Mazur and Tate proposed a conjecture which compares the Mordell-Weil rank of an elliptic curve overQwith the order of vanishing of Mazur-Tate elements, which are analogues of Stickelberger elements. Under some relatively mild assumptions, we prove this conjecture. Our strategy of the proof is to study divisibility of certain derivatives of Kato’s Euler system. CONTENTS

In this paper, we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module an abelian variety. [G. Banaszak and P. Krasoń, On étale K-groups curves, J. K-Theory Appl. Algebra Geom. Topol. 12 (2013) 183–201], G. author obtained sufficient condition validity [Formula: see text]-theory curve. This fact has been established by means analysi...

Abstract Answering a question of Zureick-Brown, we determine the cubic points on modular curves $X_0(N)$ for $N \in \{53,57,61,65,67,73\}$ as well quartic $X_0(65)$. To do so, develop “partially relative” symmetric Chabauty method. Our results generalise current theorems and improve upon them by lowering involved prime bound. For our number novelties occur. We prove “higher-order” theorem to de...

These notes are based on lectures given at the “Arithmetic of Hyperelliptic Curves” workshop, Ohrid, Macedonia, 28 August–5 September 2014. They offer a brief (if somewhat imprecise) sketch of various methods for computing the set of rational points on a curve, focusing on Chabauty and the Mordell–Weil sieve.

We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasi-orthogonality in the Mordell-Weil lattice ([Sil6], [GS], [He]). We apply our results to break previous bounds on the number of elliptic curves of given conductor and the size of the 3-torsion part of...

Shioda described in his article [6] a method to compute the Lefschetz number of a Delsarte surface. In one of his examples he uses this method to compute the rank of an elliptic curve over kptq. In this article we find all elliptic curves over kptq for which his method is applicable. For these curves we also compute the maximal Mordell-Weil rank.

Computing a lower bound for the canonical height is a crucial step in determining a Mordell–Weil basis for elliptic curves. This paper presents an algorithm for computing such a lower bound for elliptic curves over number fields without searching for points. The algorithm is illustrated by some examples.

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...

We refine necessary and sufficient conditions for the generating series of a weighted model quarter plane walk to be differentially algebraic. In addition, we give algorithms based on theory Mordell–Weil lattices, that, each model, yield polynomial weights determining this property associated series.