ) = \frac{1}{240} + ∑ σ_{3}(n)q^{n}.

We now determine a and b. For this, we first note the relation

a + b = \frac{5}{12}(p − 1)^{2} (3.29)

which follows from (3.27) by comparing constant coefficients. To obtain another relation, let us evaluate s_{p}(p^{r}). By definition and Corollary (1.5) we have

s_{p}(p^{r}) = ∑_{j=0}^{r−1} ∑_{{k=1 \atop p|̸ k}}^{p^{r−j}−1} d_{p}(k)d_{p}(p^{r−j} − k) = ∑_{j=0}^{r−1} s′′(p^{r−j})

= ∑_{j=0}^{r−1} α(p)p^{3(r−j)} + (−1)^{r−j}β(p)p^{r−j} = α(p)σ_{3}(p^{r}) + O(p^{r}).

On the other hand, since σ_{3}(p^{r−1}) = (σ_{3}(p^{r}) − 1)/p^{3} and since by Miyake [Mi], Theorem 4.6.17, f′_{i}(p^{r}) = f′_{i}(p)^{r} = (±p)^{r}, (3.30)

we obtain from (3.28) that s_{p}(p^{r}) = (a + \frac{b}{p^{3}} ) σ_{3}(p^{r}) + O(p^{r}), and so it follows that

a + \frac{b}{p^{3}} = α(p) = \frac{5}{12} \frac{(p − 1)^{2}(p + 1)}{p(p^{2} + 1)}. (3.31)

Solving the linear equations (3.29) and (3.31) yields

a = \frac{5}{12} \frac{(p − 1)^{2}}{p^{2} + 1} , b = \frac{5}{12}(p − 1)^{2} \frac{p^{2}}{p^{2} + 1} = p^{2}a

and so (3.25) holds with

R_{1}(n) = ∑_{i=1}^{s} c′_{i}f′_{i}(n).

Moreover, by the Ramanujan-Petersson conjecture (which was proved by Deligne; cf. [Mi], p. 150) we have

|f′_{i}(n)| ≤ d(n)n^{3/2} , if p |̸ n.

Combining this with (3.30) yields (by multiplicativity) the estimate

|f′_{i}(n)| ≤ d(m)p^{r}m^{3/2} ≤ d(n)^{3/2} , if n = p^{r}m , p |̸ m

from which (4.5.2) follows with c′ = ∑_{i=1}^{s} |c′_{i}|.

Finally, we note that if p = 2 or 3 then s = 0, so R_{1}(p, n) = 0 in this case.

Remark 3.5 a) For p = 2 or 3 the above Proposition 3.4 reduces to the identity

∑_{k=1}^{n−1} d_{p}(k)d_{p}(n − k) = \frac{p − 1}{12} [σ_{3}(n) + (5p − 6)σ_{3} (\frac{n}{p}) − d_{p}(n)] , (3.32)

which generalizes another identity of Glaisher (1883) (cf. Dickson [Di], p. 294): to be precise, Glaisher’s identity is the case p = 2, n ≡ 1(2) of (3.32).

Moreover, for p = 5 we have the identity

∑_{k=1}^{n−1} d_{p}(k)d_{p}(n − k) = \frac{10}{39}σ_{3}(n) + \frac{250}{39}σ_{3} (\frac{n}{p}) − \frac{1}{3}d_{p}(n) + \frac{1}{13}a(n), (3.33)

where the numbers a(n) are given by the relation

∑_{n≥1} a(n)t^{n} = t ∏_{n≥1}(1 − t^{n})^{4}(1 − t^{5n})^{4}.

(The identity (3.33) follows from the above formula (3.28) (and (3.34) below) by noting that (cf. Shimura [Sh], Ex. 2.28, p. 49)

f(τ) = ∑_{m≥1} a(m)q^{m} = η(τ)^{4}η(5τ)^{4} ∈ S_{4}(Γ_{0}(5)),

and hence f(τ) is the unique normalized newform of S_{4}(Γ_{0}(5)).)

b) As in Remark 3.3 we can bound the constant c′ exponentially by a function of p. Contrary to the constant c(E), however, the constant c′ must grow with p, for we have the lower bound

c′ ≥ ∑_{i=1}^{s} c′_{i} = c − a = \frac{(p − 1)(p − 2)(p − 3)}{12(p^{2} + 1)}, (3.34)

where the indicated equality follows by taking n = 1 in (3.28). Note that c′ = c − a if c′_{i} ≥ 0 for all 1 ≤ i ≤ s, and that this is the case for s ≤ 1 (i.e. for p ≤ 7; cf. Table A in Miyake [Mi]).

