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 $\pi$ of the form

$\sum_{n = 0}^{\infty} A(n)x_0^n = \frac{c}{\pi^k}$

where $A(n)$ is a polynomial $n$ with algebraic coefficients and $x_0$ and $c$ are algebraic numbers. van Hamme first noticed Ramanujan-type supercongruences, or congruences of the form

$\sum_{n=0}^{p-1}A(n)x_0^n \equiv 0\hspace{2mm} ({\rm mod } p^k)$

and

$\sum_{n=0}^{\frac{p-1}{2}}A(n)x_0^n \equiv 0 \hspace{2mm}({\rm mod } p^k)$

for almost all primes $p$ .

Ramanujan-type supercongruences also come up when computing points on certain CM elliptic curves mod $p$ in the following sense: let $E_{\lambda}$ be the curve $y^2 = x(x-1)(x-\lambda)$ with $\lambda$ so that $E$ is CM. Define the hypergeometric series

$\ _2F_1(\frac{1}{2},\frac{1}{2};1;x) = \sum_{n=0}^{\infty} \frac{\left(\frac{1}{2}\right)_n^2}{n!^2}x^n$

Using period relations of the elliptic curves, there are various ways to write $\frac{1}{\pi}$ in terms of $\ _2F_1(\frac{1}{2},\frac{1}{2};1;\lambda)$ . The associated supercongruence is:

$\ _2F_1(\frac{1}{2},\frac{1}{2};1;\lambda)_{p-1} \equiv a_p(\lambda) \hspace{2mm} ({\rm mod } p)$

Where $\frac{1}{\pi}$ in terms of $\ _2F_1(\frac{1}{2},\frac{1}{2};1;\lambda)_{p-1}$ is the hypergeometric series truncated at $p-1$ and $a_p(\lambda) = \# E_{\lambda}({\bf F}_p) - p - 1$ .

There is a similar construction for K3 surfaces which gives rise to various ways to write $\frac{1}{\pi^2}$ in terms of another hypergeometric series. This has an associated supercongruence mod $p$ . 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 $p^2$ and $p^3$ .

seminar/nt/201208

Ramanujan-type supercongruences

Location

Abstract