Timo Keller’s blog

August 26, 2022: My quadratic Chabauty conference

I have been back to Bayreuth to my quadratic Chabauty conference (it had to take place there because of my funding).

The teams.

We had three remote working groups, one in the US and two in Europe, working on ongoing projects. Five of us worked in person starting something new. We will continue working on our problems after the conference.

August 12, 2022: Last day of ANTS

ANTS-XV.

Here are the slides of my 25 minutes talk I gave at ANTS-XV (Fifteenth Algorithmic Number Theory Symposium), taking place at the University of Bristol this year. This is the largest international conference on this topic.

My talk.

I talked about the completion of the determination of the \(\mathbf{Q}\)-points on the Atkin-Lehner quotients \(X_0(N)^*\) which are hyperelliptic. There are exactly \(64\) of them, and they were determined by Hasegawa in 1997. Here is our paper and our Magma code with log files.

Our methods.

To do this, we used a combination of well-established (Chabauty–Coleman method in its implementation by Balakrishnan–Tuitman, elliptic curve Chabauty in its implementation by Bars–González–Xarles) and recent developments in the quadratic Chabauty method by Balakrishnan–Dogra–Müller–Tuitman–Vonk.

The quadratic Chabauty condition.

Looking at the root number, you expect that most \(X_0(N)^*\) with \(N\) square-free have \(r = g\) (\(g\) the genus of the curve and \(r\) the Mordell-Weil rank of its Jacobian \(J\)) because the corresponding space of cusp forms is \(S_2(\Gamma_0(N))^{w_d = +1 : d \mid N, (d,N/d)=1}\), so the analytic ranks of the \(L(f,s)\) are odd and in fact should conjecturally be \(1\) in most of the times. In these cases, the Chabauty–Coleman method will not be applicable (except if the \(\mathbf{Z}_p\)-rank of the closure of \(J(\mathbf{Q})\) in \(J(\mathbf{Q}_p)\) will be less than \(g\)), and you need quadratic Chabauty.

Fake residue discs.

As David Harvey asked, we usually have many fake residue discs coming from the various Chabauty methods. These need to be ruled out using the Mordell-Weil sieve. For genus \(g > 2\), one would not expect fake residue discs from quadratic Chabauty (in fact, we didn’t have this problem in our genus \(4,5,6\) paper) because there are more Chabauty functions if \(g-1 = \operatorname{rk}\, \mathrm{NS}(J) - 1 \geq 2\). If there are, there should be a geometric reason explaining them.

Ongoing work.

We (Nikola Adžaga, Shiva Chidambaram, Oana Padurariu, and me) and others will work on non-hyperelliptic \(X_0(N)^*\) and on quotients of Shimura curves on my conference in two weeks.

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 an 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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