Hypergeometric L-functions in average polynomial time, II

Abstract

We describe an algorithm for computing, for all primes p ≤ X , the trace of Frobenius at p of a hypergeometric motive over Q in time quasilinear in X. This involves computing the trace modulo p e for suitable e; as in our previous work treating the case e = 1 , we combine the Beukers–Cohen–Mellit trace formula with average polynomial time techniques of Harvey and Harvey–Sutherland. The key new ingredient for e > 1 is an expanded version of Harvey’s “generic prime” construction, making it possible to incorporate certain p-adic transcendental functions into the computation; one of these is the p-adic Gamma function, whose average polynomial time computation is an intermediate step which may be of independent interest. We also provide an implementation in Sage and discuss the remaining computational issues around tabulating hypergeometric L-series.