Computing zeta functions of arithmetic schemes

We present new algorithms for computing zeta functions of algebraic varieties over finite fields. In particular, let X be an arithmetic scheme (scheme of finite type over Z), and for a prime p let zeta_{X_p}(s) be the local factor of its zeta function. We present an algorithm that computes zeta_{X_p}(s) for a single prime p in time p^(1/2+o(1)), and another algorithm that computes zeta_{X_p}(s) for all primes p < N in time N (log N)^(3+o(1)). These generalise previous results of the author from hyperelliptic curves to completely arbitrary varieties.Comment: 23 pages, to appear in the Proceedings of the London Mathematical Societ

Point counting on curves using a gonality preserving lift

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

Computational tools for quadratic Chabauty

http://math.bu.edu/people/jbala/2020BalakrishnanMuellerNotes.pdfhttp://math.bu.edu/people/jbala/2020BalakrishnanMuellerNotes.pdfFirst author draf

Computational tools for quadratic Chabauty

Lecture note