Contemporary Mathematics Volume 00, 0000

The Phragmén-Lindelöf Theorem and Modular Elliptic Curves

LIEM MAI^{1} AND M. RAM MURTY^{2}

1. Introduction

In 1957, Rademacher [4,5] wrote two papers on the Phragmén-Lindelöf theorem. The significance of these papers is that they provide (to date) precise inequalities satisfied by a general L-function inside the critical strip. For the Riemann zeta function or for the classical Dirichlet L-functions, special methods of analytic number theory give better estimates than the convexity argument supplied by the Phragmén-Lindelöf theorem. However, in the absence of such special methods, as in the case of certain Artin L-functions or L-functions of elliptic curves, Rademacher’s work gives sufficiently accurate estimates useful in many applications (see for example [3]).

Ever since G. Frey [1] had the brilliant idea of linking Fermat’s “Last Theorem” with the Taniyama conjecture (that every elliptic curve over ℚ is modular), a variety of conjectures have arisen in the theory of elliptic curves. Two of these conjectures are the height and degree conjectures (see section 3 for definitions). The purpose of this paper is to show that for modular elliptic curves, these two conjectures are equivalent. Since Frey had shown the equivalence of the height conjecture and the famous ABC conjecture, our result establishes the equivalence of the degree and ABC conjectures, modulo the Taniyama conjecture. Our main tool will be Rademacher’s version of the Phragmén-Lindelöf theorem.

After stating the form of Rademacher’s theorem in a slightly general context, we review the relevant facts and conjectures in the theory of elliptic curves. Our main theorems appear in section 4.

1991 Mathematics Subject Classification. 11G05, 14H52, 30C80.

1 Research partially supported by an NSERC Postdoctoral Fellowship.

2 Research partially supported by NSERC and FCAR grants.

This paper is in final form and no version of it will be submitted for publication elsewhere.

Acknowledgement. It is a pleasure to thank G. Frey without whose encouragement this paper would not have been written.

2. Rademacher’s Theorem

Let

L(s) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}}

be a Dirichlet series absolutely convergent for ℜ(s) > 1 and admitting (i) an Euler product : for ℜ(s) > 1

L(s) = ∏_{p} L_{p}(s)

where L_{p}(s) is the inverse of a polynomial of degree d_{p} in p^{−s};

(ii) a positive integer d such that d_{p} ≤ d for all p and equality holds for all but finitely many p;

(iii) an analytic continuation to an entire function and satisfying a functional equation of the form :

Λ(s) = wΛ^{∗}(1 − s)

where w ∈ ℂ, |w| = 1 and

Λ(s) = A^{s/2} ∏_{i=1}^{m} Γ(α_{i}s + r_{i})L(s)

and

Λ^{∗}(s) = A^{s/2} ∏_{i=1}^{m} Γ(α_{i}s + r_{i})L^{∗}(s)

with

\begin{aligned}L_{p}(s) & = ∏_{i=1}^{d_{p}}(1 − α_{i}p^{−s}), |a_{i}| = 1 \\ L_{p}^{∗}(s) & = ∏_{i=1}^{d_{p}}(1 − \overline{α_{i}}p^{−s}) \end{aligned}

and we have set

L^{∗}(s) = ∏_{p} L_{p}^{∗}(s).

Here A > 0, α_{i}, r_{i} ∈ ℝ and m is a natural integer.

Then Rademacher’s version of the Phragmén-Lindelöf theorem immediately gives :

THE PHRAGMÉN-LINDELÖF THEOREM AND MODULAR ELLIPTIC CURVES 3

Proposition. For 0 ≤ σ ≤ 1,

|L(σ + it)| ≪ (A(|t| + 2)^{d})^{(1−σ)/2}(log(A(|t| + 2)^{d}))^{d}

for any L(s) satisfying (i),(ii) and (iii) above. The implied constants depend only on the r_{i}’s.

