Papers

The second moment of the number of integral points on elliptic curves is bounded
with Levent Alpöge

We give a new upper bound for the number of integral points on an integral short Weierstrass model of an elliptic curve depending only on its rank and the square divisors of its discriminant. This upper bound comes from understanding a bijection, first observed by Mordell, between integral points on elliptic curves and certain types of binary quartic forms, and then bounding the number of solutions to Thue equations. We also prove a more general such bound for S-integral points on elliptic curves over any number field.

We then "average" that upper bound and apply Bhargava-Shankar's result on the average size of 5-Selmer groups to show that the second moment of the number of integral points on elliptic curves over ℚ is bounded. (Alpoge has shown that the average is bounded in previous work. In fact, he shows that the r-th moment for 0 < r < log₃ 5 is bounded, and we now extend that to any 0 < r ≤ log₂ 5.) We similarly show these same moments for the number of S-integral points on elliptic curves over any fixed number field are bounded.

We use analogous ideas to bound moments for the number of integral points in families of elliptic curves with one or two marked points. In order to handle some special cases (e.g., when r = log₂ 5 above), we introduce a method to count large-weighted orbits in the geometry-of-numbers methods used in arithmetic statistics.

On average sizes of Selmer groups and ranks in families of elliptic curves having marked points
with Manjul Bhargava

We determine the average size of 2- and/or 3-Selmer groups of elliptic curves in certain families, such as the family of elliptic curves with a marked rational point or a marked 2- or 3-torsion point. We thus obtain upper bounds for the average ranks of the elliptic curves in these families. We also show that the average size of the 2-torsion subgroup of the Tate-Shafarevich group is infinite for the family of elliptic curves with a marked rational 2-torsion point.

The main idea is to apply geometry-of-numbers methods and sieve techniques to count the relevant integral orbits of the corresponding representations (using the relationship between the rational orbits for these representations and Selmer elements of the elliptic curves from the paper Coregular spaces and genus one curves discussed below).

Quadratic enrichment of the logarithmic derivative of the zeta function (arXiv version)
with Margaret Bilu, Padmavathi Srinivasan, Isabel Vogt, and Kirsten Wickelgren
Trans. Amer. Math. Soc. Ser. B 11 (2024), 1183-1225

We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck--Witt group. This enrichment is related to the topology of the real points of a lift. We show a rationality result for cellular schemes over a field, and compute several examples, including toric varieties.

Splitting Brauer classes using the universal Albanese (arXiv version has less exposition)
with Max Lieblich
Enseign. Math. 67 (2021), 209--224

We show that for any Brauer class over a field, there exists a torsor for an abelian variety over that field splitting the class; in other words, given a Brauer-Severi variety, there exists such a torsor with a morphism to the Brauer-Severi variety. In fact, for any nice curve of genus at least 1 splitting the Brauer class, the Albanese of the curve splits the class (unless the index of the class is congruent to 2 mod 4, in which case one may need to take a product with an extra genus one curve).

Everywhere local solubility for hypersurfaces in products of projective spaces (arXiv version, journal pdf)
with Tom Fisher and Jennifer Park
Res. Number Theory 7 (2021), no. 6

Poonen and Voloch conjecture that for the family of degree d hypersurfaces in ℙⁿ, the Hasse principle is satisfied (asymptotically) 100% of the time or 0% of the time when d < n + 1 and when d > n + 1, respectively (i.e., in the Fano case and in the general type case, respectively). We extend their conjecture to hypersurfaces in products of projective spaces (in a way that does not just depend on the positivity of the canonical bundle).

One case that is not covered by our conjecture is the family of bidegree (2,2) curves in ℙ¹ x ℙ¹, which are of arithmetic genus one. We compute that the proportion of (2,2) curves that are everywhere locally soluble is approximately 87.39%. (Bhargava, Cremona, and Fisher previously studied the analogous problem for plane cubics, a case not covered by the Poonen-Voloch heuristics.)

Galois closures of non-commutative rings and an application to Hermitian representations (arXiv version)
with Matthew Satriano
Int. Math. Res. Not. IMRN (2020), no. 21, 7944–7974

We define and study a general notion of Galois closure for possibly non-commutative rings, generalizing the definition for Galois closure given in Bhargava-Satriano for commutative rings (which also built on work of Grothendieck, Katz-Mazur, and Gabber). For a so-called degree n algebra A, we define the Galois closure as a quotient of the n-fold tensor product of A; while it not naturally a ring, it is a left (or right) S_n-equivariant A^⊗n-module.

We compute many examples and study basic properties its behavior under products and base change. As an application, we study "Hermitian representations" that make use of Galois closures. Our motivation comes from arithmetic invariant theory, namely to use these Hermitian representations to describe moduli spaces of interesting arithmetic or algebraic data.

