Symmetric Powers of Elliptic Curve L-Functions

Abstract

The conjectures of Deligne, Beȋlinson, and Bloch-Kato assert that there should be relations between the arithmetic of algebrogeometric objects and the special values of their L-functions. We make a numerical study for symmetric power L-functions of elliptic curves, obtaining data about the validity of their functional equations, frequency of vanishing of central values, and divisibility of Bloch-Kato quotients.

Some Heuristics about Elliptic Curves

“…Similar to questions about the vanishing of L(E, s), we can ask about the vanishing of the symmetric power L-functions L(Sym 2k−1 E, s). We refer the reader to [22] for more details about this, but mention that, due to conjectures of Deligne and more generally Bloch and Beȋlinson [24], we expect that we should have a formula similar to that of Birch and Swinnerton-Dyer,…”

“…It should be said that this heuristic will likely mislead us about curves with complex multiplication, for which the symmetric power L-function factors (it is imprimitive in the sense of the Selberg class), with each factor having a 50% chance of having odd parity. However, even ignoring CM curves, the data of [22] find a handful of curves for which the 9th, 11th and even the 13th symmetric powers appear (to 12 digits of precision) to have a central zero of order 2. We find this surprising, and casts some doubt about the validity of our methodology of modelling of vanishings.…”

Extremal primes for elliptic curves without complex multiplication

“…For p a prime of potentially good reduction, the value of ǫ n (p) depends on the inertia group of the local extension G p = Gal(Q p (E ℓ )/Q p ), and the congruence of n modulo 12. The values of ǫ n (p) in all cases that arise are given in Table 1 of [MW06], and we always have 0 ≤ ǫ n (p) ≤ n + 1. The wild conductors δ n (2) and δ n (3) are given in Tables 2 and 3 of [MW06], and we have that δ n (2) ≤ 2(n + 1) and δ n (3) ≤ (n + 1)/2.…”

“…We also remark that the computation of the conductor N E,n in [MW06] is the idea presented in [Rou07, Section 5] applied to the special case of elliptic curves.…”

On elliptic curves and random matrix theory

“…This enumeration is quite fast, but the process of finding suitable representatives (A, B, C) ∈ S β (−d, N ) is comparatively quite time-consuming. 7 We borrowed the code for periods, conductors, root numbers, and Fourier coefficients from the SYMPOW package of [28], while the torsion was a command-line option to the programme. The composition of forms would often yield coefficients with more than 32 bits, and so it was useful to have a 64-bit processor.…”