Note also that by an argument similar to that of Remark 3.3c) we have that

c′_{i} = 0 if F′_{i}(τ) ∉ S_{4}^{−}(Γ_{0}(p)),

so the equality c′ = c − a holds as long as dim S_{4}^{−}(Γ_{0}(p)) ≤ 1.

We are now ready to prove the desired order of magnitude of the function s_{p}^{∗}(n).

Proposition 3.6 If n = p^{r}m, where p |̸ m, then we have

s_{p}^{∗}(n) = \frac{5}{12} \frac{(p − 1)^{2}}{p^{2} + 1} [σ_{3}(p^{r}) + p^{2}σ_{3}(p^{r−1})] sl(m) − \frac{p − 1}{12}σ(p^{r})δ(m) + R_{2}(n) (3.35)

where δ(1) = 1 and δ(m) = 0 for m > 1, and the error term R_{2}(n) satisfies the estimate R_{2}(n) = O(n^{3/2+ε}). (3.36)

Moreover, if p = 2 or 3 then R_{2}(n) = 0.

Proof. Let s′_{p}(n) = ∑_{k=1}^{n−1} d_{p}(k(n − k)). Then as in the proof of Theorem 1 we have

s_{p}^{∗}(n) = ∑_{d|n} μ(d)s′_{p}(n/d) and s′_{p}(n) = ∑_{d|n} μ(d)ds_{p}(n/d), (3.37)

from which we obtain the formula

s_{p}^{∗}(n) = ∑_{d|n} u_{p}(d)s_{p}(n/d), where u_{p}(n) = ∑_{{d|n \atop p |̸ d}} μ(d)dμ(n/d). (3.38)

Thus, substituting (3.25) in (3.38) yields

s_{p}^{∗}(n) = a ∑_{d|n} u_{p}(d) [σ_{3}(n/d) + p^{2}σ_{3} (\frac{n}{pd}) − cd_{p}(n/d)] + R_{2}(n), (3.39)

where a and c are as in (3.25) and

R_{2}(n) = ∑_{d|n} u_{p}(d)R_{1}(n/d). (3.40)

Write n = p^{r}m with p |̸ m. Then from (1.9) we deduce the identity

∑_{d|n} u_{p}(d)σ_{k}(n/d) = σ_{k}(p^{r})mJ_{k−1}(m),

and so (3.39) reduces to (3.35) since mJ_{2}(m) = sl(m) and J_{0}(m) = δ(m). It remains to establish the estimate (3.36). From (3.40) and (3.26) we obtain

|R_{2}(n)| ≤ ∑_{t|n} |u_{p}(t)||R_{1}(n/t)| ≤ c′ ∑_{t|n} |u_{p}(t)|d(n/t)(n/t)^{3/2} =: c′R′_{2}(n),

and so (3.36) follows once we have shown that for each ε > 0 there is a constant c′′(ε) such that R′_{2}(n) ≤ c′′(ε)n^{3/2+ε}. (3.41)

For this we first note that

R′_{2}(n) ≤ d(n)n^{3/2} ∏_{{q|n \atop q≠p}} (1 + \frac{1}{q^{3/2}} ) (1 + \frac{1}{q^{1/2}} ) (3.42)

which follows easily from the fact that R′_{2}(n) is multiplicative; note that |u_{p}(q)| = q +1, |u_{p}(q^{2})| = q and u_{p}(q^{r}) = 0 for r ≥ 3. Now by a variant of the usual argument for estimating d(n) we obtain

d(n) ∏_{q|n} (1 + \frac{1}{q^{1/2}} ) ≤ c′(ε)n^{ε} (3.43)

with c′(ε) = ∏_{q^{ε}<3} \frac{2}{ε log q} (1 + \frac{1}{\sqrt{q}} ). On the other hand,

