About

Frank Calegari is an Associate Chair and Professor in the Department of Mathematics at The University of Chicago. His research is in the area of algebraic number theory, with particular interest in the Langlands programme, especially the notion of reciprocity linking Galois representations and motives to automorphic forms. He is also very interested in the cohomology of arithmetic groups, especially in torsion classes. Calegari has been recognized for his contributions to mathematics, being elected to the American Academy of Arts and Sciences in April 2025. He has also delivered a plenary talk at the International Congress of Mathematicians in June 2021 and has been involved in extending significant mathematical bridges through collaborations.

Research topics

• Pure mathematics
• Discrete mathematics
• Mathematics
• Geometry
• Combinatorics
• Arithmetic

Selected publications

• Arithmetic holonomy bounds and effective Diophantine approximation

  ArXiv.org · 2025-10-05

  preprintOpen access1st authorCorresponding

  In this paper, we explore several threads arising from our recent joint work on arithmetic holonomy bounds, which were originally devised to prove new irrationality results based on the method of Apéry limits. We propose a new method to address effective Diophantine approximation on the projective line and the multiplicative group. This method, and all our other results in the paper, emerged from quantifying our holonomy bounds in a way that directly yields effective measures of irrationality and linear independence. Applying these to a dihedral algebraic construction, we derive good effective irrationality measures for high order roots of an algebraic number, in an approach that might be considered a multivalent continuation of the classical hypergeometric method of Thue, Siegel, and Baker. A well-known Dirichlet approximation argument of Bombieri allows one to derive from this the classical effective Diophantine theorems, hitherto only approachable by Baker's linear forms in logarithms or by Bombieri's equivariant Thue--Siegel method. These include the algorithmic resolution of the two-variable $S$-unit equation, the Thue--Mahler equation, and the hyperelliptic and superelliptic equations, as well as the Baker--Feldman effective power sharpening of Liouville's theorem. We also give some other applications, including irrationality measures for the classical $L(2,χ_{-3})$ and the $2$-adic $ζ(5)$, and a new proof of the transcendence of $π$. Due to space limitations, a full development of these ideas will be deferred to future work.

  PublisherOA PDFDOI

• The Ramanujan and Sato–Tate Conjectures for Bianchi modular forms

  Forum of Mathematics Pi · 2025-01-01 · 1 citations

  articleOpen access

  Abstract We prove the Ramanujan and Sato–Tate conjectures for Bianchi modular forms of weight at least $2$ . More generally, we prove these conjectures for all regular algebraic cuspidal automorphic representations of $\operatorname {\mathrm {GL}}_2(\mathbf {A}_F)$ of parallel weight, where F is any CM field. We deduce these theorems from a new potential automorphy theorem for the symmetric powers of $2$ -dimensional compatible systems of Galois representations of parallel weight.

  PublisherDOI

• Fields of definition for triangle groups as Fuchsian groups

  arXiv (Cornell University) · 2025-01-03

  preprintOpen access1st authorCorresponding

  The compact hyperbolic triangle group $Δ(p,q,r)$ admits a canonical representation to $\mathrm{PSL}_2(\mathbf{R})$ with discrete image which is unique up to conjugation. The trace field of this representation is \[K = \mathbf{Q}(\cos(π/p), \cos(π/q), \cos(π/r)).\] We prove that there are exactly eleven such groups which are conjugate to subgroups of $\mathrm{PSL}_2(K)$. Moreover, we prove that there are no additional compact hyperbolic triangle groups which are conjugate to subgroups of $\mathrm{PSL}_2(L)$ for any totally real field $L$. This answers a question first raised by Waterman and Machlachlan, and also resolves (in the positive) five (interrelated) recent conjectures of McMullen.

  PublisherOA PDFDOI

• Cuspidal cohomology classes for GL_{h}(𝐙)

  Journal of the American Mathematical Society · 2024-10-04 · 2 citations

  preprint

  We prove the existence of a cuspidal automorphic representation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi> π </mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L 79 slash bold upper Q"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>GL</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>79</mml:mn> </mml:mrow> </mml:msub> <mml:mo> ⁡ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {GL}_{79}/\mathbf {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of level one and weight zero. We construct <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi> π </mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> using symmetric power functoriality and a change of weight theorem, using Galois deformation theory. As a corollary, we construct the first known cuspidal cohomology classes in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript asterisk Baseline left-parenthesis upper G upper L Subscript n Baseline left-parenthesis bold upper Z right-parenthesis comma bold upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>GL</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^*(\operatorname {GL}_{n}(\mathbf {Z}),\mathbf {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n &gt; 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

  PublisherDOI

• The adelic closure of triangle groups

  arXiv (Cornell University) · 2024-07-29 · 1 citations

  preprintOpen access1st authorCorresponding

  Motivated by questions arising from billiard trajectories in the regular $n$-gon, McMullen defined a pair of functions $κ$ and $δ$ on the cusps $c$ of the corresponding triangle group $Δ_n$ inside $\mathrm{SL}_2({\mathcal{O}})$, where ${\mathcal{O}} = \mathbf{Z}[ζ_n+ ζ^{-1}_n]$. McMullen asks for which $n$ these functions are congruence, that is, when they only depend on the image of the cusp $c \in \mathbf{P}^1(\mathcal{O})$ in $\mathbf{P}^1(\mathcal{O}/d)$ for some integer $d$. In this note, we answer McMullen's questions. We obtain our results by computing the exact closure of $Δ_n \subset \mathrm{SL}_2({\mathcal{O}})$ inside $\mathrm{SL}_2(\widehat{\mathcal{O}})$, where $\widehat{\mathcal{O}}$ is the profinite completion of ${\mathcal{O}}$.

