## Number Theory - Payman Eskandari (University of Toronto)

### Tuesday, October 23rd, 2018

**Time:** 10:00-11:00 a.m. **Place:** Jeffery Hall 422

**Speaker:** Payman Eskandari (University of Toronto)

**Title:** On the transcendence degree of the field genarated by quadratic periods of a smooth curve over a number field.

**Abstract:** Grothendieck's period conjecture predicts the transcendence degree of the field generated by the periods of a smooth projective variety (or more generally, a pure motive) over a number field, in terms of the dimension of its Mumford-Tate group. For mixed motives a similar conjecture was made by Andre. The upper bound part of Grothendieck's conjecture was proved in the case of abelian varieties by Deligne (as a consequence of his "Hodge implies absolute Hodge" theorem for abelian varieties). The lower bound part of Grothendieck's conjecture is known for a CM elliptic curve, thanks to a theorem of G. V. Chudnovsky.

This talk is a report on an aspect of a work in progress with Kumar Murty, in which we use Hodge theoretic methods and Tannakian formalism to study quadratic and higher periods of a punctured curve. We start by some background material and motivation. In the end, we prove the upper bound part of Andre's conjecture for quadratic periods of a punctured elliptic curve, defined over a subfield of $\mathbb{R}$. The argument is quite formal, and in fact, applies to any extension of $H^1\otimes H^1$ by $H^1$.