NCMW - Elliptic Curves, Elliptic Functions and Trancendence

Venue: Harish-Chandra Research Institute, Prayagraj

Dates: 24 Nov 2022 to 3 Dec 2022

First four days, we shall cover the preliminaries required for the workshop themes. We shall introduce (1) Modular functions, mainly, modular j-function and its properties, its applications to space of meromorphic functions and Picard’s theorem. (2) Basic Elliptic curve properties and Complex multiplication theory, mainly, (3rd chapter of Cox book on ‘Primes of the form x2 +ny2 ’) (3) Introduction to Elliptic functions and its properties, Elliptic curve over C.

In the workshop part, we shall consider three themes: (1) The Schneider Lang Theorem: statement, corollaries and proof, The St ́ephanois Theorem on the transcendence of J(q) and Nesterenko’s Theorem: statement, corollaries and proof. (2) Gelfond’s transcendence criterion and its development with proof, Gelfond’s algebraic independence method: Small transcendence degree, Chudnovsky’s algebraic independence theorem (ω1, ω2, η1, η2) using: (i) the transcendence measure of π, (ii) Gelfond’s algebraic independence method combined with his zeros estimate and Elliptic analog of Brownawell-Waldschmidt theorem. The third theme is: Iwasawa theory for CM Elliptic curves.