Time：14:00-16:00, Friday, October 18 2024
Venue: E10-211
Speaker：Marc Hindry, Université Paris Cité
Biography: Marc Hindry is a professor at the Université Paris Cité. His research interests include diophantine arithmetic of algebraic varieties on finite and global fields, elliptic curves, abelian and jacobian varieties, algebraic surfaces, Algebraic and analytical number theory, distribution of prime Numbers, algebraic cycles, Functions zeta and so on.
Title: Integral points on elliptic curves
Abstract: We will discuss a generalisation of a classical problem initiated by Mordell (1964) : «which integer can be written as a product of two consecutive integers and as a product of three consecutive integers?» by choosing and integer A (at least 1) and replacing consecutive by consecutive in an arithmetic progression with common difference A.
The problem translates into the problem of finding all integral points on an elliptic curve (with parameter A).
We will review the general theory of elliptic curves (group law, heights, Mordell-Weil theorem, Siegel theorem) and sketch the proof of our main result - the complete solution of the problem when the rank of the elliptic curve is equal to two (it is always at least two). The proof uses many Diophantine tools (heights, reduction modulo a prime, linear forms of elliptic logarithms).
