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Creators/Authors contains: "Calegari, Frank"

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  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 . 
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    Free, publicly-accessible full text available February 6, 2026
  2. ; ; (, Annales scientifiques de l'École Normale Supérieure)
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  3. ; ; ; ; ; ; ; ; ; (, Annals of Mathematics)
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  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 . 
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  5. ; ; (, Journal of the Institute of Mathematics of Jussieu)
    Abstract Let $$n$$ be either  $$2$$ or an odd integer greater than  $$1$$ , and fix a prime  $p>2(n+1)$ . Under standard ‘adequate image’ assumptions, we show that the set of components of $$n$$ -dimensional $$p$$ -adic potentially semistable local Galois deformation rings that are seen by potentially automorphic compatible systems of polarizable Galois representations over some CM field is independent of the particular global situation. We also (under the same assumption on  $$n$$ ) improve on the main potential automorphy result of Barnet-Lamb et al.  [Potential automorphy and change of weight, Ann. of Math. (2)   179 (2) (2014), 501–609], replacing ‘potentially diagonalizable’ by ‘potentially globally realizable’. 
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  6. ; ; ; (, 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. 
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  7. ; (, Mathematische Annalen)
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    We prove that, for fixed level $(N,p) = 1$ and $p > 2$, there are only finitely many Hecke eigenforms f of level $$\Gamma _1(N)$$ and even weight with $$a_p(f) = 0$$ which are not CM. 
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  8. ; (, Duke Mathematical Journal)
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  9. ; ; (, Mathematics of Computation)
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  10. ; ; (, Open Book Series)
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