Jack A. Thorne
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View article: Inductive construction of supercuspidal $L$-packets
Inductive construction of supercuspidal $L$-packets Open
Genestier--Lafforgue and Fargues--Scholze have constructed a semisimple local Langlands paramterization for reductive groups over equicharacteristic local fields. Assuming a version of the stable twisted trace formula for function fields, …
View article: The Ramanujan and Sato–Tate Conjectures for Bianchi modular forms
The Ramanujan and Sato–Tate Conjectures for Bianchi modular forms Open
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}}_…
View article: Cyclic base change of cuspidal automorphic representations over function fields
Cyclic base change of cuspidal automorphic representations over function fields Open
Let $G$ be a split semisimple group over a global function field $K$ . Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$ , there is a cyclic base change l…
View article: An LLL algorithm with symmetries
An LLL algorithm with symmetries Open
We give a generalisation of the Lenstra-Lenstra-Lovász (LLL) lattice-reduction algorithm that is valid for an arbitrary (split, semisimple) reductive group $G$. This can be regarded as `lattice reduction with symmetries'. We make this algo…
View article: Reduction theory for stably graded Lie algebras
Reduction theory for stably graded Lie algebras Open
We define a reduction covariant for the representations a la Vinberg associated to stably graded Lie algebras. We then give an analogue of the LLL algorithm for the odd split special orthogonal group and show how this can be combined with …
View article: 100% of odd hyperelliptic Jacobians have no rational points of small height
100% of odd hyperelliptic Jacobians have no rational points of small height Open
We study the universal family of odd hyperelliptic curves of genus $g \geq 1$ over $\mathbb{Q}$. We relate the heights of $\mathbb{Q}$-points of Jacobians of curves in this family to the reduction theory of the representation of $\mathrm{S…
View article: A p$p$‐adic approach to the existence of level‐raising congruences
A p$p$‐adic approach to the existence of level‐raising congruences Open
We construct level‐raising congruences between ‐ordinary automorphic representations, and apply this to the problem of symmetric power functoriality for Hilbert modular forms. In particular, we prove the existence of the symmetric power li…
View article: Non-abelian base change for symmetric power liftings of holomorphic modular forms
Non-abelian base change for symmetric power liftings of holomorphic modular forms Open
Let $f$ be a non-CM Hecke eigenform of weight $k \geq 2$. We give a new proof of some cases of Langlands functoriality for the automorphic representation $π$ associated to $f$. More precisely, we prove the existence of the base change lift…
View article: The Ramanujan and Sato-Tate Conjectures for Bianchi modular forms
The Ramanujan and Sato-Tate Conjectures for Bianchi modular forms Open
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 $\mathrm{GL}_2(\mathbf{A}_F)$ of …
View article: Elliptic curves and modularity
Elliptic curves and modularity Open
We survey results and conjectures concerning the modularity of elliptic curves over number fields.
View article: Potential automorphy over CM fields
Potential automorphy over CM fields Open
Let F be a CM number field. We prove modularity lifting theorems for regular n-dimensional Galois representations over F without any self- duality condition. We deduce that all elliptic curves E over F are poten- tially modular, and furthe…
View article: Automorphy lifting with adequate image
Automorphy lifting with adequate image Open
Let F be a CM number field. We generalise existing automorphy lifting theorems for regular residually irreducible p -adic Galois representations over F by relaxing the big image assumption on the residual representation.
View article: Symmetric power functoriality for Hilbert modular forms
Symmetric power functoriality for Hilbert modular forms Open
Let $F$ be a totally real field. We prove the existence of all symmetric power liftings of those cuspidal automorphic representations of $\mathrm{GL}_2(\mathbf{A}_F)$ associated to Hilbert modular forms of regular weight.
