Jack Thorne's homepage

Jack Thorne

I am the Kuwait Professor of Number Theory and Algebra at the University of Cambridge and a Fellow of Trinity College.

Here are some of my papers. (Please note that these may not be final versions. In particular, they may not include changes made before publication.)

100% of odd hyperelliptic Jacobians have no rational points of small height.
Preprint. With Jef Laga. [pdf]

Reduction theory for stably graded Lie algebras.
Preprint. [pdf]

Congruences between modular forms (survey article).
Proc. Indian Acad. Sci. Math. Sci. 134 (2024), no. 1, Paper No. 9, 31 pp. [pdf]

Non-abelian base change for symmetric power liftings of holomorphic modular forms.
Preprint. With Laurent Clozel and James Newton. [pdf]

The Ramanujan and Sato--Tate conjectures for Bianchi modular forms.
Preprint. With George Boxer, Frank Calegari, Toby Gee, and James Newton. [pdf]

Symmetric power functoriality for Hilbert modular forms.
Preprint. With James Newton. [pdf]

A p-adic approach to the existence of level-raising congruences.
Proc. Lond. Math. Soc. (3) 128 (2024), no. 2, Paper No. e12584, 56 pp. [pdf]

Reciprocity and symmetric power functoriality (survey article).
Current Developments in Mathematics 2021 (2023), pp. 95-162. [pdf]

Cyclic base change of cuspidal automorphic representations over function fields.
To appear in Compositio Math. With Gebhard Böckle, Tony Feng, Michael Harris and Chandrashekhar Khare. [pdf]

Automorphy lifting with adequate image.
Forum Math. Sigma 11 (2023), Paper No. e8, 31 pp. With Konstantin Miagkov. [pdf]

On the vanishing of adjoint Bloch--Kato Selmer groups of irreducible automorphic Galois representations.
Pure Appl. Math. Q. 18 (2022), no. 5, pp. 2159-2202. [pdf]

The power of 2: small primes in number theory (survey article).
Eur. Math. Soc. Newsl. No. 118 (2020), pp. 34-39. [pdf]

Elliptic curves and modularity (survey article).
In European Congress of Mathematics (2023), pp. 643-662, EMS Press. [pdf]

Modularity of PGL2(Fp)-representations over totally real fields.
PNAS August 17, 2021 118 (33). With Patrick B. Allen and Chandrashekhar Khare. [pdf]

Symmetric power functoriality for holomorphic modular forms, II.
Publ. Math. Inst. Hautes Études Sci. 134 (2021), pp. 117-152. With James Newton. [pdf]

Symmetric power functoriality for holomorphic modular forms, I.
Publ. Math. Inst. Hautes Études Sci. 134 (2021), pp. 1-116. With James Newton. [pdf]

Adjoint Selmer groups of automorphic Galois representations of unitary type.
J. Eur. Math. Soc. 25 (2023), no. 5, pp. 1919-1967. With James Newton. [pdf]

Raising the level of automorphic representations of GL2n of unitary type.
J. Inst. Math. Jussieu 21 (2022), no. 4, pp. 1421-1444. With Christos Anastassiades. [pdf]

Automorphy lifting for residually reducible l-adic Galois representations, II.
Compos. Math. 156 (2020), no. 11, pp. 2399-2422. With Patrick B. Allen and James Newton. [pdf]

Modularity of GL2(Fp)-representations over CM fields.
Camb. J. Math. 11 (2023), no. 1, pp. 1-158. With Patrick B. Allen and Chandrashekhar Khare. [pdf]

A local Langlands parameterization for generic supercuspidal representations of p-adic G2.
Ann. Sci. de l'ENS (4) 56 (2023), no. 1, pp. 257-286. With Michael Harris and Chandrashekhar Khare. [pdf]

Potential automorphy over CM fields.
Ann. of Math. (2) 197 (2023), no. 3, pp. 897-1113. With P. Allen, A. Caraiani, F. Calegari, T. Gee, D. Helm, B. Le Hung, J. Newton, P. Scholze, and R. Taylor. [pdf]

Potential automorphy of Ĝ-local systems.
In Proceedings of the International Congress of Mathematicians, Rio de Janeiro 2018. Vol. II. Invited lectures, pp. 415-434, World Sci. Publ. [pdf]

E8 and the average size of the 3-Selmer group of the Jacobian of a pointed genus-2 curve.
Proc. Lond. Math. Soc. (3) 122 (2021), no. 5, pp. 678-723. With Beth Romano. [pdf]

