ANTS XVII in Groningen next week!

One the best conference series in computational number theory (including many L-function-related themes) is the biannual ANTS series. The 2024 edition was held at MIT, and next week ANTS XVII is happening in Groningen. Personally, I haven't been to a real math conference for years, but this time I'll attend ANTS, and I'm super-excited! So if you happen to read this and you're attending in Groningen, find me and say hello!

For this post, I want to highlight some interesting papers from the upcoming conference, as an invitation to browse and explore both this year's program and the program from 2024.

Banwait and Huang: On the identification of elliptic curves that admit infinitely many twists satisfying the Birch–Swinnerton-Dyer conjecture. This paper has a nice introduction to the current status and some recent advances on the BSD conjecture. For any elliptic curve L-function, one can obtain a new L-function by twisting with a Dirichlet character, which in RH Saga language means taking the tensor product with one of our friends $L_D$. This gives an infinite family of L-functions, one for each fundamental discriminant $D$. Like many ANTS papers, this paper has a good mix of theory and computation, and one highlight is a numerical investigation of a prediction from random matrix theory on the size of the order of Sha (the Tate-Shafarevich group, a key invariant both for the BSD conjecture and the abc conjecture).

Batubara, Garzella, Huang, Mellberg: Newton strata realization for hypersurfaces via explicit p-adic cohomology. If you recall the Weil Conjectures story from Episode 11, you may remember the idea of cohomology groups and the Frobenius endomorphism, which can be thought of as just a matrix. The classical theory developed by Grothendieck and his school used $\ell$-adic cohomology, but in this context it is virtually impossible to compute an actual matrix that represents Frobenius. An important method in the computation of zeta functions over finite field is to replace $\ell$-adic cohomology by something called $p$-adic cohomology, where such a computation is still not easy, but far more accessible, using techniques pioneered by Kiran Kedlaya. Learning to perform such computations could be a concrete and computational entry point to learning about the otherwise rather abstract theme of cohomology theories in general. Anyway, the paper linked above has a nice introduction and gives some examples of cutting-edge computations of this type.

Finally, not directly related to the RH Saga, but perhaps a fun thing to see: Elsenhals has a paper on Numerical verification of the Collatz conjecture for billion-digit numbers!