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 $\{4,5,6\}$ .

Why this matters. 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 by work of Momose with $X_0(125)^+$ being solved in Arul–Müller.

Specialization of Mordell–Weil groups. The rank records how many independent rational points an elliptic curve (or, more generally, an abelian variety) has; controlling how it varies in families is a central and largely open problem. 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 map from the Mordell–Weil group $A(K)$ to the Mordell–Weil groups of 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.