skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


  • Home
  • Search Results
  • Page 1 of 1

Search for: All records

Creators/Authors contains: "Shankar, Arul"

Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher. Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?

Some links on this page may take you to non-federal websites. Their policies may differ from this site.

  1. ; ; (, Inventiones mathematicae)
    Abstract The Dedekind zeta functions of infinitely many non-Galois cubic fields have negative central values. 
    more » « less
    Free, publicly-accessible full text available October 7, 2026
  2. ; ; ; (, Algebra & Number Theory)
    Free, publicly-accessible full text available May 14, 2026
  3. ; ; (, Proceedings of the London Mathematical Society)
    Abstract In this article, we combine Bhargava's geometry‐of‐numbers methods with the dynamical point‐counting methods of Eskin–McMullen and Benoist–Oh to develop a new technique for counting integral points on symmetric varieties lying within fundamental domains for coregular representations. As applications, we study the distribution of the 2‐torsion subgroup of the class group in thin families of cubic number fields, as well as the distribution of the 2‐Selmer groups in thin families of elliptic curves over . For example, our results suggest that the existence of a generator of the ring of integers with small norm has an increasing effect on the average size of the 2‐torsion subgroup of the class group, relative to the Cohen–Lenstra predictions. 
    more » « less
    Free, publicly-accessible full text available April 10, 2026
  4. ; ; ; (, Forum of Mathematics, Pi)
    Abstract Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $$X_{\overline {K}}$$ has infinitely many rational curves or X has infinitely many unirational specialisations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K . 
    more » « less
    Full Text Available 
  5. ; ; ; (, Journal of the European Mathematical Society)
    Full Text Available 
  6. ; ; (, Duke Mathematical Journal)
    Full Text Available