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.
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.
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
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) Sn-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.
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.
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.
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
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.
Orbit Parametrizations of Curves (Ph.D. thesis), also available on ProQuest.
The m-step, same-step, and any-step competition graphs
Discrete Appl. Math. 152 (2005), no. 1–3, 159–175
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.