Point counting on curves using a gonality preserving lift

Abstract

We study the problem of lifting curves from finite fields to number fields in a genus and gonality preserving way. More precisely, we sketch how this can be done efficiently for curves of gonality at most four, with an in-depth treatment of curves of genus at most five over finite fields of odd characteristic, including an implementation in Magma. We then use such a lift as input to an algorithm due to the second author for computing zeta functions of curves over finite fields using p-adic cohomology.

Counting points on curves using a map to P1, II

“…In some (quite general) special cases we can almost always find a suitable lift, for example for curves of genus at most 5 and most nondegenerate, trigonal or tetragonal curves [7]. Finally, the lifting problem can also be circumvented by starting from a curve that is already defined over a number field, which is still very interesting from the point of view of computing zeta functions.…”

Lifting low-gonal curves for use in Tuitman's algorithm

“…Expecting our curve to be given in the above form is essentially equivalent to assuming knowledge of a simply branched F q -rational degree d ≤ 5 morphism C → P 1 . This contrasts with [8], but for most practical applications this seems not much of a restriction.…”

“…A partial approach to lifting curves having gonality at most four was sketched in [8], with concrete details being limited to curves of genus five. In the current paper we present a different method, which is faster, is more rigorous, works for curves of gonality at most five, and is easier to implement.…”

“…All subsequent arithmetic in Tuitman's algorithm is done in the p-adic completion Z q of O K , up to some finite p-adic precision. But for the lifting step it suffices to work over O K ; this has some implementation-technical advantages, pointed out in [8,Rmk. 2].…”

Sato-Tate groups of abelian threefolds

“…(As a reminder, it remains unclear whether this holds even in dimension 3.) Consequently, some new work would be needed in order to compute the L-functions of sufficiently many abelian fourfolds to have a reasonable chance of finding examples of all possible groups; reasonably good algorithms and implementations now exist for general curves [Tui16,Tui18,CT18], but applying these to an abelian variety would require relating that variety to a Jacobian in some fashion that does not blow up the dimension too much (e.g., using the Prym construction).…”