∏_{q|n} (1 + \frac{1}{q^{3/2}} ) < ∏_{q} (1 + \frac{1}{q^{3/2}} ) = ∑_{n=1}^{∞} \frac{|μ(n)|}{n^{3/2}} = \frac{ζ(3/2)}{ζ(3)}

(cf. [Mc], p. 227) is bounded, and so (3.42) holds with

c′′(ε) = c′(ε)ζ(3/2)ζ(3)^{−1} (1 + \frac{1}{\sqrt{p}})^{−1} . (3.44)

We thus obtain

|R_{2}(n)| ≤ c′c′′(ε)n^{3/2+ε},

which proves (3.36). Finally we note that if p = 2 or 3 then R_{1}(n) = 0 for all n and hence by (3.40) we also have that R_{2}(n) = 0 in this case.

Proof of Theorem 3. If p |̸ N then by Propositions (3.2) and (3.6) we obtain

r(E_{1}E, N) = \frac{12}{p − 1}s_{p}^{∗}(N) + R_{0}(N)
= 5\frac{p − 1}{p^{2} + 1}sl(N) + \frac{12}{p − 1}R_{2}(N) + R_{0}(N),

and so

n(E, E, N) = (1 − 5\frac{p − 1}{p^{2} + 1}) sl(N) − R(N). (3.45)

where R(N) = R_{0}(N) + \frac{12}{p−1}R_{2}(N). This proves (3), and the estimate (4) follows from the estimates (3.4) and (3.36). Finally, if p = 2 or 3 then R_{0}(N) = R_{2}(N) = 0 and so also R(N) = 0.

Remark 3.7 The fact that for a supersingular curve E we have

n(E, E, N) = 0 if p = 2 or 3 and p |̸ N

also follows from the results of [IKO], as F. Oort has kindly pointed out to the author. (This observation may be viewed as an impotant check on the validity of the somewhat complicated formula (2.1) for n(E, E′, N).)

Even though the constant N_{0}(p) of Theorem 3 may be explicitly calculated for each given p (cf. Remarks 3.3 and 3.5b), combined with the fact that it possible to compute a basis of newforms for S_{k}(Γ_{0}(p)), k = 2, 4), this does not lead directly to any practical bounds which are of use for verifying Theorem 4. Nevertheless, by refining the method for p = 5, 11 and 17, we can determine a sufficiently samll upper bound for N_{0}(p) such that the remaining cases N ≤ N_{0}(p) can be checked by hand (or by a small computer).

Proposition 3.8 If E is a supersingular curve in characteristic p = 5, then for every ε > 0 we have

n(E, E, N) > \frac{75}{52π^{2}} (N^{3} − \frac{4}{5}c′(ε)N^{3/2+ε}) , if p |̸ N, (3.46)

where c′(ε) is as in (3.43). Moreover, n(E, E, N) > 0 for all N ≥ 2 with p |̸ N.

Proof. Since R_{0}(N) = 0 by Proposition 3.2, we see that (3.45) reduces to

n(E, E, N) = \frac{3}{13}sl(N) − 3R_{2}(N). (3.47)

Now on the one hand we have for 5 |̸ N the lower bound

\frac{sl(N)}{N^{3}} = ∏_{p|N} (1 − \frac{1}{p^{2}} ) > ∏_{p≠5} (1 − \frac{1}{p^{2}} ) = (1 − \frac{1}{5^{2}} )^{−1} ζ(2)^{−1} = \frac{25}{4π^{2}} . (3.48)

On the other hand we have the upper bound R_{2}(N) ≤ c′c′′(ε)N^{3/2+ε} (cf. (4.7.13)). Here, the constant c′ of Proposition 3.4 is (by Remark 3.5b)) c′ = \frac{(p−1)(p−2)(p−3)}{12(p^{2}+1)} = \frac{1}{13}.

Moreover, since ζ (\frac{3}{2}) ζ(3)^{−1} (1 + \frac{1}{\sqrt{5}})^{−1} = 1.50168 … < 1.5198 … = \frac{15}{π^{2}} , we see from (3.44) that

c′′(ε) < \frac{15}{π^{2}}c′(ε).

Substituting these bounds into (3.47) yields the lower bound (3.46).

