We investigate the Mordell constant of certain families of lattices, in particular, of lattices arising from totally real fields. We define the almost sure value κμ of the Mordell constant with respect to certain homogeneous measures on the space of lattices, and establish a strict inequality κμ1 < κμ2 when the μi are finite and supp(μ1) supp(μ2). In combination with known results regarding the...

We introduce certain integrals of a product of the Bernoulli polynomials and logarithms of Milnor’s multiple sine functions. It is shown that all the integrals are expressed by the Mordell-Tornheim zeta values at positive integers and that the converse is also true. Moreover, we apply the theory of the integral to obtain various new results for the Mordell-Tornheim zeta values.

1 Gross’ formula for special values of L-series . . . . . . . . . . . . . . . 4 2 Bad reduction of Shimura curves . . . . . . . . . . . . . . . . . . . 5 3 Heegner points and connected components . . . . . . . . . . . . . . . 7 4 Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 A rigid analytic Gross-Zagier formula . . . . . . . . . . . . . . . . 11 6 Kolyvagin cohomolog...

We discuss the Mordell-Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the Mordell-Weil sieve algorithm and...

RÉSUMÉ. Nous prouvons que toute somme de Mordell-Tornheim avec des arguments entiers positifs peut s’écrire comme une combinaison linéaire rationnelle de valeurs prises par des fonctions multi-zêta ayant le même poids et la même profondeur. Selon un résultat de Tsumura, il s’ensuit que toute somme de Mordell-Tornheim ayant un poids et une profondeur de parité différente peut s’exprimer comme un...

 The Mordell-Weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎In our previous paper, H‎. ‎Daghigh‎, ‎and S‎. ‎Didari‎, On the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎Bull‎. ‎Iranian Math‎. ‎Soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using Selmer groups‎, ‎we have shown that for a prime $p...

This paper deals with the reverse inequalities of Erdös-Mordell type. Our result contains as special case the following reverse Erdös-Mordell inequality: R1 +R2 +R3 < √ 2 (ρ1 +ρ2 +ρ3) , where Ri and ρi (i=1, 2, 3) denote respectively the distances from an interior point Q of A1A2A3 to the vertexes A1, A2, A3 and to the circumcenters of A2QA3 , A3QA1 , A1QA2 . Some other closely related inequali...

Using the Skolem-Mahler-Lech theorem, we prove a dynamical Mordell-Lang conjecture for semiabelian varieties.