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 $1$ most of the time. 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.