Let us now take ε = 1 in (3.46). Then by definition (cf. (3.43))

c′(ε) = \frac{2}{log(2)} (1 + \frac{1}{\sqrt{2}}) \frac{2}{log(3)} (1 + \frac{1}{\sqrt{3}}) = 14.14421 …,

and so n(E, E, N) > 0 if N^{1/2} > \frac{4}{5}c′(1) = 11.3153 …, i.e. if N ≥ 129. The remaining cases (N ≤ 128) are easily checked by hand (or by computer), using the formula

n(E, E, N) = sl(N) − 3s_{5}^{∗}(N)

which follows from (3.3). In particular, we have the following (partial) table for n = n(E, E, N):

\begin{matrix}N & 2 & 3 & 4 & 6 & 7 & 8 & 9 & 11 & 12 & 13 & 14 & … & 126 & 127 & 128 & 129 \\ n & 3 & 6 & 6 & 30 & 78 & 96 & 156 & 300 & 276 & 516 & 462 & … & 301506 & 472278 & 362496 & 440292 \end{matrix}

Proposition 3.9 If p = 11 or 17 and j(E) = 0, then

n(E, E, N) > (\frac{p − 6}{p − 1} − \frac{37}{2} \frac{p − 5}{p − 1}N^{−1/3}) sl(N), if p |̸ N. (3.49)

Moreover, n(E, E, N) > 0 for all N ≥ 2 with p |̸ N.

The basic method for proving this proposition is that outlined in Remark 3.3a). However, in order to obtain sharper estimates, we refine the argument as follows.

Lemma 3.10 Let ε, ε′ > 0 and suppose c_{1} = c_{1}(ε) and c_{2} = c_{2}(ε, ε′) are such that for all n ≥ 1 we have

∑_{{k=1 \atop (k,n)=1}}^{n} d(k(n − k)) ≤ c_{1}ϕ(n)n^{ε} and ∑_{t|n} \frac{ψ(t)}{t^{ε}}ϕ(\frac{n}{t}) ≤ c_{2}sl(n)n^{ε′−2}. (3.50)

Then, in the situation of Proposition 3.2, we have with ε′′ := ε + ε′ − 1 that

n(E, E, N) > (\frac{p − 6}{p − 1} − \frac{1}{4}c_{1}c_{2}c(E)N^{ε′′}) sl(N), (3.51)

where c(E) is as in Remark 3.3b). Moreover, if p = 11 or 17 and j(E) = 0 and ε′′ > 0, then

n(E, E, N) > 0, if N ≥ (\frac{3(p − 5)}{2(p − 6)}c_{1}c_{2})^{1/ε′′} . (3.52)

Furthermore, the inequalities (3.50) are valid with

c_{1} = \frac{c_{ε/2}}{2^{ε}} and c_{2} = ∏_{p<M} max(1, f_{p}(r_{p}), f_{p}(r_{p} + 1)), (3.53)

where c_{ε} is as defined in (3.18), M = max(2^{1/ε}, (1 + ε/ε′)^{1/ε}, (7/4)^{1/ε′}) and

f_{p}(r) = (\frac{1}{p^{ε′r}} ) (\frac{p}{p + 1} + \frac{1}{p^{εr}(p − 1)} + \frac{1 − p^{−εr}}{p^{ε} − 1} ) ,
r_{p} = max (1, [log ((1 + \frac{ε}{ε′} ) \frac{(p − p^{ε})(p + 1)}{(p^{ε+1} + 1)(p − 1)}) / log (p^{ε})]) .

Proof. Since n(E, E, N) = sl(N) − \frac{12}{p−1}s_{p}^{∗}(N) − R_{0}(N) by (3.11), and since s_{p}^{∗}(N) < \frac{5}{12}sl(N) (cf. Remark 3.3a)), it is enough to show that |R_{0}(N)| ≤ \frac{1}{4}c_{1}c_{2}c(E)N^{ε′′}sl(N). But since |R_{0}(N)| ≤ \frac{1}{2}c(E) max_{j} |f̃_{j}(N)|, this follows immediately from the following inequality which is derived in the same way as (3.19):

|f̃_{j}(N)| ≤ \frac{1}{2}c_{1}c_{2}N^{ε+ε′−1}sl(N), for 1 ≤ j ≤ g.

