Andy Booker

L-functions from nothing

Long before the proof of the modularity theorem, evidence for it was amassed in the 1972 "Antwerp IV" tables comparing elliptic curves and modular forms of small conductor; in particular, D. J. Tingley's calculation of spaces of modular forms was essential for filling some gaps in the elliptic curve table, and gave a de facto complete list of all elliptic curves of conductor up to 200. In the decades that followed this list was extended by John Cremona and is now a major part of the L-functions and Modular Forms Database (LMFDB). Elliptic curves are abelian varieties of dimension 1. In recent decades similar efforts have been undertaken in dimension 2, including a 2015 tabulation of genus 2 curves corresponding to abelian surfaces that can be found in the LMFDB. But we lack an analogue of Tingley's table of modular forms, and with it any proof of completeness or evidence of gaps. Directly computing the relevant spaces of modular forms is prohibitively difficult, even for small conductors. In the talk I will describe joint work with Andrew Sutherland extending a method of Farmer, Koutsoliotas, and Lemurell that makes it possible to compute these spaces indirectly. The result is a provably complete tabulation of L-functions of modular forms of conductor up to 1500, and partial results for some conductors as large as 3125. For instance, we obtain a complete classification of modular abelian surfaces with good reduction away from 7.