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...

We show that there is a bound depending only on g, r and [K : Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its Jacobian is at most g − 3. If K = Q, an explicit bound is 8rg + 33(g − 1) + 1. The proof is based on Chabauty’s method; the new ingredient is an estimate for the number of zeros of an abelian log...

Let E be an optimal elliptic curve over Q of conductor N having analytic rank one, i.e., such that the L-function LE(s) of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split and such that the L-function of E over K vanishes to order one at s = 1. Suppose there is another optimal elliptic curve over Q of the same conductor N whose Mord...

Much success in finding rational points on curves has been obtained by using Chabauty’s Theorem, which applies when the genus of a curve is greater than the rank of the Mordell-Weil group of the Jacobian. When Chabauty’s Theorem does not directly apply to a curve C, a recent modification has been to cover the rational points on C by those on a covering collection of curves Di, obtained by pullb...

Let E be an optimal elliptic curve over Q of conductor N having analytic rank zero, i.e., such that the L-function LE(s) of E does not vanish at s = 1. Suppose there is another optimal elliptic curve over Q of the same conductor N whose Mordell-Weil rank is greater than zero and whose associated newform is congruent to the newform associated to E modulo a power r of a prime p. The theory of vis...

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.