Now suppose that p = 11 or 17 and that j(E) = 0. Then (cf. Remark 3.3b)) we have

c(E) = a(E) − \frac{24}{p − 1} = 6 − \frac{24}{p − 1} = 6 \frac{p − 5}{p − 1}, (3.54)

and so we see that the factor in front of sl(n) in (3.51) is non-negative if and only if N^{ε′′} ≥ (\frac{3(p−5)}{2(p−6)}c_{1}c_{2}), which proves (3.52).

Next we observe that from (3.9) we have

∑_{{k=1 \atop (k,n)=1}}^{n} d(k(n−k)) ≤ ∑_{{k=1 \atop (k,n)=1}}^{n} c_{ε/2}(k(n−k))^{ε/2} ≤ ∑_{{k=1 \atop (k,n)=1}}^{n} c_{ε/2}(n^{2}/4)^{ε/2} = \frac{c_{ε/2}}{2^{ε}} ϕ(n)n^{ε},

which shows that (the first equality of ) (3.50) holds with c_{1} = c_{ε/2}/2^{ε}. Finally, to verify that (3.50) holds with c_{2} as in (3.53), we first note the formula

∑_{t|n} \frac{ψ(t)}{t^{ε}}ϕ(\frac{n}{t}) = ( ∏_{p^{r}||n} f_{p}(r)) sl(n)n^{ε′−2},

which is easily checked by observing that both sides are multiplicative functions.

Viewing f_{p}(r) as a function of a real variable r, we see by computing its derivative that f_{p}(r) assumes its maximum value at

r̃_{p} = log ((1 + \frac{ε}{ε′} ) \frac{(p − p^{ε})(p + 1)}{(p^{ε+1} + 1)(p − 1)}) / log (p^{ε}) ,

and so it follows that

∏_{p^{r}||n} f_{p}(r) ≤ ∏_{p} max(1, f_{p}(r_{p}), f_{p}(r_{p} + 1))

where, as above, r_{p} = max(1, [r̃_{p}]). Thus, to finish the proof, it remains to show that max(1, f_{p}(r_{p}), f_{p}(r_{p} + 1)) = 1 if p ≥ M. Indeed, if p ≥ M, then in particular p^{ε} ≥ (1 + ε/ε′), so log((1 + ε/ε′)((p − p^{ε})/(p − 1))((p + 1)/(p^{1+ε} + 1))) ≤ log(p^{ε}), which means that r̃_{p} ≤ 1 and r_{p} = 1. Thus, since also p^{ε} ≥ 2 (so p ≥ 3) and p^{ε′} ≥ \frac{7}{4}, we obtain

f_{p}(r_{p} + 1) ≤ f_{p}(r_{p}) = (\frac{p}{p + 1} + \frac{1}{p^{ε}(p − 1)} + \frac{1}{p^{ε}} ) (\frac{1}{p^{ε′}} ) ≤ (1 + \frac{1}{4} + \frac{1}{2}) (\frac{4}{7}) = 1,

which proves that max(1, f_{p}(r_{p}), f_{p}(r_{p} + 1)) = 1, as desired.

Proof of Proposition 3.9. We shall apply Lemma 3.10 with ε = 5/9 and ε′ = 1/9. Then, using formula (3.23) for c_{5/18}, we obtain

c_{1} = \frac{c_{5/18}}{2^{5/9}} = \frac{1}{2^{5/9}} (\frac{2}{77} (2^{16}3^{8}5^{13}7^{13}11^{13})^{\frac{1}{18}} ) ≐ 3.928381768 …,
c_{2} = f_{2}(2)f_{3}(1)f_{5}(1)f_{7}(1) ≐ 3.10599597 …

Thus c_{1}c_{2} = 12.2015 … < 12\frac{1}{3}, and so the estimate (3.49) follows from (3.51). (Recall that c(E) = 6(p − 5)/(p − 1); cf. (3.54).)

Moreover, using (3.52) we therefore have that

n(E, E, N) > 0, if N ≥ (\frac{3(p − 6)}{2(p − 5)}c_{1}c_{2})^{1/3} ≐ \{ \begin{matrix}10594.0 & if p = 11 \\ 7959.5 & if p = 17 \end{matrix} .

