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
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Binary quartic forms having bounded invariants, and the
boundedness of the average rank of elliptic curves
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with M. Bhargava, published
in the Annals of Mathematics.
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Ternary cubic forms having bounded invariants, and the existence
of a positive proportion of elliptic curves having rank 0
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with M. Bhargava, published in the
Annals of Mathematics.
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The average number of elements in the 4-Selmer groups of
elliptic curves is 7
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with M. Bhargava.
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The average size of the 5-Selmer group of elliptic curves is 6,
and the average rank is less than 1
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with M. Bhargava.
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Average size of the 2-Selmer group of Jacobians of monic
even hyperelliptic curves
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with X. Wang, published in Compositio Mathematica.
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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
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On the Davenport-Heilbronn theorems and second order terms
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with M. Bhargava and J. Tsimerman, published in Inventiones Mathematicae.
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Counting S5 fields with a power saving error term
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with J. Tsimerman,
published in the Forum of Mathematics, Sigma.
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- Geometry of numbers
methods over global fields I: Prehomogeneous vector spaces
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with M. Bhargava and X. Wang.
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- Geometry of numbers methods over global fields II: Coregular vector spaces
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with M. Bhargava and X. Wang, preprint available on request.
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- Geometry of numbers methods in the cusp with applications to class
groups
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with A. Siad, A. A. Swaminathan,
and I. Varma.
Torsion in class groups of number fields
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Odd degree number fields with odd class number
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with W. Ho
and I. Varma,
published in Duke Math. Journal.
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Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves
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with M. Bhargava, T. Taniguchi,
F. Thorne,
J. Tsimerman, and
Y. Zhao,
published in the
Journal of the American Mathematical Society.
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The mean number of 2-torsion elements in the class groups of n-monogenized cubic fields
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with M. Bhargava and J. Hanke.
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- Counting integral points on symmetric varieties with
applications to arithmetic statistics
- with A. Siad and A. A. Swaminathan.
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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
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The number of quartic D4-fields ordered by conductor
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with A. Altug, and K. Wilson, and
and I. Varma,
published in the Journal of the European Mathematical Society.
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Heuristics for the asymptotics of the number of Sn-number fields
- with J. Tsimerman, published in the Journal of the London Mathematical Society.
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- An improvement on Schmidt's bound on the number of number fields of bounded discriminant and small degree
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with M. Bhargava and X. Wang,
published in Forum of Math Sigma.
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- Asymptotics
for the number of cubic fields ordered by general invariants
- with F. Thorne.
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- Malle's conjecture for octic D4 fields
- with I. Varma,
in preparation.
Uniformity estimates and squarefree sieves in arithmetic statistics
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- Squarefree values of polynomial discriminants I
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with M. Bhargava and X. Wang,
published in Inventiones Mathematicae.
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- Squarefree values of polynomial discriminants II
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with M. Bhargava and X. Wang.
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Large families of elliptic curves ordered by conductor
- with A. N. Shankar and
X. Wang,
published in Compositio Mathematica.
Families of L-functions
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Sato-Tate equidistribution of certain families of Artin L-functions
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with A. Södergren
and N. Templier,
published in Forum of Math. Sigma.
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- Central values of zeta functions of non-Galois cubic fields
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with A. Södergren
and N. Templier.
Reductions of K3 surfaces and abelian surfaces modulo primes
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Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields
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with A. N. Shankar,
Y. Tang,
and S. Tayou,
published in Forum of Math Pi.
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Abelian surfaces over finite fields with prescribed groups
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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.