View article: A $p$-adic approach to the existence of level-raising congruences
A $p$-adic approach to the existence of level-raising congruences Open
We construct level-raising congruences between $p$-ordinary automorphic representations, and apply this to the problem of symmetric power functoriality for Hilbert modular forms. In particular, we prove the existence of the $n^\text{th}$ s…
View article: On the vanishing of adjoint Bloch--Kato Selmer groups of irreducible automorphic Galois representations
On the vanishing of adjoint Bloch--Kato Selmer groups of irreducible automorphic Galois representations Open
Let $ρ$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic, polarizable automorphic representation of $GL_n$. Assuming only that $ρ$ satisfies an irreducibility condition, we prove the vanishing of the adjoint …
View article: Cyclic base change of cuspidal automorphic representations over function fields
Cyclic base change of cuspidal automorphic representations over function fields Open
Let $G$ be a split semi-simple group over a global function field $K$. Given a cuspidal automorphic representation $Π$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lift…
View article: Adjoint Selmer groups of automorphic Galois representations of unitary type
Adjoint Selmer groups of automorphic Galois representations of unitary type Open
Let \rho be the p -adic Galois representation attached to a cuspidal, regular algebraic automorphic representation of \operatorname{GL}_n of unitary type. Under very mild hypotheses on \rho , we prove the vanishing of the (Bloch–Kato) adjo…
View article: Automorphy lifting with adequate image
Automorphy lifting with adequate image Open
Let $F$ be a CM number field. We generalize existing automorphy lifting theorems for regular residually irreducible $p$-adic Galois representations over $F$ by relaxing the big image assumption on the residual representation.
View article: Modularity of PGL <sub>2</sub> (𝔽 <i>p</i> )-representations over totally real fields
Modularity of PGL <sub>2</sub> (𝔽 <i>p</i> )-representations over totally real fields Open
Significance The connection between modular forms and Galois representations plays a significant role in modern algebraic number theory. J.-P. Serre made an influential conjecture relating mod modular forms and mod representations of t…
RAISING THE LEVEL OF AUTOMORPHIC REPRESENTATIONS OF OF UNITARY TYPE Open
We use the endoscopic classification of automorphic representations of even-dimensional unitary groups to construct level-raising congruences.
Automorphy lifting for residually reducible -adic Galois representations, II Open
We revisit the paper [ Automorphy lifting for residually reducible $l$ - adic Galois representations , J. Amer. Math. Soc. 28 (2015), 785–870] by the third author. We prove new automorphy lifting theorems for residually reducible Galois re…
View article: Automorphy lifting for residually reducible $l$-adic Galois representations, II
Automorphy lifting for residually reducible $l$-adic Galois representations, II Open
We prove new automorphy lifting theorems for residually reducible Galois representations of unitary type in which the residual representation is permitted to have an arbitrary number of irreducible constituents.
View article: Raising the level of automorphic representations of $\mathrm{GL}_{2n}$ of unitary type
Raising the level of automorphic representations of $\mathrm{GL}_{2n}$ of unitary type Open
We use the endoscopic classification of automorphic representations of even-dimensional unitary groups to construct level-raising congruences.
View article: Symmetric power functoriality for holomorphic modular forms
Symmetric power functoriality for holomorphic modular forms Open
Let $f$ be a cuspidal Hecke eigenform of level 1. We prove the automorphy of the symmetric power lifting $\mathrm{Sym}^n f$ for every $n \geq 1$. We establish the same result for a more general class of cuspidal Hecke eigenforms, including…
View article: E 8 and the average size of the 3‐Selmer group of the Jacobian of a pointed genus‐2 curve
E 8 and the average size of the 3‐Selmer group of the Jacobian of a pointed genus‐2 curve Open
We prove that the average size of the 3‐Selmer group of a genus‐2 curve with a marked Weierstrass point is 4. We accomplish this by studying rational and integral orbits in the representation associated to a stably Z / 3 Z ‐graded simple L…
View article: Modularity of $\operatorname{GL}_2(\mathbb{F}_p)$-representations over CM fields
Modularity of $\operatorname{GL}_2(\mathbb{F}_p)$-representations over CM fields Open
We prove that many representations $\overlineρ : \operatorname{Gal}(\overline{K} / K) \to \operatorname{GL}_2(\mathbb{F}_3)$, where $K$ is a CM field, arise from modular elliptic curves. We prove similar results when the prime $p = 3$ is r…