The sub-leading coefficient of the L-function of an elliptic curve
Christian Wuthrich
August 23, 2016
Abstract
We show that there is a relation between the leading term at s = 1 of an L-function of an elliptic
curve defined over an number field and the term that follows.
Let E be an elliptic curve defined over a number field K. We will assume that the L-function
L(E, s) admits an analytic continuation to s = 1 and that it satisfies the functional equation. By
modularity [1], we know that this holds when K = Q. The conjecture of Birch and Swinnerton-Dyer
predicts that the behaviour at s = 1 is linked to arithmetic information. More precisely, ifL(E, s) = ar (s − 1)r + ar+1 (s − 1)r+1 + · · ·is the Taylor expansion at s = 1 with ar 6= 0, then r should be the rank of the Mordell-Weil group
E(K) and the leading term ar is equal to a precise formula involving the Tate-Shafarevich group ofE. It seems to have passed unnoticed that the sub-leading coefficient ar+1 is also determined by thefollowing formula.
Theorem 1. With the above assumption, we have the equalityar+1 = [K : Q] · (γ + log(2π)) − 21 log(N ) − log |∆K | · ar(1)

where γ = 0.577216 . . . is Euler’s constant, N is the absolute norm of the conductor ideal of E/Kand ∆K is the absolute discriminant of K/Q.
In particular, the conjecture of Birch and Swinnerton-Dyer also predicts completely what the subleading coefficient ar+1 should be. One consequence for K = Q is that for all curves with conductor
N > 125, and this is all but 404 isomorphism classes of curves, the sign of ar+1 is the opposite of ar .
Of course, it is believed that ar is positive for all E/Q.
√Proof. Set f (s) = B s · Γ(s)n with n = [K : Q] and B = N · |∆K |/(2π)n . Then Λ(s) = f (s) · L(E, s)
is the completed L-function, which satisfies the functional equation Λ(s) = (−1)r · Λ(2 − s), see [3].
diFor i ≡ r + 1 (mod 2) it follows that dsi Λ(s) s=1 = 0. Hence for i = r + 1, we obtain that(r + 1) · f 0 (s) ·drdr+1
L(E, s) + f (s) · r+1 L(E, s)
rdsds

is zero at s = 1. Therefore (r + 1) f0 (1) r! ar + f (1) (r + 1)! ar+1 = 0. It remains to note that f (1) = B
and f 0 (1) = B · log(B) + n · Γ0 (1) together with Γ0 (1) = −γ.
Obviously a similar formula holds for the L-function of a modular form of weight 2 for Γ0 (N ).
More generally, for any L-function with a functional equation there is a relation between the leading
and the sub-leading coefficient of the Taylor expansion of the L-function at the central point.
Sub-leading coefficients of Dirichlet L-functions have been investigated; for instance Colmez [2]makes a conjecture, which is partially known. However these concern the much harder case when s
is not at the centre but the boundary of the critical strip of the L-function.

1References[1] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic
curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939.
[2] Pierre Colmez, Périodes des variétés abéliennes à multiplication complexe, Ann. of Math. (2) 138(1993), no. 3, 625–683.
[3] Dale Husemöller, Elliptic curves, second ed., Graduate Texts in Mathematics, vol. 111, SpringerVerlag, New York, 2004, With appendices by Otto Forster, Ruth Lawrence and Stefan Theisen.

2