Welcome to the Maryland Distinguished Lectures in Algebra and Number Theory! This is an annual lecture series concerning recent developments in Algebra and Number Theory, delivered by the world's foremost experts. It is organized by Thomas Haines.

2021 Wei Zhang

Cycles on products of elliptic curves and a conjecture of Bloch-Kato

Wednesday, December 1 at 3:15pm
University of Maryland - Kirwan Hall Rm 3206

Abstract: One classical way to study rational or integral solutions of a polynomial equation is to look at the simpler questions of the various congruence equations modulo n for all integer n. The conjecture of Birch and Swinnerton-Dyer predicts that, for elliptic curves (defined by equations of the form y^2=x^3+ax+b with integer coefficients), the data from these congruence equations mod n should actually encode much information on the solutions in rational numbers. In the first talk we will discuss a generalization of the question to the product of several elliptic curves, where, instead of rational points, we look for algebraic cycles (i.e., parameter solutions) modulo suitable equivalence relations (rational equivalence, Abel—Jacobi or its p-adic version). In particular, I'll report some recent results on a conjecture of Bloch-Kato.

Periods of automorphic forms and L-values

Thursday, December 2 at 11:00am
University of Maryland - Kirwan Hall Rm 3206

Abstract: In the second and the third talk I will report some of the ingredients in the proof of the results presented in the first talk. For a reductive group G over a number field and H a suitable subgroup, a fundamental question is to study the period integral of an automorphic form on G along the subgroup H. I’ll give examples with an emphasis on the cases that are related to special values of L-functions.

Heights of special cycles on Shimura varieties and L-derivatives

Friday, December 3 at 2:00pm
University of Maryland - Kirwan Hall Rm 3206

Abstract: In the last talk, I will move to an arithmetic analog of period integrals from the second talk. Here we consider Shimura varieties defined by a group G and special algebraic cycles defined by a subgroup H of G. The height pairing of such special cycles is often related to the special value of the first derivative of L-function; the first of such instance was proved by Gross and Zagier in 1980s. We report the progress on the arithmetic Gan-Gross-Prasad conjecture, one of the generalizations of Gross—Zagier theorem to high dimensional varieties.

2019 Ngô Bảo Châu

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