2025 Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation Annual Meeting

This meeting had three goals: To take stock of the achievements over the last seven years, to highlight specific accomplishments of early-career researchers involved with the collaboration, and to suggest avenues for future research and development.

Joseph Silverman reviewed quantitative measures of the number of rational points of bounded height, including a proposed trichotomy: abundant rational points (as seen on projective space), logarithmically-growing rational (for abelian varieties), and sparse rational points (as expected on varieties of general type based on the Bombieri–Lang conjecture). Silverman discussed work with Hector Pasten extending this framework to orbits in arithmetic dynamics and presented examples — morphisms on projective space and K3 surfaces — supporting this perspective.

Jennifer Balakrishnan presented a survey of the Chabauty–Coleman method along with quadratic extensions due to Kim and others. She discussed both the technical tools — p-adic heights and integration, correspondences — as well as the motivating questions on rational points of modular curves. Extensions to more general number fields and computational tools for large-scale surveys — of curves from the L-functions and Modular Forms Database (LMFDB) — were also addressed.

Noam Elkies presented recent work with Zev Klagsbrun: The first example of an elliptic curve over Q with rank at least 29. Elkies presented the history of the problem over the last century, in the context of characterizing all possible groups of rational points, with the torsion case resolved by Mazur in 1978. Elliptic surfaces over the projective line with large rank — especially K3 surfaces — play an important role. He discussed how promising candidates might be found and the analysis of their fibers.

Bjorn Poonen discussed two topics in sequence. The first was to characterize the integers arising as the number of rational points on an abelian variety of dimension d over a finite field of q elements. This is joint work with van Bommel, Costa, Li, and Smith. He reviewed the history of the Weil bounds and their refinements and focused on identifying long intervals in which all such integers arise. The techniques are used in Smith’s recent work on the Schur–Siegel–Smyth trace problem. The second part focused on the classification of tetrahedra in R3 with rational angles and generalizations to larger sets of vectors (with Kedlaya, Kolpakov, and Rubinstein).

Sam Schiavone reported on new progress for the inverse Galois problem: How to realize a given finite group as the Galois group of an extension of Q? Until recently, the transitive permutation group 17T7 was the simplest open case. Once we have a candidate polynomial, computing the Galois group is straightforward with computer algebra systems. But finding a suitable candidate involves an intricate analysis with Hilbert modular forms.

John Voight presented his high-level view on automorphic forms from the perspective of the Langlands program, explaining how this informs the global architecture of the LMFDB, both in its presentation of various flavors of modular forms and the interconnections among varieties with modularity properties.

Early-career researchers gave presentations of several projects directly connected to the Collaboration:

• Santiago Arango-Piñeros, Galois Groups of Low-Dimensional Abelian Varieties over Finite Fields: new developments on angle rank for abelian varieties over finite fields
• Kate Finnerty, Quadratic Points on Modular Curves in the LMFDB: computation of bi-elliptic modular curves with rank 1 and 2 over Q
• Sachi Hashimoto, Rational Points on X0(N)∗ when N is Non-Squarefree: computations extended beyond hyperelliptic examples
• David Lowry-Duda, Rigorous Maass Forms: ingredients for the first non-heuristic computation of Maass forms in the LMFDB
• Asimina Hamakiotes, Families of modular curves in the LMFDB: new genus formulas for modular curves

The final talk was a joint presentation by Andrew Sutherland and Brendan Hassett. Sutherland spoke on successes and future challenges in the arithmetic of genus-two curves, taking Poonen’s 1996 survey Computational aspects of curves of genus at least 2, laying out questions for future research, as the point of departure. While many problems have been solved, several remain the topic of current research. A simple but surprisingly difficult example is the characterization of genus-two curves over Q whose Jacobian has good reduction away from 2; more generally, one would like to enumerate all curves up to a given conductor bound. Hassett discussed a problem on K3 surfaces building on work of Kedlaya–Sutherland, Taelman, and Ito: When does a Weil polynomial arise from the cohomology of a K3 surface — or related variety — over a finite field? Recent work of Auel–Kulkarni–Petok– Weinbaum points to the utility of higher-dimensional hyperkähler manifolds for this realization problem.

The Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation has supported dozens of researchers and hundreds of publications. It has catalyzed dramatic increases in the public availability of arithmetic data, opening the door to a new age of data-driven research in the field. At the same time, this rapid progress leaves in stark relief fundamental mathematical questions that will motivate research over the coming decades.