To deal with the remaining values (N < 10594), we proceed as follows. Although it might be possible to calculate n(E, E, N) in this range explicitly, it is more expedient to improve the above bounds. While the bound on c_{2} is sharp (it is assumed for n = 2^{2} · 3 · 5 · 7 = 420), the bound for c_{1} is quite far from the truth, for numerical calculations show that

t(n) := ∑_{{k=1 \atop (k,n)=1}}^{n} d(k(n − k)) ≤ c_{1}^{∗}ϕ(n)n^{\frac{5}{9}} , if 1 ≤ n ≤ 11000,

where c_{1}^{∗} = t(17)/(ϕ(17)17^{5/9}) = \frac{67}{136}17^{4/9} ≐ 1.735409945 . . .. (In fact, it is likely that this estimate holds for all n because t(n) grows much more slowly than ϕ(n)n^{5/9}.)

Thus, if we use c_{1}^{∗} in place of c_{1}, then we obtain (for N ≤ 11000) that

n(E, E, N) > 0 if N ≥ (\frac{3(p − 6)}{2(p − 5)}c_{1}^{∗}c_{2})^{1/3} ≐ \{ \begin{matrix}913.3 & if p = 11 \\ 686.2 & if p = 17 \end{matrix} .

For remaining values (i.e. N ≤ 1000) we compute n(E, E, N) directly by using a (small) computer. To do this, we shall use the formula (cf. (3.11) and (3.7))

n(E, E, N) = sl(N) − \frac{12}{p − 1}s_{p}^{∗}(N) − 3\frac{p − 5}{p − 1}f̃(N),

where f̃ = f̃_{1} is defined as in (3.12) using the unique normalized cusp form F (τ) = F_{1}(τ) = ∑_{n≥1} f(n)q^{n} of weight 2 on Γ_{0}(p). For p = 11, the coefficients of F (τ) are given by the product expansion

∑_{n=1}^{∞} f(n)q^{n} = q ∏_{n=1}^{∞}(1 − q^{n})^{2}(1 − q^{11n})^{2},

but for p = 17 it is not so easy to generate them. However, for both p = 11 and p = 17 the coefficients f(ℓ) of F for primes ℓ < 1000 are given in Table B of Miyake [Mi], pp. 302–303, from which all the coefficients f(n) with n ≤ 1000 are readily calculated. (Note, however, that in both tables the coefficient f(p) has the wrong sign: it should be f(p) = 1 in both cases, for otherwise F (τ) ∈ S_{2}^{+}(Γ_{0}(p)); cf. Remark 3.3c) and [SB], Theorem 3.) Checking these values for N ≤ 1000, we have that n(E, E, N) > 0 in all cases; this is partly justified by the following tables, in which n_{p}(N) = n(E, E, N) for E/K with j(E) = 0 and char(K) = p:

\begin{matrix}N & n_{11}(N) & n_{17}(N) & N & n_{11}(N) & n_{17}(N) & N & n_{11}(N) & n_{17}(N) \\ 2 & 3 & 3 & 3 & 24 & 24 & 4 & 24 & 24 \\ 5 & 60 & 96 & 6 & 90 & 108 & 7 & 204 & 240 \\ 8 & 210 & 210 & 9 & 432 & 522 & 10 & 306 & 504 \\ 11 & – & 948 & 12 & 660 & 828 & 13 & 1308 & 1578 \\ 14 & 1110 & 1260 & 15 & 1812 & 2196 & 16 & 1788 & 1914 \\ 17 & 2916 & – & 18 & 2292 & 2766 & 19 & 3924 & 4896 \\ 20 & 3036 & 4044 & 21 & 5256 & 6006 & 22 & – & 5472 \\ 23 & 7272 & 8844 & 24 & 5532 & 6468 & 25 & 8364 & 10992 \\ 26 & 7542 & 9246 & 27 & 10920 & 13140 & 28 & 9144 & 10584 \\ 29 & 14448 & 17844 & 30 & 9996 & 12660 & 31 & 17544 & 21468 \\ 32 & 14364 & 16842 & 33 & – & 23394 & 34 & 16716 & – \\ 35 & 23184 & 29268 & 36 & 18372 & 22452 & 7 & 29892 & 36576 \\ 38 & 23052 & 28824 & 39 & 32712 & 38844 & 40 & 26556 & 32436 \\ 41 & 40524 & 49992 & 42 & 28932 & 34626 & 43 & 46968 & 57516 \\ 44 & – & 44292 & 45 & 46092 & 57984 & 46 & 42270 & 51120 \\ 47 & 61560 & 75180 & 48 & 44304 & 52692 & 49 & 68688 & 82986 \end{matrix}

