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Ramanujan-type supercongruences

Speaker: Alyson Deines of University of Washington

Location

3:30pm in Padelford C401 on December 8, 2011.

Abstract

Ramanujan's work features various formulas for of the form

where is a polynomial with algebraic coefficients and and are algebraic numbers. van Hamme first noticed Ramanujan-type supercongruences, or congruences of the form

and

for almost all primes .

Ramanujan-type supercongruences also come up when computing points on certain CM elliptic curves mod in the following sense: let be the curve with so that is CM. Define the hypergeometric series

Using period relations of the elliptic curves, there are various ways to write in terms of . The associated supercongruence is:

Where in terms of is the hypergeometric series truncated at and .

There is a similar construction for K3 surfaces which gives rise to various ways to write in terms of another hypergeometric series. This has an associated supercongruence mod . At WIN2, under Ling Long's direction and with other group members Gabriel Nebe, Sara Chisholm, and Holly Swisher, we examined these K3 surfaces and their associated supercongruence mod and .

2013-05-11 18:34