  PublisherOA PDFDOI

• The unbounded denominators conjecture

  Journal of the American Mathematical Society · 2024-11-13 · 3 citations

  preprint1st authorCorresponding

  We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L 2 left-parenthesis bold upper Z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>SL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {SL}_2(\mathbf {Z})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L 2 left-parenthesis bold upper Z left-bracket 1 slash p right-bracket right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>SL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {SL}_2(\mathbf {Z}[1/p])</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and a close description of the Fuchsian uniformization <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D left-parenthesis 0 comma 1 right-parenthesis slash normal upper Gamma Subscript upper N"> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">D(0,1)/\Gamma _N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the Riemann surface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper C minus mu Subscript upper N"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">C</mml:mi> </mml:mrow> <mml:mo> ∖ </mml:mo> <mml:msub> <mml:mi> μ </mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {C} \smallsetminus \mu _N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

  PublisherDOI

• The linear independence of $1$, $ζ(2)$, and $L(2,χ_{-3})$

  arXiv (Cornell University) · 2024-08-27

  preprintOpen access1st authorCorresponding

  We prove the irrationality of the classical Dirichlet L-value $L(2,χ_{-3})$. The argument applies a new kind of arithmetic holonomy bound to a well-known construction of Zagier. In fact our work also establishes the $\mathbf{Q}$-linear independence of $1$, $ζ(2)$, and $L(2,χ_{-3})$. We also give a number of other applications of our method to other problems in irrationality.

  PublisherOA PDFDOI

• Finite groups whose real irreducible representations have unique dimensions

  arXiv (Cornell University) · 2024-07-30

  preprintOpen access

  We determine the finite groups whose real irreducible representations have different degrees.

  PublisherOA PDFDOI

• Bloch groups, algebraic $K$-theory, units, and Nahm's Conjecture

  Annales Scientifiques de l École Normale Supérieure · 2023-01-01 · 23 citations

  article1st author

  Given an element of the Bloch group of a number field F and a natural number n, we construct an explicit unit in the field F n = F (e 2πi/n ), well-defined up to n-th powers of nonzero elements of F n .The construction uses the cyclic quantum dilogarithm, and under the identification of the Bloch group of F with the K-group K 3 (F ) gives (up to an unidentified invertible scalar) a formula for a certain abstract Chern class from K 3 (F ).The units we define are conjectured to coincide with numbers appearing in the quantum modularity conjecture for the Kashaev invariant of knots (which was the original motivation for our investigation), and also appear in the radial asymptotics of Nahm sums near roots of unity.This latter connection is used to prove Nahm's conjecture relating the modularity of certain q-hypergeometric series to the vanishing of the associated elements in the Bloch group of Q.

  PublisherDOI

• Cuspidal cohomology classes for GL_n(Z)

  arXiv (Cornell University) · 2023-09-27

  preprintOpen access

  We prove the existence of a cuspidal automorphic representation $π$ for $GL_{79}/\mathbf{Q}$ of level one and weight zero. We construct $π$ using symmetric power functoriality and a change of weight theorem, using Galois deformation theory. As a corollary, we construct the first known cuspidal cohomology classes in $H^*(GL_{n}(\mathbf{Z}),\mathbf{C})$ for any $n &gt; 1$.

  PublisherOA PDFDOI

Recent grants

• New Approaches To Modularity

  NSF · $330k · 2017–2021

• New Directions in Modularity

  NSF · $245k · 2015–2018

• Modularity of Genus Two Curves

  NSF · $650k · 2020–2026

• Families of p-adic modular forms

  NSF · $194k · 2007–2011

• CAREER: Arithmetic of Cohomological Automorphic Forms

  NSF · $400k · 2009–2015

Frequent coauthors

• Matthew Emerton

  University of Chicago

  23 shared

• Stavros Garoufalidis

  Southern University of Science and Technology

  17 shared

• Toby Gee

  15 shared

• Shiva Chidambaram

  Massachusetts Institute of Technology

  11 shared

• George Boxer

  8 shared

• David Geraghty

  Seattle University

  7 shared

• Kevin Buzzard

  Imperial College London

  6 shared

• Ana Caraiani

  Imperial College London

  5 shared

Labs

• Frank Calegari's LabPI

  Students of Frank Calegari

Awards & honors

• Elected to American Academy of Arts and Sciences (2025)