We introduce a notion of visibility for Mordell-Weil groups, make a conjecture about visibility, and support it with theoretical evidence and data. These results shed new light on relations between Mordell-Weil and Shafarevich-Tate groups. 11G05, 11G10, 11G18, 11Y40

We discuss the extension of inequality RA c a rb + b a rc to the plane of triangle ABC . Based on the obtained extension, in regard to all three vertices of the triangle, we get the extension of Erdös-Mordell inequality, and some inequalities of Erdös-Mordell type. Mathematics subject classification (2010): 51M16, 51M04, 14H50.

We prove that the elliptic surface y 2 = x 3 + 2(t 8 + 14t 4 + 1)x + 4t 2 (t 8 + 6t 4 + 1) has geometric Mordell-Weil rank 15. This completes a list of Kuwata, who gave explicit examples of elliptic K3-surfaces with geometric Mordell

Let E/Q be an elliptic curve defined over Q of conductor N and let Gal(Q/Q) be the absolute Galois group of an algebraic closure Q of Q. For an automorphism σ ∈ Gal(Q/Q), we let Q be the fixed subfield of Q under σ. We prove that for every σ ∈ Gal(Q/Q), the Mordell-Weil group of E over the maximal Galois extension of Q contained in Q σ has infinite rank, so the rank of E(Q σ ) is infinite. Our ...

Fix the data (p,K,E) where p is a prime number, K a number field, and E an elliptic curve over Q. Let K∞/K denote the maximal Zp-power extension of K. Recent work provides, in some instances, detailed information about p-adic completions of Mordell-Weil groups and their associated p-adic height pairings, and the p-primary Shafarevich-Tate groups and their associated Cassels pairings, over inter...

Very rough notes for a lecture to be given October 5, 2013 at the Quebec/Maine Number Theory Conference. I’ll discuss diophantine questions that take on a somewhat different flavor when one deals with varying number fields rather than restricts to Q as a base field: an on-going joint project with Maarten Derickx and Sheldon Kamienny regarding Mordell-Weil torsion, and some recent work with Zev ...

Let K be a number field and A an abelian variety over K. The K-rational points of A are known to constitute a finitely generated abelian group (Mordell-Weil theorem). The problem studied in this paper is to find an explicit upper bound for the rank r of its free part in terms of other invariants of A/K. This is achieved by a close inspection of the classical proof of the so-called ‘weak Mordell...

These notes continue the notes of Anand Pillay on model theory and diophantine geometry. In my lectures I describe a model theoretic approach to some analogues of the Mordell-Lang conjecture for Drinfeld modules. Many questions remain open and algebraic proofs along the lines of the proof of the Manin-Mumford conjecture described by Pillay may be possible. We discuss these questions and potenti...

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.