Congruences for critical values of higher derivatives of twisted Hasse–Weil L-functions, III

Abstract

Let A be an abelian variety defined over a number field k, let p be an odd prime number and let $F/k$ be a cyclic extension of p-power degree. Under not-too-stringent hypotheses we give an interpretation of the p-component of the relevant case of the equivariant Tamagawa number conjecture in terms of integral congruence relations involving the evaluation on appropriate points of A of the ${\rm Gal}(F/k)$ -valued height pairing of Mazur and Tate. We then discuss the numerical computation of this pairing, and in particular obtain the first numerical verifications of this conjecture in situations in which the p-completion of the Mordell–Weil group of A over F is not a projective Galois module.

Numerical Evidence for a refinement of Deligne's Period Conjecture for Jacobians of Curves

“…where '≈' means 'equal to 10 significant figures'. We find that Θ 11 (A F ) belongs to Z (5) [G], and thus that claim (i) of Conjecture 1.1 is valid. Explicitly,…”

“…Integral refinements of Deligne's Period Conjecture similar to that of Conjecture 1.1, both for values of the form (3) and for analogous elements constructed from derivatives of Hasse-Weil-Artin L-series, have also been numerically investigated in the articles [1,2,4,5,12]. However, as alluded to above, the investigations in these articles were undertaken exclusively for elliptic curves, satisfying moreover stricter hypotheses on reduction types than those that are in place in Conjecture 1.1.…”

“…In this section we will provide a guide to the tables in the following section. Let S be the set of pairs (p, q) of odd primes {(3, 7), (3,13), (3,19), (3,31), (5,11), (5, 31), (7, 29)}, which all satisfy q ≡ 1 mod p, let F p,q be the degree p subfield of Q(ζ q ) and write G = Gal(F p,q /Q). For each of the 38 abelian varieties A of conductor at most 500 that arise as Jacobians of genus 2 curves over Q that are listed in the LMFDB [17] and each pair (p, q) ∈ S, we calculated the p-tuple of modified L-values L q (A, ψ) : ψ ∈ G .…”

“…In recent years, there has been much interest in the formulation and study of integral refinements of Deligne's Period Conjecture for the equivariant L-values that are associated to the base change of an abelian variety through a Galois extension of number fields. We refer the reader to [1,2,3,4,5,11,12,15,20]. However, as far as we are aware, any theoretical or numerical evidence obtained for such refinements has been restricted to the case of elliptic curves.…”