If f(z) is a normalized new form of weight 2 on Γ_{0}(N), let f(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz}

be its Fourier expansion at i∞. Writing a_{p}/\sqrt{p} = α_{p} + \overline{α_{p}} with |α_{p}| = 1, the symmetric square L-function associated to f is defined by

L_{p}(s, Sym^{2}(f)) = (1 − \frac{α_{p}^{2}}{p^{s}} )^{−1}(1 − \frac{α_{p}\overline{α_{p}}}{p^{s}} )^{−1}(1 − \frac{\overline{α_{p}}^{2}}{p^{s}} )^{−1}

for p |̸N and for p|N, it can be defined appropriately (see [9]) and we set

L(s, Sym^{2}(f)) = ∏_{p} L_{p}(s),

which converges for ℜ(s) > 1. By Shimura [9], we know that L(s, Sym^{2}(f)) extends to an entire function and satisfies a functional equation of the form :

\begin{aligned}Λ(s) & = A^{s}Γ(\frac{s + k − 1}{2})Γ(\frac{s + k}{2} )Γ(\frac{s + 1}{2})L(s, Sym^{2}(f)) \\ & = wΛ(1 − s) \end{aligned}

where |w| = 1, and log A = O(log N).

Hence, by Rademacher’s version of the Phragmén-Lindelöf, L(1, Sym^{2}(f)) = O((log N)^{3}).

On the other hand, the Rankin-Selberg method shows that

L(1, Sym^{2}(f)) = 288π^{3} ∏_{p^{2}|N}(1 − \frac{1}{p^{2}} )\frac{⟨f, f⟩}{N}

where ⟨, ⟩ denotes the Petersson inner product. This yields

Proposition. log⟨f, f⟩ = O(log N)

3. The height and degree conjectures

Let E be an elliptic curve defined over ℚ given by a Weierstrass equation :

(*)  y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}

with a_{i} ∈ ℤ. Define

\begin{aligned}c_{4} & = (a_{1}^{2} + 4a_{2})^{2} − 24(a_{1}a_{3} + 2a_{4}) \\ c_{6} & = −(a_{1}^{2} + 4a_{2})^{2} + 36(a_{1}^{2} + 4a_{2})(a_{1}a_{3} + 2a_{4}) − 216(a_{3}^{2} + 4a_{6}) \end{aligned}

and

Δ = \frac{c_{4}^{3} − c_{6}^{2}}{1728} .

(*) is called a minimal model of E if Δ has minimal absolute value compared with all discriminants of Weierstrass equations describing E. A minimal model is unique up to a change of coordinates of the form :

\begin{aligned}x & = u^{2}x′ + r \\ y & = u^{3}y′ + u^{2}sx′ + t \end{aligned}

with u, r, s, t ∈ ℤ, u ≠ 0. Consequently, the discriminant of any minimal model is the same. We will denote by Δ_{E} the discriminant of any minimal model. One knows that Δ_{E} is an integer. Let N_{E} be the conductor of E. (See Silverman [10] for a precise definition). One knows that N_{E} divides Δ_{E} and has the same prime divisors as Δ_{E}. For p ≠ 2, 3, we have v_{p}(N_{E}) ≤ 2 (where v_{p}(n) is defined such that p^{−v_{p}(n)}n is coprime to p).

Let N be a positive integer and let X_{0}(N) be a modular curve of level N. An elliptic curve E over ℚ is called modular if there is a non-constant ℚ-rational morphism

ϕ : X_{0}(N) ⟶ E.

A famous conjecture of Taniyama (see [10]) predicts that every elliptic curve over ℚ is modular. Ribet [7] has show that Taniyama’s conjecture implies Fermat’s “Last Theorem”.

Henceforth, E shall be a modular elliptic curve. Let

ω_{E} = \frac{dx}{2y + a_{1}x + a_{3}}

be the Néron differential of E and

ω_{E}^{∗} = ϕ^{∗}(ω_{E})

be its pull-back to X_{0}(N_{E}).

Then ω_{E}^{∗} is a holomorphic differential of X_{0}(N_{E}) and so by the theory of modular forms,

ω_{E}^{∗} = f̃_{E}(z)dz

where f̃_{E}(z) is a cusp form of weight 2 on Γ_{0}(N_{E}). Moreover, by a theorem of Shimura [9] one knows that f̃_{E}(z) is an eigenform for all the Hecke operators. Thus f̃_{E}(z) = c_{E}f_{E}(z) where f_{E}(z) is a normalized new form of weight 2 and level N_{E} and c_{E} is a constant. One knows that c_{E} ∈ ℤ.