Jennifer Balakrishnan
Boston University

Quadratic Chabauty for Modular Curves
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By Faltings’ theorem, the set of rational points on a curve of genus 2 or more is finite. We give a survey of the quadratic Chabauty method, which is used to determine a finite superset of the set of rational points for certain curves of genus 2 or more. In particular, we discuss what aspects of quadratic Chabauty can be made practical for certain modular curves and highlight several examples.

This is based on joint work with Alexander Betts, Netan Dogra, Daniel Hast, Aashraya Jha, Steffen Müller, Jan Tuitman, and Jan Vonk.
 

Noam Elkies
Harvard University

Elliptic Curves $E/{\bf Q}$ of High Rank
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Mordell (1922) proved that the rational points of an elliptic curve $E / {\bf Q}$ form a finitely-generated abelian group. It is still not known which finitely-generated abelian groups can occur as $E({\bf Q})$. Mazur (1977) proved that the possible torsion subgroups $T$ are the cyclic groups of order $1, 2, \ldots, 10$, and $12$, and the products of cyclic groups of orders $2$ and $2k$ with $k=1,2,3,4$. For each of these fifteen $T$, it is still an open problem which ranks occur.

For small $T$, the current records all come from elliptic fibrations of K3 surfaces; the most recent such record is $29$ for $|T| = 1$, found in August 2024 and giving the first improvement since 2006 for curves with trivial torsion (and indeed for curves with arbitrary torsion structure). We describe how we find elliptic K3’s over $\bf Q$ whose Mordell-Weil rank is as high as possible given the torsion subgroup, and how we search for fibers of even higher rank on such a surface.

This joint work with Zev Klagsbrun.
 

Asimina Hamakiotes
University of Connecticut

Families of Modular Curves in the LMFDB
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For each open subgroup $H\leq GL_2(\widehat{\mathbb{Z}})$, there is a modular curve $X_H$, defined as a quotient of the full modular curve $X(N)$, where $N$ is the level of $H$. The general genus formula of a modular curve is well known for $X_0(N)$, $X_1(N)$, $X(N)$, $X_{\mathrm{sp}}(N)$, $X_{\mathrm{ns}}(N)$, and $X_{S_4}(p)$ for $p$ prime; however, the genus formulas for $X_{\mathrm{sp}}^+(N)$, $X_{\mathrm{ns}}^+(N)$, $X_{\text{arith}}(N)$, and $X_{\text{arith},1}(N)$ cannot be found in the literature. In this talk, we share the genus formulas for these families of modular curves and will showcase their addition to the LMFDB.

Brendan Hassett
Brown University

K3 Surfaces, Computation, and Future Challenges

The last decade has seen breakthroughs in the geometry of K3 surfaces, including results on moduli spaces from the standpoint of Shimura varieties, criteria for good reduction, derived equivalence, and the Kuga-Satake construction. We explore implications of these ideas for computational and effective results.
 

Bjorn Poonen
MIT

Abelian Varieties and Tetrahedra
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This talk will cover two unrelated topics. The first is constructing abelian varieties over finite fields realizing nearly every possible order in the allowable range (joint work with Raymond van Bommel, Edgar Costa, Wanlin Li, and Alexander Smith). The second is classifying tetrahedra satisfying various geometric conditions, such as tiling space, being scissors-congruent to a cube, or having rational dihedral angles: some of these lead to problems of a familiar type, such as solving a multivariable polynomial equation in roots of unity, but others lead to unsolved unlikely intersection problems in search of a framework

This is joint work with Kiran Kedlaya, Alexander Kolpakov, and Michael Rubinstein, with further work by MIT undergraduates Abdellatif Anas Chentouf and Yihang Sun.
 

Joseph H. Silverman
Brown University

Height Density and Dynamical Propagation of Rational Points on Varieties
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We start with a 30-year old trichotomy conjecture for the height density of rational points on varieties. We will explain how this conjecture suggests an orbit propagation principle which says that if the rational points are Zariski dense, then the orbits under an endomorphism are widely spaced. As time permits, we will discuss proofs of various versions of the principle for projective spaces, abelian varieties, and surfaces. (Joint work with Hector Pasten)
 

Sam Schiavone
Massachusetts Institute of Technology

Explicit Inverse Galois Theory via Hilbert Modular Forms
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We construct an explicit polynomial realizing the transitive permutation group 17T7 as a Galois group over the rationals. Our construction uses the 2-torsion of particular abelian fourfolds with real multiplication. We compute such fourfolds using the Eichler-Shimura construction for Hilbert modular forms.
 

