Papers, preprints, and works in preperation

Ranks and Selmer groups of elliptic and hyperelliptic curves

1. Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves
   with M. Bhargava, published in the Annals of Mathematics.
2. Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0
   with M. Bhargava, published in the Annals of Mathematics.
3. The average number of elements in the 4-Selmer groups of elliptic curves is 7
   with M. Bhargava.
4. The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
   with M. Bhargava.
5. Average size of the 2-Selmer group of Jacobians of monic even hyperelliptic curves
   with X. Wang, published in Compositio Mathematica.
6. The second moment of the 2-Selmer groups of elliptic curves
   with M. Bhargava and A. A. Swaminathan.

Expanding and refining geometry-of-numbers methods

1. On the Davenport-Heilbronn theorems and second order terms
   with M. Bhargava and J. Tsimerman, published in Inventiones Mathematicae.
2. Counting S₅ fields with a power saving error term
   with J. Tsimerman, published in the Forum of Mathematics, Sigma.
3. Geometry of numbers methods over global fields I: Prehomogeneous vector spaces
   with M. Bhargava and X. Wang.
4. Geometry of numbers methods over global fields II: Coregular vector spaces
   with M. Bhargava and X. Wang, preprint available on request.
5. Geometry of numbers methods in the cusp with applications to class groups
   with A. Siad, A. A. Swaminathan, and I. Varma.

Torsion in class groups of number fields

1. Odd degree number fields with odd class number
   with W. Ho and I. Varma, published in Duke Math. Journal.
2. Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves
   with M. Bhargava, T. Taniguchi, F. Thorne, J. Tsimerman, and Y. Zhao, published in the Journal of the American Mathematical Society.
3. The mean number of 2-torsion elements in the class groups of n-monogenized cubic fields
   with M. Bhargava and J. Hanke.
4. Counting integral points on symmetric varieties with applications to arithmetic statistics
   with A. Siad and A. A. Swaminathan.
5. Nontrivial bounds on 2-, 3-, 4-, and 5-torsion in class groups of number fields, conditional on standard L-function conjectures
   with J. Tsimerman.

Asymptotics of families of number fields

1. The number of quartic D4-fields ordered by conductor
   with A. Altug, and K. Wilson, and and I. Varma, published in the Journal of the European Mathematical Society.
2. Heuristics for the asymptotics of the number of S_n-number fields
   with J. Tsimerman, published in the Journal of the London Mathematical Society.
3. An improvement on Schmidt's bound on the number of number fields of bounded discriminant and small degree
   with M. Bhargava and X. Wang, published in Forum of Math Sigma.
4. Asymptotics for the number of cubic fields ordered by general invariants
   with F. Thorne.
5. Malle's conjecture for octic D4 fields
   with I. Varma, in preparation.

Uniformity estimates and squarefree sieves in arithmetic statistics

1. Squarefree values of polynomial discriminants I
   with M. Bhargava and X. Wang, published in Inventiones Mathematicae.
2. Squarefree values of polynomial discriminants II
   with M. Bhargava and X. Wang.
3. Large families of elliptic curves ordered by conductor
   with A. N. Shankar and X. Wang, published in Compositio Mathematica.

Families of L-functions

1. Sato-Tate equidistribution of certain families of Artin L-functions
   with A. Södergren and N. Templier, published in Forum of Math. Sigma.
2. Central values of zeta functions of non-Galois cubic fields
   with A. Södergren and N. Templier.

Reductions of K3 surfaces and abelian surfaces modulo primes

1. Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields
   with A. N. Shankar, Y. Tang, and S. Tayou, published in Forum of Math Pi.
2. Abelian surfaces over finite fields with prescribed groups
   with C. David, D. Garton, Z. Scherr, E. Smith, L. Thompson, published in the Bulletin of the London Mathematical Society.

Current courses

• I am teaching MAT301H5F: Groups and Symmetries and MAT302H5F: Introduction to Algebraic Cryptography in Fall 2023. The course websites are on Quercus.