July 30, 2022: First blog entry, PCMI

This is the first entry of my blog started as announced in my Marie Skłodowska-Curie fellowship starting in June 2023 with Steffen Müller in Groningen.

PCMI: collaboration. It is the end of the second week at PCMI “Number Theory Informed by Computation” organized by the Princeton Institute of Advanced Study, which I enjoy very much. I could continue many collaborations and start new ones, for example with Ross Paterson and Carlo Pagano. When there are preprints available at arXiv, I will write a blog post here.

PCMI: my talk. Here are the slides of my 25 minutes talk I gave in the Research Program. The talk was about the exact verification of the strong Birch–Swinnerton-Dyer (BSD) conjecture for some abelian surfaces with real multiplication over $\mathbf{Q}$ . There were several interesting questions by Levent Alpöge, Sam Frengley, Bjorn Poonen, and Akshay Venkatesh. For example, I hope I can prove strong BSD for an example of a Jacobian of a genus $2$ curve of level $3200$ provided by Sam, which has rank $0$ and $7$ -torsion in the Shafarevich–Tate group. (All examples I computed had order of Sha equal to $1,2,4$ .) Akshay suggested to formulate a refinement of an equivariant BSD formula. I did so over Heegner fields and I will check whether this formula holds in my examples. The question of Levent was about the Galois representations, and Bjorn asked for a heuristic explanation why the Shafarevich–Tate group is trivial for all Atkin-Lehner quotients with Jacobian absolutely simple and modular of dimension $2$ .

ANTS-XV. After the third week, I’m at ANTS-XV, where I will give a talk about our paper on the determination of the $\mathbf{Q}$ -points on hyperelliptic Atkin-Lehner quotients of modular curves accepted there.

Future blog entries. As the audience of this blog is also the interested mathematical public, I will post some introductory entries on curves and abelian varieties besides the research topics.

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