Timo Keller’s blog

May 14, 2023: New preprint on quadratic points on \(X_0(N)\)

The preprint Computing Quadratic Points on Modular Curves \(X_0(N)\) is joint work with Nikola Adžaga, Philippe Michaud-Jacobs, Filip Najman, Ekin Ozman, and Borna Vukorepa. We started to work on this at the 2022 MIT Modular curves workshop, whose aim was to extend the LMFDB with data on modular curves.

The \(\mathbf{Q}\)-rational points on \(X_1(N)\) and \(X_0(N)\) for all \(N\) have been computed by Mazur and Kenku. For the modular curve \(X_1(N)\) classifying (generalized) elliptic curves with a point of order \(N\), there is a bound \(N(d)\) by Merel and Kamienny such that \(X_1(N)(K)\) contains only cusps if \(N > N(d)\) and \(K\) runs through all number fields of degree \(d\). This has been used to compute all degree \(d\) points for \(d \leq 7\) by combined work of Kamienny–Kenku–Momose (\(d=2\)), Derickx–Etropolski–van Hoeij–Morrow–Zureick-Brown (\(d = 3\)), and for \(4\leq d\leq 7\) and prime values of \(N\) by Derickx–Kamienny–Stein–Stoll.

However, there is no \(d > 1\) such that one knows all degree-\(d\) points on \(X_0(N)\) (classifying elliptic curves with a cyclic \(N\)-isogeny) for all \(N\)! Conjecturally, the only quadratic points are cusps and CM points for \(N \gg 0\).

In our preprint, we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves \(X_0(N)\) of genus up to \(8\), and genus up to \(10\) with \(N\) prime, for which they were previously unknown. This extends previous work by Bruin–Najman, Ozman–Siksek, Box, and Najman–Vukorepa. Our methods apply to more than these curves; we restricted to these to make the computations feasible.

We use the following three methods:

  1. Going down: If \(f\colon X \to Y\) is a finite morphism of curves and \(Y(K)\) is known and finite, one can compute \(X(K)\) by taking fibers. Problems arise if there are infinitely many quadratic points on \(Y\). We use this for \(N \in \{58, 68, 76 \}\).

  2. Rank \(0\): If \(J_0(N)(\mathbf{Q}) = J_0(N)(\mathbf{Q})_\mathrm{tors}\) is known (we verify the generalized Ogg conjecture in several cases), one can compute the quadratic points on \(X_0(N)\) using a variant of the Mordell–Weil sieve. We use this for \(N \in \{ 80, 98, 100 \}\).

  3. Atkin–Lehner sieve: This is the most involved method and a variant of the Mordell–Weil sieve, which, if applicable, reduces the problem to considering fixed points of an Atkin–Lehner involution and the rational points on a given Atkin–Lehner quotient. We use this for \(N \in \{ 74, 85, 97, 103, 107, 109, 113, 121, 127 \}\).

We also give optimized algorithms to compute models of Atkin–Lehner quotients of \(X_0(N)\) and the \(j\)-invariant morphism.

Note that there can be infinitely many quadratic points on a curve \(X\); this happens if and only if \(X\) is hyperelliptic (plug in infinitely many values for \(x\) in \(y^2 = f(x)\)) or bielliptic with elliptic curve of rank \(> 0\).

We get evidence for the fact that non-cuspidal, non-CM points are rare.

Update (2023-06-22).

Here are my slides for a talk at a conference in Dubrovnik.

Update (2023-08-23).

The article has been accepted for publication in Mathematics of Computation.

February 13, 2023: Two articles accepted

Recently, two articles have been accepted:

Quadratic Chabauty for modular curve quotients.

In Quadratic Chabauty for Atkin-Lehner Quotients of Modular Curves of Prime Level and Genus 4, 5, 6 (with Nikola Adžaga, Vishal Arul, Lea Beneish, Mingjie Chen, Shiva Chidambaram, and Boya Wen), accepted for publication in Acta arithmetica, we apply the quadratic Chabauty method to determine all \(\mathbf{Q}\)-rational points of Atkin–Lehner quotients \(X_0(p)^+\) of prime level \(p\) such that the genus is in \(\{4,5,6\}\).

Since the non-cuspidal points of \(X_0(p)\) correspond to elliptic curves with a \(p\)-isogeny and since the Atkin–Lehner (or Fricke) involution \(w_p\) maps such an isogeny to its dual, our result classifies elliptic curves with an unordered pair of \(p\)-isogenies. Computing the points on \(X_0(p)^+ := X_0(p)/w_p\) has been called an “extremely interesting arithmetic question” by Mazur in his seminal Eisenstein paper. For composite levels \(N\), one can mod out (potentially) more Atkin–Lehner involutions and go down to the quotient \(X_0(N)^*\). The \(\mathbf{Q}\)-points on the hyperelliptic \(X_0(N)^*\) have been determined in our ANTS paper, see my blog post from August 12, 2022 below.

Our article proves a conjecture of Galbraith for those curves. The main difficulty in computing the set of rational points was to find suitable plane models of those modular curves such that the existing implementation of the Quadratic Chabauty algorithm determined the set of rational points. It was surprising to the experts that we could go up to (relatively high) genus \(6\). This article originated from a project started at the 2020 Arizona Winter School.

Those of smaller genus have been tackled before, for example in the article Quadratic Chabauty for modular curves: Algorithms and examples by Balakrishnan–Dogra–Müller–Tuitman–Vonk. Those of composite level and genus \(\leq 6\) are known known by work of Momose with \(X_0(125)^+\) being solved in Arul–Müller.

Specialization of Mordell–Weil groups.

In Specialization of Mordell-Weil ranks of abelian schemes over surfaces to curves, accepted for publication in International Journal of Number Theory, I prove the following surjectivity result for specialization morphisms: Recall that Silverman’s specialization theorem (with generalizations by Wazir) roughly says that for an abelian variety \(A\) over a function field \(K\) over a global field, outside a set of bounded height, the specialization theorem from the Mordell–Weil group \(A(K)\) to the Mordell–Weil group to closed points is injective modulo torsion, i.e., the rank does not drop.

It is a challenging question whether there exist infinitely many specializations that the rank does also not jump! For example, when applied to a family of elliptic curves over \(\mathbf{Q}(T)\) with generic rank \(r\) (such families exist for small \(r\)), a positive answer to this question would imply that there are infinitely many non-isomorphic elliptic curves over \(\mathbf{Q}\) with rank exactly equal to \(r\). The latter is known for \(r = 0,1\) (explicit constructions or the Gross–Zagier formula combined with Kolyvagin’s Heegner point Euler system), but open already for \(r = 2\).

My article does not solve this question, but shows that for an abelian scheme over a surface, infinitely many specializations to curves have the same rank.

The proof uses a recent result of Ambrosi on a specialization theorem of Néron–Severi ranks and the Shioda–Tate formula to go from Néron–Severi ranks to Mordell–Weil ranks. The result holds more generally over infinite finitely generated fields.

My original motivation to prove such a result was to prove the reduction of the analog of the Birch–Swinnerton-Dyer conjecture over higher-dimensional bases to that over curves, with the reduction to surfaces as a basis established in my Documenta article and the reduction to curves by my recent article.

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


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 \(2\).


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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