We present a Mordell-Lang statement for Drinfeld modules. In the case of Drinfeld modules of finite characteristic, we show the connection between our Mordell-Lang statement and the study of the ring of quasi-endomorphisms for a certain minimal group in the theory of separably closed fields associated to the Drinfeld module.

Three lectures on elliptic surfaces and curves of high rank Noam D. Elkies Over the past two years we have improved several of the (Mordell–Weil) rank records for elliptic curves over Q and nonconstant elliptic curves over Q(t). For example, we found the first example of a curve E/Q with 28 independent points P i ∈ E(Q) (the previous record was 24, by R. Martin and W. McMillen 2000), and the fi...

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

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...