On subquotients of the étale cohomology of Shimura varieties.
In T. Haines & M. Harris (Eds.), Shimura Varieties (London Mathematical Society Lecture Note Series), pp. 306-334. With Christian Johansson. [pdf]

On the arithmetic of simple singularities of type E.
Research Number Theory 4 (2018), No. 2, Art. 21. With Beth Romano. [pdf]

Beyond the Taylor--Wiles method.
Notes for lectures at the workshop "Deformation theory, Completed Cohomology, Leopoldt Conjecture and K-theory" (not for publication). [pdf]

Ĝ-local systems on smooth projective curves are potentially automorphic.
Acta Math. 223 (2019), No. 1, pp. 1-111. With Gebhard Böckle, Michael Harris, and Chandrashekhar Khare. [pdf]

On the average number of 2-Selmer elements of elliptic curves over Fq(X) with two marked points.
Documenta Math. 24 (2019), pp. 1179-1223. [pdf]

On the GL(n) eigenvariety and a conjecture of Venkatesh.
Selecta Math. (N.S.) 23 (2017), No. 2, pp. 1205-1234. With David Hansen. [pdf]

Elliptic curves over Q are modular.
J. Eur. Math. Soc. 21 (2019), no. 7, 1943-1948. [pdf]

Torsion Galois representations over CM fields and Hecke algebras in the derived category.
Forum Math. Sigma 4 (2016), e21, 88 pp. With James Newton. [pdf]

Automorphy of some residually S5 Galois representations.
Math. Z. 286 (2017), No. 1-2, pp. 399-429. With Chandrashekhar Khare. [pdf]

A 2-adic automorphy lifting theorem for unitary groups over CM fields.
Math. Z. 285 (2017), No. 1-2, pp. 1-38. [pdf]

Equidistribution of Frobenius eigenvalues.
IMRN 21 (2015), pp. 11388-11403. [pdf]

Potential automorphy and the Leopoldt conjecture.
Amer. J. Math. 139 (2017), no. 5, 1205-1273. With Chandrashekhar Khare. [pdf]

Arithmetic invariant theory and 2-descent for plane quartic curves.
Algebra Number Theory 10 (2016), No. 7, pp. 1373-1413. With an appendix by Tasho Kaletha. [pdf]

A remark on the arithmetic invariant theory of hyperelliptic curves.
Mathematical Research Letters 21 (2014), No. 6, pp. 1451-1482. [pdf]

Automorphy of some residually dihedral Galois representations.
Mathematische Annalen 364 (2016), No. 1-2, pp. 589-648 [pdf]

E6 and the arithmetic of a family of non-hyperelliptic curves of genus 3.
Forum of Mathematics Pi 3 (2015), e1 [pdf]

Level-raising and symmetric power functoriality, III.
Duke Math. J. 166 (2017), No. 2, pp. 325-402. With Laurent Clozel. [pdf]

On the rigid cohomology of certain Shimura varieties.
Res. Math. Sci. 3 (2016), Paper No. 37, 308 pp. With Michael Harris, Kai-Wen Lan, and Richard Taylor. [pdf]

On the φ-Selmer groups of the elliptic curves y2 = x3 - D x.
Math. Proc. Cambridge Philos. Soc. 163 (2017), No. 1, pp. 71-93. With Daniel Kane. [pdf]

Level-raising and symmetric power functoriality, II.
Annals of Mathematics 181 (2015), No. 1, pp. 303-359. With Laurent Clozel. [pdf]

Level-raising and symmetric power functoriality, I.
Compositio Mathematica 150 (2014), No. 5, pp. 729-748. With Laurent Clozel. [pdf]

Raising the level for GL(n).
Forum of Mathematics Sigma 2 (2014), e16. [pdf]

Automorphy lifting for residually reducible l-adic Galois representations.
J. Amer. Math. Soc. 28 (2015), No. 3, pp. 785-870. [pdf]

Vinberg's representations and arithmetic invariant theory.
Algebra & Number Theory 7 (2013), No. 9, pp. 2331-2368. This is a revised version of my thesis, The Arithmetic of Simple Singularities. [pdf]

On the automorphy of l-adic Galois representations with small residual image.
Journal of the Inst. of Math. Jussieu 11 (2012), no. 4, pp.855-920. [pdf]

Adequate subgroups.
Appendix to the above paper. With Robert Guralnick, Florian Herzig and Richard Taylor. [pdf]

On the Tate-Shafarevich groups of certain elliptic curves.
Journal of Number Theory 130 (2010), No. 9, pp. 2092-2098. [pdf]

CV (last updated September 2023): [pdf]