We study Rubin’s variant of the p-adic Birch and Swinnerton-Dyer conjecture for CM elliptic curves concerning certain special values of the Katz two-variable p-adic L-function that lie outside the range of p-adic interpolation.

[3] , Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero, Math. Finiteness results for modular curves of genus at least 2, Amer.

We survey the history of the Tate conjecture on algebraic cycles. The conjecture is closely related with other big problems in arithmetic and algebraic geometry, including the Hodge and Birch–Swinnerton-Dyer conjectures. We conclude by discussing the recent proof of the Tate conjecture for K3 surfaces over finite fields.

Abstract This article proves a case of the p -adic Birch and Swinnerton–Dyer conjecture for Garrett L -functions (Bertolini et al. in On analogues -functions, 2021), imaginary dihedral exceptional zero setting extended analytic rank 4.

The Birch and Swinnerton--Dyer conjecture famously predicts that the rank of an elliptic curve can be computed from its $L$-function. In this article we consider a weaker version called parity prove following. Let $E_1$ $E_2$ two curves defined over number field $K$ whose 2-torsion groups are isomorphic as Galois modules. Assuming finiteness Shafarevich-Tate $E_2$, show Swinnerton-Dyer correctl...

Let N be a prime and let A be a quotient of J0(N) over Q associated to a newform f such that the special L-value of A (at s = 1) is non-zero. Suppose that the algebraic part of special L-value is divisible by an odd prime q such that q does not divide the numerator of N−1 12 . Then the Birch and Swinnerton-Dyer conjecture predicts that q divides the algebraic part of special L value of A, as we...

We provide two proofs that the conjecture of Artin-Tate for a fibered surface is equivalent to Birch-Swinnerton-Dyer Jacobian generic fibre. As byproduct, we obtain new proof theorem Geisser relating orders Brauer group and Tate-Shafarevich group.

Assuming finiteness of the Tate–Shafarevich group, we prove that Birch–Swinnerton–Dyer conjecture correctly predicts parity rank semistable principally polarised abelian surfaces. If surface in question is Jacobian a curve, require curve has good ordinary reduction at 2-adic places.

Let E/Q be an optimal elliptic curve of analytic rank zero. It follows from the Birch and Swinnerton-Dyer conjecture for curves zero that order torsion subgroup divides product Shafarevich–Tate group E/Q, (global) Tamagawa number at infinity. This consequence was noticed by Agashe Stein in 2005. In this paper, we prove divisibility statement unconditionally many cases, including case where is s...