# Joel Bellaiche

Degrees

University of Paris XI, Ph.D.University of Paris XI, Ed.M.

Ecole Normale Superieure, M.A.

Ecole Normale Superieure, B.A.

Expertise

Number Theory, Algebra, Algebraic GeometryProfile

My main research interest lies in number theory, both algebraic and, more recently, analytic. I am also interested in the closely related fields of group theory, representation theory, and algebraic geometry.A large part of my work has involved the topic of Galois representations, which provides a unifying language to study many classical and new questions in number theory. Since Galois representations are often difficult to construct, and even more, to study, I develop and use tools belonging to the so-called Langlands program, a large field of current research aiming at understanding deep relations between Galois representations and some more analytic objects called automorphic representations (of which on of the most famous avatar are the modular forms).

Courses Taught

MATH | 131a | Algebra I |

MATH | 141b | Complex Analysis |

MATH | 200a | Second-Year Seminar |

MATH | 201a | Topics in Algebra |

MATH | 202a | Algebraic Geometry I |

MATH | 203a | Number Theory |

MATH | 203b | Topics in Number Theory |

MATH | 250a | Complex Algebraic Geometry I |

Awards and Honors

American Mathematical Society Centennial Fellowship (2010 - 2011)

Scholarship

Bellaiche, Joel. __The Eigenbook. Eigenvarieties, p-adic families of Galois representations, p-adic L-functions__. first ed. Springer-Birkhauser, 2021. (forthcoming)

Bellaiche, Joel. "Image of pseudo-representations and coefficients of modular forms modulo p." __Advances in Math.__ 353. 1 (2020): 647-721.

Bellaïche, Joël; Green, Ben; Soundararajan, Kannan. "Nonzero coefficients of half-integral weight modular forms mod ℓ.." __Res. Math. Sci. 5 (2018), n__ no 1. Paper No. 6, 10 pp. 11F37 (special issue in honor of Don Zagier) (2018): 10pp.

Bellaiche, Joel and Nicolas, Jean-Louis. "Parité des coefficients de formes modulaires." __Ramanujan Journal__ 40 (2016). 1 (2017): 1-44.

Bellaiche, Joel. "A Remark on the Error Term in Chebotarev's Density Theorem." __Math. Research Letters__ 23 (2017). 1 (2017): 10 pages.

Bellaiche, Joel & Dasgupta, Samit. "the p-adic L function of critical Eisenstein series." __Composite Mathematicae__ 151 (2015). 6 (2016): 999-1040.

Bellaiche, Joel & Dimitrov, Mladen. "On the eigencurve at classical weight 1 points.." __Duke Mathematical Journal__ 165. 2 (2016): 245-266.

Bellaiche, Joel. "Théorème de Chebotarev et complexité de Littlewood.." __Annales Scientifiques de L'École Normale Supérieure__ 49 (2016). 3 (2016): 579--632.

Bellaiche, Joel & Soundararajan, Kannan. "The number of nonzero coefficients of modular forms (mod p)." __Algebra and Number Theory__ 9. 8 (2015): 1825-1856.

Bellaiche, Joel. "Unitary Eigenvarieties at Isobaric Points.." __Canad. J. Math.__ 67. 2 (2015): 315-32

Bellaiche, Joel; Khare, Chandrashekhar. "Level 1 Hecke algebras of modular forms modulo p." __Compositio Mathematicae__ 151. 3 (2015): 397-415.

Bellaiche, Joel and Chenevier, Gaetan. "Sous-groupes de Gl(2) et arbres." __Journal of Algebra__ Reference: YJABR14652. 1 (2014): 10.1016/j.jalgebra.2014.02.005.

Bellaiche, Joel. "Critical p-adic L-function." __Inventiones Math__ 189. 1 (2012): 1-60.

Bellaiche, Joel. "Pseudodeformations." __Mathematsche Zeitchrift__ 270. 3-4 (2012): 1163-1180.

Bellaiche, Joel. "Ranks of Selmer groups in an analytic family." __Trans. Amer. Math. Soc.__ 364. 9 (2012): 4735–4761.

Bellaiche, Joel. "Une représentation galoisienne universelle attachée aux formes modulaires modulo 2." __Comptes-rendus de l'academie des sciences__ 2012. 9-10 (2012): 443-449.

Bellaiche, Joel and Chenevier Gaëtan. "The sign of Galois representations attached to automorphic forms for unitary groups." __Compositio Mathematicae__ 147. 5 (2011): 1337-1352.

Bellaiche, Joel. "New examples of p-adically rigid automorphic forms." __Mathematical Research Letters__ 17 (2010). 4 (2010): 781–792.

Bellaiche, Joel. "On the path-dependence of economic growth." __Journal of Mathematical Economics__ 46. (2010): 163-178.