In this paper we establish a Hasse principle concerning the linear dependence over Z of nontorsion points in the Mordell-Weil group of an abelian variety over a number field.

The canonical height ĥ on an abelian variety A defined over a global field k is an object of fundamental importance in the study of the arithmetic of A. For many applications it is required to compute ĥ(P ) for a given point P ∈ A(k). For instance, given generators of a subgroup of the Mordell-Weil group A(k) of finite index, this is necessary for most known approaches to the computation of gen...

The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to high order at the center point of their functional equation. Conjectures of Birch and Swinnerton-Dyer, Bloch, and Beilinson relate the orders of vanishing of ...

Arguing by induction on the dimension, it is easy to show that the same holds for higher dimensional base varieties, more precisely, let V be a smooth variety over C and let T be a closed subset of V . Define Fg(V, T ) as the set of non-isotrivial families of smooth curves of genus g ≥ 2 over V − T . Then Fg(V, T ) is finite. Partial motivation for the Shafarevich conjecture was that, as Parshi...

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find m...

Let C : Y 2 = anX n + · · · + a 0 be a hyperelliptic curve with the a i rational integers, n ≥ 5, and the polynomial on the right irreducible. Let J be its Jacobian. We give a completely explicit upper bound for the integral points on the model C, provided we know at least one rational point on C and a Mordell–Weil basis for J(Q). We also explain a powerful refinement of the Mordell–Weil sieve ...

This paper extends tools developed in [10, 8] to study character polylogarithms. These objects are used to compute Mordell-Tornheim-Witten character sums and to explore their connections with multiple-zeta values (MZVs) and with their character analogues [17].

We prove a dynamical version of the Mordell–Lang conjecture in the context of Drinfeld modules. We use analytic methods similar to those employed by Skolem, Chabauty, and Coleman for studying diophantine equations.

Bachmann, Takeyama and Tasaka introduced the Q-linear map ϕ on harmonic algebra H1, which we call symmetrization in this paper. They calculated ϕ(w) explicitly for an element w H1 related to multiple zeta values of Mordell–Tornheim type. In paper, introduce its t-adic generalization ϕˆ calculate ϕˆ(w) elements H1〚t〛 constructed from theory 2-colored rooted trees.