Andrew Sutherland
MIT

Computational Aspects of Curves of Genus at Least 2: A Retrospective
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At the second Algorithmic Number Theory Symposium in 1996, Poonen proposed a dozen “reasonable projects for the near future?” related to computational aspects of curves of genus at least 2. In this talk, Andrew Sutherland will survey the progress that has been made on these projects over the last twenty-eight years, including by members of the Simons Collaboration in Arithmetic Geometry, Number Theory, and Computation. Sutherland will then discuss what remains to be done (some of which might not be reasonable), as well as new projects for the near future that might be reasonable.
 

John Voight
University of Sydney

Computing Modular Forms and the LMFDB: Past, Present, and Future
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Computing modular forms remains a topic of central importance in number theory and arithmetic geometry—both in theory and in practice. In this talk, we survey this topic: after a brief look at the past, we discuss the current state-of-the-art methods and their limitations and then provide some possible future directions. Looking ahead, we consider how collaborative efforts can expand resources like the LMFDB to deepen our explicit understanding of modular forms in the context of the Langlands program.

Santiago Arango-Piñeros
Emory University

Galois Groups of Low-Dimensional Abelian Varieties over Finite Fields
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In joint work with Sam Frengley and Sameera Vemulapalli, we consider three isogeny invariants of abelian varieties over finite fields: the Galois group, Newton polygon, and the angle rank. Motivated by work of Dupuy, Kedlaya, and Zureick-Brown, we define a new invariant called the \emph{weighted permutation representation} which encompasses all three of these invariants and use it to study the subtle relationships between them. We use this permutation representation to classify the triples of invariants that occur for abelian surfaces and simple abelian threefolds. (https://arxiv.org/abs/2412.03358)
 

Kate Finnerty
Boston University

Quadratic Points on Modular Curves in the LMFDB
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We describe a computation of quadratic Chabauty sets of genus 2 bielliptic modular curves over $\mathbb{Q}$ of ranks 1 and 2 in the LMFDB, building on work of Balakrishnan–Dogra and Bianchi–Padurariu. The analysis produces surprising examples of points over number fields on these curves, including quadratic points on $X_{ns}^+(15)$. This talk will describe the analysis and briefly describe the results.
 

Sachi Hashimoto
Brown University

Rational Points on X₀(N)^* when N is Non-Squarefree
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The rational points of the modular curve $X_0(N)$ classify pairs $(E,C_N)$ of elliptic curves over $Q$ together with a rational cyclic subgroup of order $N$. The curve $X_0(N)^*$ is the quotient of $X_0(N)$ by the full group of Atkin-Lehner involutions. Elkies showed that the rational points on this curve classify elliptic curves over the algebraic closure of $Q$ that are isogenous to their Galois conjugates. In ongoing joint work with Timo Keller and Samuel Le Fourn, we study the rational points on the family $X_0(N)^*$ for $N$ non-squarefree. In particular we report on an integrality result for $X_0(N)^*$.
 
 

David Lowry-Duda
Institute of Computational and Experimental Research in Mathematics

Rigorous Maass Forms
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Maass forms are non-holomorphic modular forms that are eigenfunctions of the Laplace Beltrami operator. Despite being building blocks for all classical modular forms, they’re challenging to compute. In this lightning talk, we’ll describe past, present, and future efforts towards collections of rigorous Maass forms.
 

Asimina Hamakiotes
University of Connecticut

Abelian Extensions Arising from Elliptic Curves with Complex Multiplication

Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f \geq 1$. Let $E$ be an elliptic curve with complex multiplication by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j(E))$, where $j(E)$ is the $j$-invariant of $E$. Let $N\geq 2$ be an integer. The extension $\mathbb{Q}(j(E), E[N])/\mathbb{Q}(j(E))$ is usually not abelian; it is only abelian for $N=2,3$, and $4$. Let $p$ be a prime and let $n\geq 1$ be an integer. In this talk, we will classify the maximal abelian extension contained in $\mathbb{Q}(E[p^n])/\mathbb{Q}$.