Speaker: Alyson Deines of University of Washington
3:30pm in Padelford C401 on December 8, 2011.
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
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 .