Odd degree number fields with odd class number (arXiv version)
with Arul Shankar and Ila Varma
Duke Math. J. 167 (2018), no. 5, 995–1047

The Cohen-Lenstra and Cohen-Martinet heuristics give precise predictions for the distribution of ideal class groups in families of numbers fields over ℚ (with modifications by Malle to account for roots of unity in the base field). We show that for any odd n ≥ 3 and nonnegative integers (r, s) with r + 2s = n, the average number of 2-torsion elements in the ideal class group in a certain family of degree n number fields of signature (r,s) is indeed the value 1+2^1-r-s predicted by the Cohen-Martinet- Malle heuristics, conditional on an expected tail estimate for n ≥ 5 (without the tail estimate, we obtain the same number as an upper bound for the limsup).

Our number fields are precisely those arising from binary n-ic forms, and we obtain the same result whether ordering by the size of the coefficients of the binary n-ic forms or by the Julia invariant (a non-polynomial invariant of binary n-ics). We have similar results for narrow class groups, which agree with the heuristics of Dummit-Voight, and when replacing degree n fields by degree n orders arising from binary n-ic forms.

As a corollary, we find that for every odd integer n ≥ 3, there exist infinitely many number fields of degree n and associated Galois group S_n whose class number is odd.

Orbit parametrizations for K3 surfaces (arXiv version)
with Manjul Bhargava and Abhinav Kumar
Forum of Mathematics, Sigma 4 (2016), e18 (86 pages)
The arXiv version has some more hyperlinks in one of the figures, which might make the paper easier to navigate.

We study numerous orbit parametrizations of K3 surfaces with additional data, namely specified lattices contained in their Néron-Severi groups; these generalize Mukai's descriptions of moduli spaces of low degree polarized K3 surfaces as complete intersections in homogeneous spaces. We have examples where the rank of the lattice (the Picard number of the generic K3 surface in the moduli space) is as low as 2 or as high as 18. The mere existence of these parametrizations shows that these moduli spaces are unirational; they also lead to applications in dynamics, e.g., we find many families of lattice-polarized K3 surfaces with fixed-point-free positive entropy automorphisms.

Coregular spaces and genus one curves (arXiv version not up-to-date)
with Manjul Bhargava
Cambridge Journal of Mathematics 4 (2016), no. 1, 1–119

We construct 20 moduli spaces related to genus one curves (with additional data such as line bundles or vector bundles) using orbit spaces of representations of algebraic groups. All these representations are coregular, meaning that the ring of polynomial invariants is a polynomial ring, so the corresponding coarse moduli spaces are naturally open subsets of weighted projective spaces. For many cases, we give a geometric interpretation of these polynomial invariants, usually as coefficients of the Jacobians of the genus one curves; the elliptic curves that occur as these Jacobians often form special families, e.g., the family of all elliptic curves with a marked rational point. We are also able to use uniform algebraic and geometric constructions to study many of these cases together.

Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks (arXiv version)
with Jennifer Balakrishnan, Nathan Kaplan, Simon Spicer, William Stein, and James Weigandt
LMS J. Comput. Math. 19 (2016), issue A, 351–370
The raw data for this project may be accessed (soon) on the LMFDB and in CoCalc.
Slides from my talk at ANTS-XII in Kaiserslautern have some extra graphs.

We created an exhaustive database of approximately 240 million elliptic curves over ℚ, up to naive height 2.7 x 10¹⁰; for each curve E, we computed the rank, size of the 2-Selmer group and Ш(E/ℚ)[2], torsion subgroup, root number, conductor, and other invariants (the rank conditional on GRH and BSD for approximately 20% of the curves). We also computed samples of 100,000 curves at height 10^k for k = 11,...,16. We found that the average rank appears to be decreasing on a large scale (slowly) as height increases. We also noted interesting behavior by the 2-Selmer and Tate-Shafarevich groups.

Zeta functions of a class of Artin-Schreier curves with many automorphisms (arXiv version)
with Irene Bouw, Beth Malmskog, Renate Scheidler, Padmavathi Srinivasan, and Christelle Vincent
Directions in Number Theory: Proceedings of the 2014 WIN3 Workshop, Springer, 2016, pp. 87–124

We describe a class of Artin-Schreier curves with large automorphism group and compute the zeta functions; we obtain new examples of maximal curves.

How many rational points does a random curve have?
Bull. Amer. Math. Soc. 51 (2014), no. 1, 27–52
This is an expanded version of the notes in the Current Events Bulletin of the 2013 Joint Mathematics Meetings.