\begin{matrix}N & n_{11}(N) & n_{17}(N) & N & n_{11}(N) & n_{17}(N) \\ 100 & 414828 & 517248 & 200 & 3372516 & 4147248 \\ 300 & 10174812 & 12532368 & 400 & 27116592 & 33289392 \\ 500 & 52864272 & 65052708 & 600 & 81576552 & 100056480 \\ 700 & 142217880 & 174867600 & 800 & 217196520 & 266662704 \\ 900 & 274994532 & 338067840 & 1000 & 424297008 & 520689948 \end{matrix}

Proof of Theorem 4. By Theorem 1 we only have to check the case that E = E′ is supersingular with j(E) = 0. Since this means in particular that p ≡ 2(mod 3), we see that we only have to consider the cases that p = 5, 11 or 17 and so the theorem follows from Propositions 3.8 and 3.9.

References

[AL] A. Atkin, J. Lehner: Hecke Operators on Γ_{0}(m). Math. Ann. 185 (1970), 134–160.

[De] M. Deuring: Die Anzahl der Typen von Maximalordnungen einer definiten Quaternionenalgebra mit primer Grundzahl. Jahresber. deut. Math.-Ver. 54(1950), 24–41.

[Di] L. E. Dickson: History of the Theory of Numbers, vol. I, Carnegie Institute, Washington, 1919 (Chelsea Reprint, 1971).

[Ei] M. Eichler: Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion. Arch. Math. 5(1954), 355–366.

[Fr] G. Frey: On elliptic curves with isomorphic torsion structures and corresponding curves of genus 2; in: “Conference on Elliptic Curves and Modular Forms”, Hong Kong, December 18-21, 1993 International Press, 1995, pp. 79-98.

[FK] G. Frey, E. Kani: Curves of genus 2 covering elliptic curves and an arithmetical application; in: “Arithmetic Algebraic Geometry” (G. van der Geer, F. Oort, J. Steenbrink, eds.) Progress in Math. 89, Birkhäuser, Boston, 1991, pp. 153-176.

[Gl] J. W. L. Glaisher: On the square of the series in which the coefficients are the sums of the divisors of the exponents. Messenger of Math. 14 (1884/5), 156–163.

[He] E. Hecke: Analytische Arithmetik der positiven quadratischen Formen. Kgl. Danske Viden. Selskab. Math.-fys. Med. XIII 12 (1940) = Mathematische Werke, Vandenhoeck & Ruprecht, Göttingen, 1993, pp. 789–918.

[IKO] T. Ibukiyama, T. Katsura, F. Oort: Supersingular curves of genus two and class numbers. Comp. Math. 57 (1986), 127–152.

[Ka] E. Kani: The number of curves of genus two with elliptic differentials. Preprint.

[Ki] I. Kiming: On certain problems in the analytical arithmetic of quadratic forms arising from the theory of curves of genus 2 with elliptic differentials. Manuscripta Math. 87 (1995), 101–129.

[La] S. Lang: Introduction to Modular Forms. Springer-Verlag, New York, 1976.

[Mc] P. J. McCarthy: Introduction to Arithmetical Functions. Universitext Springer Verlag, New York, 1986.

[Mi] T. Miyake: Modular Forms. Springer-Verlag, Berlin, 1986.

[Ra] S. Ramanujan: Collected Papers. Cambridge University Press, Cambridge, 1927 (Chelsea reprint, 1962).

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

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

[Si] J. Silverman: The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.

[SB] H. P. F. Swinnerton-Dyer, B. J. Birch: Elliptic Curves and Modular Functions; in: “Modular Functions of One Variable IV” (B. J. Birch, W. Kuyk, eds.) Springer Lecture Notes 476 (1975), pp. 2-32.