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

Award ID contains: 2001097

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. ; ; (, Journal of the American Mathematical Society)
    We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of SL 2 ⁡<#comment/> ( Z ) \operatorname {SL}_2(\mathbf {Z}) . Our result includes also Mason’s generalization of the original conjecture to the setting of vector-valued modular forms, thereby supplying a new path to the congruence property in rational conformal field theory. The proof involves a new arithmetic holonomicity bound of a potential-theoretic flavor, together with Nevanlinna second main theorem, the congruence subgroup property of SL 2 ⁡<#comment/> ( Z [ 1 / p ] ) \operatorname {SL}_2(\mathbf {Z}[1/p]) , and a close description of the Fuchsian uniformization D ( 0 , 1 ) / Γ<#comment/> N D(0,1)/\Gamma _N of the Riemann surface C ∖<#comment/> μ<#comment/> N \mathbf {C} \smallsetminus \mu _N . 
    more » « less
    Free, publicly-accessible full text available February 6, 2026
  2. ; ; ; ; ; ; ; ; ; (, Annals of Mathematics)
    Full Text Available 
  3. ; ; (, Annales scientifiques de l'École Normale Supérieure)
    Full Text Available 
  4. ; (, Proceedings of the American Mathematical Society)
    Let  ρ ¯ : G Q → GSp 4 ⁡ ( F 3 ) \overline {\rho }: G_{\mathbf {Q}} \rightarrow \operatorname {GSp}_4(\mathbf {F}_3) be a continuous Galois representation with cyclotomic similitude character. Equivalently, consider ρ ¯ \overline {\rho } to be the Galois representation associated to the  3 3 -torsion of a principally polarized abelian surface  A / Q A/\mathbf {Q} . We prove that the moduli space  A 2 ( ρ ¯ ) \mathcal {A}_2(\overline {\rho }) of principally polarized abelian surfaces  B / Q B/\mathbf {Q} admitting a symplectic isomorphism  B [ 3 ] ≃ ρ ¯ B[3] \simeq \overline {\rho } of Galois representations is never rational over  Q \mathbf {Q} when  ρ ¯ \overline {\rho } is surjective, even though it is both rational over  C \mathbf {C} and unirational over  Q \mathbf {Q} via a map of degree  6 6 . 
    more » « less
    Full Text Available 
  5. ; ; ; (, Publications mathématiques de l'IHÉS)
    Abstract We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse–Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces  $$A$$ A over  $${\mathbf {Q}}$$ Q with  $$\operatorname{End}_{ {\mathbf {C}}}A={\mathbf {Z}}$$ End C A = Z . We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields. 
    more » « less
    Full Text Available 
  6. ; (, Duke Mathematical Journal)
    Full Text Available 
  7. ; ; (, Open Book Series)
    null (Ed.)
    Full Text Available