There is a result of Raynaud [6] that if N_{E} is squarefree, log |c_{E}| is bounded. Manin conjectures that in fact, |c_{E}| = 1.

The ( Faltings ) height of E is by definition

h(E) = −\frac{1}{2} log(\frac{1}{2π} ∫_{E(ℂ)} |ω_{E} ∧ \overline{ω_{E}}|).

The height conjecture for elliptic curves states that h(E) = O(log N_{E}). This conjecture is equivalent to the famous ABC conjecture (see Frey [1]).

If E is a modular elliptic curve, the preceding remarks imply that

h(E) = −\frac{1}{2} log(\frac{c_{E}^{2}}{2π deg ϕ} ∫_{H^{∗}/Γ_{0}(N)} |f_{E}|^{2}dz)

where deg ϕ is the degree of the modular parametrization ϕ of E. The degree conjecture is that

log(\frac{deg ϕ}{c_{E}^{2}}) = O(log g(X_{0}(N_{E})) ),

where g(X_{0}(N_{E})) is the genus of the modular curve X_{0}(N_{E}). We have, for 36 |̸N_{E} (see [8]),

g(X_{0}(N_{E})) = 1 + \frac{μ}{12} − \frac{μ_{2}}{4} − \frac{μ_{3}}{3} − \frac{μ_{∞}}{2}

where

\begin{aligned}μ & = N_{E} ∏_{p|N_{E}}(1 + \frac{1}{p}) \\ μ_{2} & = ∏_{p|N_{E}}(1 + (\frac{−1}{p})) \\ μ_{3} & = ∏_{p|N_{E}}(1 + (\frac{−3}{p})) \\ μ_{∞} & = ∑_{t|N_{E}} φ((t, \frac{N_{E}}{t})) \end{aligned}

As #{distinct primes dividing N_{E}} ≪ log N_{E}/ log log N_{E}, we have

log N_{E} ≪ log g(X_{0}(N_{E})) ≪ log N_{E}.

In the next section, we will show the equivalence of the height and degree conjectures for modular elliptic curves.

4. The main result

From the preceding section, we know

h(E) = \frac{1}{2} log(deg ϕ) − log |c_{E}| − log⟨f_{E}, f_{E}⟩.

From the application of the Phragmén-Lindelöf theorem in section 2, we find

log⟨f_{E}, f_{E}⟩ = O(log N_{E})

so the height and degree conjectures are equivalent. This proves :

Theorem. Suppose E is a modular elliptic curve. Then the height and degree conjectures are equivalent.

Corollary. Under Taniyama’s conjecture, the degree conjecture is equivalent to the ABC conjecture.

Remark. Recently, Wiles has announced the semi-stable case of the Taniyama conjecture. This means that when ABC is divisible by 16, the degree conjecture and the ABC conjecture are equivalent.

References

1. G. Frey, Links between solution of A-B=C and elliptic curves, Number Theory, Ulm 1987, LNM 1380.

2. , Construction and arithmetical applications of modular forms of low weight (to appear).

3. V. Kumar Murty and M. Ram Murty, A variant of the Bombieri-Vinograov theorem, Number Theory (H. Kisilevsky and J. Labute, eds.), vol. 7, CMS conference Proceedings, 1987, pp. 243-272.

4. H. Rademacher, On the Phragmén-Lindelöf theorem and some applications, Seminar on analytic functions, vol. 1, Institute for Advance Study, Princeton, N.J., 1957, pp. I-314 - I-322.

5. , On the Phragmén-Lindelöf theorem and some applications. In Memoriam Leon Lichtenstein, Mathematische Zeitschrift 72 (1959), 192-204.

6. M. Raynaud, Letter to Mestre (1985), Unpublished.

7. K. Ribet, On modular representations of Gal(\overline{ℚ}/ℚ) arising from modular forms (1987), Math. Research Institute, Berkeley, CA, preprint..

8. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton University Press, 1971.

9. , On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. 31 (1975), 79-98.

10. J. Silverman, The arithmetic of elliptic curves, 2nd ed., Springer-Verlag, New-York, 1991.

Centre de Recherches Mathematiques, University de Montreal, CP 6128-A, Montreal Quebec, H3C 3J7 Canada

Department of Mathematics, McGill University, 805 Sherbrooke St West, Montreal Quebec, H3A 2K6 Canada