Let E be an elliptic curve over Q and ` be an odd prime. Also, let K be a number field and assume that E has a semi-stable reduction at `. Under certain assumptions, we prove the vanishing of the Galois cohomology group H1(Gal(K(E[`i])/K), E[`i]) for all i ≥ 1. When K is an imaginary quadratic field with the usual Heegner assumption, this vanishing theorem enables us to extend a result of Kolyv...

Let F be a number field with ring of integers OF , and let E/OF be an abelian scheme of arbitrary dimension. In this paper, we study the class invariant homomoprhisms on E with respect to powers of a prime p of ordinary reduction of E. Our main result implies that if the p-adic Birch and Swinnerton-Dyer conjecture holds for E, then the kernels of these homomorphisms are of bounded order. It fol...

We study zeros of the L-functions L(f, s) of primitive weight two forms of level q. Our main result is that, on average over forms f of level q prime, the order of the L-functions at the central critical point s = 1 2 is absolutely bounded. On the Birch and Swinnerton-Dyer conjecture, this provides an upper bound for the rank of the Jacobian J0(q) of the modular curve X0(q), which is of the sam...

In the early 90’s, Perrin-Riou (Ann Inst Fourier 43(4):945–995, 1993) introduced an important refinement of Mazur–Swinnerton-Dyer p-adic L-function elliptic curve E over $$\mathbb {Q}$$ , taking values in its de Rham cohomology. She then formulated a analogue Birch and Swinnerton-Dyer conjecture for this L-function, which formal group logarithms global points on make intriguing appearance. The ...

We prove a fundamental conjecture of Rubin on the structure local units in anticyclotomic $\mathbb{Z}_p$-extension unramified quadratic extension $\mathbb{Q}_p$ for $p\geq 5$ prime. Rubin's underlies Iwasawa theory deformation CM elliptic curve over field at primes $p$ good supersingular reduction, notably main terms $p$-adic $L$-function. As consequence, we an inequality Birch and Swinnerton-D...

Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the padic L-function of a modular abelian variety and showing it satisfies the appropriate interpolation property. We descri...

We discuss p-adic unipotent Albanese maps for curves of positive genus, extending the theory of p-adic multiple polylogarithms. This construction is then used to relate linear Diophantine conjectures of ‘Birch and Swinnerton-Dyer type’ to non-linear theorems of Faltings-Siegel type. In a letter to Faltings [14] dated June, 1983, Grothendieck proposed several striking conjectural connections bet...

We produce explicit elliptic curves over Fp(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related elliptic surfaces) and then use zeta functions to determine the rank. In contrast to earlier examples of Shafarevitch and Tate, our curves are not isotrivial. Asymptoti...