Hypergeometric L-functions in average polynomial time, II

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...

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Bibliographic Details
Published in:Research in number theory Vol. 11; no. 1
Main Authors: Costa, Edgar, Kedlaya, Kiran S., Roe, David
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.03.2025
Subjects:
Mathematics
Mathematics and Statistics
Number Theory
Average polynomial time algorithms
Secondary 11G09
functions
Hypergeometric motives
33C20
Primary 11Y16
11M38
11T24
ISSN:2522-0160, 2363-9555
Online Access:Get full text
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Summary: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.
ISSN:2522-0160
2363-9555
DOI:10.1007/s40993-024-00593-8