Joint PU/IAS Number Theory
Eigenvalues of P-Adic Random Matrices and Zeroes of P-Adic L-Functions
Ellenberg-Jain-Venkatesh (2011) proposed a p-adic version of the Katz-Sarnak philosophy for usual L-functions: the statistics of zeroes of p-adic L-functions, averaged over families, should agree with the limiting eigenvalue statistics of p-adic random matrices. This was born out by numerical data on the number of zeroes in the unit disc, but many basic statistics, such as correlations between pairs of eigenvalues, had not been computed on the random matrix side.
This talk will explain recent results on eigenvalues of p-adic random matrices, including a surprising Jacobi theta function formula for the pair correlation, and the precise conjectures on zeroes of p-adic L-functions which come from these (joint with J. Shen, https://arxiv.org/abs/2601.06283). If time permits I will also mention ongoing numerical work on the L-function side supporting these conjectures (joint with J. Shen and H. Knospe) .