Arul Shankar
I am an associate professor in Mathematics at
the University of
Toronto. My research is in Number Theory, specifically, in
Arithmetic Statistics and surrounding areas. You can contact me at
"arul dot shnkr at gmail dot com".
Papers, preprints, and works in preperation
Ranks and Selmer groups of elliptic and hyperelliptic curves

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.

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.


The average number of elements in the 4Selmer groups of
elliptic curves is 7

with M. Bhargava.

The average size of the 5Selmer group of elliptic curves is 6,
and the average rank is less than 1

with M. Bhargava.


Average size of the 2Selmer group of Jacobians of monic
even hyperelliptic curves

with X. Wang, published in Compositio Mathematica.


The second moment of the 2Selmer groups of elliptic curves
 with M. Bhargava and A. A. Swaminathan.
Expanding and refining geometryofnumbers methods


On the DavenportHeilbronn theorems and second order terms

with M. Bhargava and J. Tsimerman, published in Inventiones Mathematicae.


Counting S_{5} fields with a power saving error term

with J. Tsimerman,
published in the Forum of Mathematics, Sigma.

 Geometry of numbers
methods over global fields I: Prehomogeneous vector spaces

with M. Bhargava and X. Wang.

 Geometry of numbers methods over global fields II: Coregular vector spaces

with M. Bhargava and X. Wang, preprint available on request.

 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


Odd degree number fields with odd class number

with W. Ho
and I. Varma,
published in Duke Math. Journal.


Bounds on 2torsion 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.


The mean number of 2torsion elements in the class groups of nmonogenized cubic fields

with M. Bhargava and J. Hanke.

 Counting integral points on symmetric varieties with
applications to arithmetic statistics
 with A. Siad and A. A. Swaminathan.


Nontrivial bounds on 2, 3, 4, and 5torsion in class groups of number fields,
conditional on standard Lfunction conjectures
 with J. Tsimerman.
Asymptotics of families of number fields


The number of quartic D4fields ordered by conductor

with A. Altug, and K. Wilson, and
and I. Varma,
published in the Journal of the European Mathematical Society.


Heuristics for the asymptotics of the number of S_{n}number fields
 with J. Tsimerman, published in the Journal of the London Mathematical Society.

 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.

 Asymptotics
for the number of cubic fields ordered by general invariants
 with F. Thorne.

 Malle's conjecture for octic D4 fields
 with I. Varma,
in preparation.
Uniformity estimates and squarefree sieves in arithmetic statistics

 Squarefree values of polynomial discriminants I

with M. Bhargava and X. Wang,
published in Inventiones Mathematicae.

 Squarefree values of polynomial discriminants II

with M. Bhargava and X. Wang.

Large families of elliptic curves ordered by conductor
 with A. N. Shankar and
X. Wang,
published in Compositio Mathematica.
Families of Lfunctions


SatoTate equidistribution of certain families of Artin Lfunctions

with A. Södergren
and N. Templier,
published in Forum of Math. Sigma.

 Central values of zeta functions of nonGalois cubic fields

with A. Södergren
and N. Templier.
Reductions of K3 surfaces and abelian surfaces modulo primes


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.


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.