In this expository article for a general mathematical audience, we explain the conjectures concerning the distribution of ranks of elliptic curves and sketch the main ideas behind recent work on bounding the average rank of elliptic curves over ℚ.

Moduli of products of stable varieties (arXiv version)
with Bhargav Bhatt, Zsolt Patakfalvi, and Christian Schnell
Compositio Mathematica 149 (2013), no. 12, 2036–2070

We show that the operation of taking a product of stable varieties (as coming from the minimal model program) induces a finite étale map on moduli spaces, generalizing van Opstall's results for curves and answering a question of Abramovich.

Genus one curves and Brauer-Severi varieties (arXiv version)
with Aise Johan de Jong
Mathematical Research Letters 19 (2012), no. 06, 1357–1359

We explicitly construct genus one curves mapping to Brauer-Severi varieties of dimension at most 4, answering a question of P. Clark. (It is not yet known whether there exists a genus one curve with a morphism to an arbitrary higher-dimensional Brauer-Severi variety.)

Orbit Parametrizations of Curves (Ph.D. thesis), also available on ProQuest.

We construct various moduli spaces of curves with specified line bundles using orbits of representations. In the first several chapters, the curves have genus one (these are some of the moduli spaces studied in the paper Coregular spaces and genus one curves described above). In later chapters, we study higher genus curves in ℙ² and ℙ¹ × ℙ¹. The last chapter focuses on genus zero curves, describing the relationship between ternary quadratic forms, quaternion algebras, and Brauer-Severi varieties over arbitrary bases.

The m-step, same-step, and any-step competition graphs
Discrete Appl. Math. 152 (2005), no. 1–3, 159–175

Cho, Kim, and Nam introduced the m-step competition graph and computed the 2-step competition numbers of paths and cycles. We extend their results in a partial determination of the m-step competition numbers of paths and cycles. In addition, we introduce two new variants of competition graphs: same-step and any-step. We classify same-step and any-step competition graphs and investigate their related competition numbers.

Table of contents

Projectivity of the moduli of curves, by Raymond Cheng, Carl Lian, and Takumi Murayama
The stack of admissible covers is algebraic, by Elsa Corniani, Neeraj Deshmukh, Brett Nasserden, Emanuel Reinecke, Nawaz Sultani, and Rachel Webb
Projectivity of the moduli space of vector bundles on a curve, by Jarod Alper, Pieter Belmans, Daniel Bragg, Jason Liang, and Tuomas Tajakka
Boundedness of semistable sheaves, by Haoyang Guo, Sanal Shivaprasad, Dylan Spence, and Yueqiao Wu
Theorem of the Base, by Raymond Cheng, Lena Ji, Matt Larson, and Noah Olander
Weil restriction for schemes and beyond, by Lena Ji, Shizhang Li, Patrick McFaddin, Drew Moore, and Matthew Stevenson
Heights over finitely generated fields, by Stephen McKean, and Soumya Sankar
An explicit self-duality, by Nikolas Kuhn, Devlin Mallory, Vaidehee Thatte, and Kirsten Wickelgren
Tannakian reconstruction of coalgebroids, by Yifei Zhao

Table of contents

From Fontaine–Mazur Conjecture to Analytic Pro-p-groups: A Survey, by Ramla Abdellatif, Supriya Pisolkar, Marine Rougnant, Lara Thomas
Orientations and Cycles in Supersingular Isogeny Graphs, by Sarah Arpin, Mingjie Chen, Kristin E. Lauter, Renate Scheidler, Katherine E. Stange, Ha T. N. Tran
Generalized Ramanujan–Sato Series Arising from Modular Forms, by Angelica Babei, Lea Beneish, Manami Roy, Holly Swisher, Bella Tobin, Fang-Ting Tu
Mock Theta Functions and Related Combinatorics, by Cristina Ballantine, Hannah Burson, Amanda Folsom, Chi-Yun Hsu, Isabella Negrini, Boya Wen
Transcendental Lattices of Certain Singular K3 Surfaces, by M. J. Bertin, O. Lecacheux
Power-Saving Error Terms for the Number of D₄-Quartic Extensions over a Number Field Ordered by Discriminant, by Alina Bucur, Alexandra Florea, Allechar Serrano López, Ila Varma
Dynamical Mahler Measure: A Survey and Some Recent Results, by Annie Carter, Matilde Lalín, Michelle Manes, Alison Beth Miller
Geometric Decomposition of Abelian Varieties of Order 1, by Toren D’Nelly-Warady, Kiran S. Kedlaya
On Markoff Type Surfaces over Number Fields and the Arithmetic of Markoff Numbers, by Seoyoung Kim, Damaris Schindler, Jyothsnaa Sivaraman
p-Adic Measures for Reciprocals of L-Functions of Totally Real Number Fields, by Razan Taha