**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).

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

**Everywhere local solubility for hypersurfaces in products of projective spaces**

with Tom Fisher and Jennifer Park

Poonen and Voloch conjecture that for the family of degree d hypersurfaces in ℙ^{n}, 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 ℙ^{1} x ℙ^{1},
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.)

One case that is not covered by our conjecture is the family of bidegree (2,2) curves in ℙ

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 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 s-th moment
for 0 < s < log_{3} 5 is bounded, and we now extend that to any 0 < s < log_{2} 5.)

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).

**Galois closures of non-commutative rings and an application to Hermitian representations** (arXiv version)

with Matthew Satriano

to appear in *Int. Math. Res. Not. IMRN*

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.

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

**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 here.

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 10^{10}; 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 ℙ^{2} and ℙ^{1} × ℙ^{1}. 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.

Haiyan Fan, Wei Ho, Scott A. Reid, A time-of-flight mass spectrometric study of laser fluence dependencies in SnO2 ablation: implications for pulsed laser deposited tin oxide thin films, Int. J. Mass Spectrometry 230 (2003), 11–17.

Scott A. Reid, Wei Ho, and F. J. Lamelas, Pulsed Laser Ablation of Sn and SnO2 target: Neutral Composition, Energetics, and Wavelength Dependence, J. Phys. Chem. B 104 (2000), no. 22, 5324–5330.