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 define certain objects associated to a modular elliptic curve E and a discriminant D satisfying suitable conditions. These objects interpolate special values of the complex L-functions associated to E over the quadratic field Q( √ D), in the same way that Bernouilli numbers interpolate special values of Dirichlet L-series. Following an approach of Mazur and Tate [MT], one can make conjecture...

The classical theory of invariants of binary quartics is applied to the problem of determining the group of rational points of an elliptic curve deened over a eld K by 2-descent. The results lead to some simpliications to the method rst presented in (Birch and Swinnerton-Dyer, 1963), and can be applied to give a more eecient algorithm for determining Mordell-Weil groups over Q, as well as being...

This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of theseconjectures relates six quantities associated to a Jacobian over the rationalnumbers. One of these six quantities is the size of the Shafarevich-Tate group.Unable to compute that, we computed the five other quantities and solved for...

This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the ShafarevichTate group. Unable to compute that, we computed the five other quantities and solved for the last one. In ...

Let p be a prime, let r and q powers of p, b relatively prime integers not divisible by p. $$C/{\mathbb {F}}_{r}(t)$$ the superelliptic curve with affine equation $$y^b+x^a=t^q-t$$ , J Jacobian C. By work Pries Ulmer (Trans Am Math Soc 368(12):8553–8595, 2016), satisfies Birch Swinnerton-Dyer conjecture (BSD). Generalizing Griffon (Pacific 305(2):597–640, 2020) we compute L-function in terms ce...

We study p-indivisibility of the central values L(1, Ed) of quadratic twists Ed of a semi-stable elliptic curve E of conductorN. A consideration of the conjecture of Birch and Swinnerton-Dyer shows that the set of quadratic discriminants d splits naturally into several families FS, indexed by subsets S of the primes dividing N. Let δS = gcdd∈FS L(1, Ed) alg, where L(1, Ed) alg denotes the algeb...