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...

Let E be an elliptic curve defined over Q. Let Γ be a free subgroup of rank r of E(Q). For any prime p of good reduction, let Γp be the reduction of Γ modulo p and Ep be the reduction of E modulo p. We prove that if E has CM then for all but o(x/ log x) of primes p ≤ x, |Γp| ≥ p r r+2 + , where (p) is any function of p such that (p)→ 0 as p→∞. This is a consequence of two other results. Denote ...

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...

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.

We study the growth and stability of Mordell–Weil group Tate–Shafarevich an elliptic curve defined over rationals, in various cyclic Galois extensions prime power order. Mazur Rubin introduced notion diophantine for $$E_{/{{\mathbb {Q}}}}$$ at a given p. Inspired by their definition group, we introduce analogous called -stability. From perspective Iwasawa theory, it benefits us to stronger grou...

This is an introductory exposition to background material useful to appreciate various formulations of the Mordell–Lang conjecture (now established by recent spectacular work due to Vojta, Faltings, Hrushovski, Buium, Voloch, and others). It gives an exposition of some of the elementary and standard constructions of algebro-geometric models (rather than model-theoretic ones) with applications (...

We will give an upper bound of Mordell-Weil rank r for relatively minimal brations of curves of genus g 1 on rational surfaces. Under the assumption that a bration is not locally trivial, we have r 4g+4. Moreover the maximal case (r = 4g + 4) will be studied in detail. We determine the structure of such brations and also the structure of their Mordell-Weil lattices introduced by Shioda.