arXiv subjects

Jack Shotton

Publications and source records attributed to Jack Shotton.

9 recordsLinked to original sources

arXiv2026

The Smallest Invariant Factor of Elliptic Curves, and Coincidences

For an elliptic curve E over Q and a natural number j, Cojocaru has shown that there is an explicit constant C_E,j giving (under GRH) the density of primes p of good reduction such that the smallest invariant factor of E(F_p) is j. For E without complex multiplication, we study the question of when C_E,j is positive (a necessary and, on GRH, sufficient condition for there to be infinitely many such p), strengthening a result by Kim. Our arguments are group-theoretic using the image of the adelic Galois representation of E. Experimentally, C_E,j appears to vanish only when there is a coincidence of division fields; we document a number of families of such coincidences arising from abelian division fields.

math.NT↗

arXiv2024

Singularities of Steinberg deformation rings

Let $l$ and $p$ be distinct primes, let $F$ be a local field with residue field of characteristic $p$, and let $\mathfrak{X}$ be the irreducible component of the moduli space of Langlands parameters for $GL_3$ over $\mathbb{Z}_l$ corresponding to parameters of Steinberg type. We show that $\mathfrak{X}$ is Cohen-Macaulay and compute explicit equations for it. We also compute the Weil divisor class group of the special fibre of $\mathfrak{X}$, motivated by work of Manning for $GL_2$. Our methods involve the calculation of the cohomology of certain vector bundles on the flag variety, and build on work of Snowden, Vilonen-Xue, and Ngo.

math.NT↗

arXiv2023

Irreducible components of the moduli space of Langlands parameters

Let $F/\mathbb{Q}_p$ be finite and let $\mathfrak{X}_G$ be the moduli space of Langlands parameters valued in $G$, in characteristic distinct from $p$. First, we determine the irreducible components of $\mathfrak{X}_G$. Then, we determine the local structure around tamely ramified points for which the image of `tame inertia' is regular. This local structure is related to the endomorphism rings of Gelfand--Graev representations, by work of Li. Lastly, we determine an open dense set in $\mathfrak{X}_M$, when $M$ is a Levi subgroup of $G$, such that the natural map of moduli stacks $[\mathfrak{X}_M/M] \to [\mathfrak{X}_G/G]$ is smooth on this set.

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arXiv2023

On endomorphism algebras of Gelfand-Graev representations II

Let $G$ be a connected reductive group defined over a finite field $\mathbb{F}_q$ of characteristic $p$, with Deligne--Lusztig dual $G^\ast$. We show that, over $\overline{\mathbb{Z}}[1/pM]$ where $M$ is the product of all bad primes for $G$, the endomorphism ring of a Gelfand--Graev representation of $G(\mathbb{F}_q)$ is isomorphic to the Grothendieck ring of the category of finite-dimensional $\overline{\mathbb{F}}_q$-representations of $G^\ast(\mathbb{F}_q)$.

math.RT↗

arXiv2020

Ihara's lemma for Shimura curves over totally real fields via patching

We prove Ihara's lemma for the mod $l$ cohomology of Shimura curves, localised at a maximal ideal of the Hecke algebra, under a large image hypothesis on the associated Galois representation. This was proved by Diamond and Taylor, for Shimura curves over $\mathbb{Q}$, under various assumptions on $l$. Our method is totally different and can avoid these assumptions, at the cost of imposing the large image hypothesis. It uses the Taylor--Wiles method, as improved by Diamond and Kisin, and the geometry of integral models of Shimura curves at an auxiliary prime.

math.NT↗

arXiv2020

Generic local deformation rings when $l \neq p$

We determine the local deformation rings of sufficiently generic mod $l$ representations of the Galois group of a $p$-adic field, when $l \neq p$, relating them to the space of $q$-power-stable semisimple conjugacy classes in the dual group. As a consequence we give a local proof of the $l \neq p$ Breuil--Mézard conjecture of the author, in the tame case.

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arXiv2020

Local deformation rings and a Breuil-Mézard conjecture when l\neq p

We compute the deformation rings of two dimensional mod l representations of Gal(Fbar/F) with fixed inertial type, for l an odd prime, p a prime distinct from p and F/Q_p a finite extension. We show that in this setting (when p is also odd) an analogue of the Breuil-Mézard conjecture holds, relating the special fibres of these deformation rings to the mod l reduction of certain irreducible representations of GL_2(O_F).

math.NT↗

arXiv2020

On the category of finitely presented mod $p$ representations of $GL_2(F)$

Let $F$ be a finite extension of $\mathbb{Q}_p$. We prove that the category of finitely presented smooth $Z$-finite representations of $GL_2(F)$ over a finite extension of $\mathbb{F}_p$ is an abelian subcategory of the category of all smooth representations. The proof uses amalgamated products of completed group rings.

math.RT↗

arXiv2017

The Breuil--Mézard conjecture when $l \neq p$

Let $l$ and $p$ be primes, let $F/\mathbb{Q}_p$ be a finite extension with absolute Galois group $G_F$, let $\mathbb{F}$ be a finite field of characteristic $l$, and let $\barρ : G_F \rightarrow GL_n(\mathbb{F})$ be a continuous representation. Let $R^\square(\barρ)$ be the universal framed deformation ring for $\barρ$. If $l = p$, then the Breuil--Mézard conjecture (as formulated by Emerton and Gee) relates the mod $l$ reduction of certain cycles in $R^\square(\barρ)$ to the mod $l$ reduction of certain representations of $GL_n(\mathcal{O}_F)$. We state an analogue of the Breuil--Mézard conjecture when $l \neq p$, and prove it whenever $l > 2$ using automorphy lifting theorems. We give a local proof when $l$ is "quasi-banal" for $F$ and $\barρ$ is tamely ramified. We also analyse the reduction modulo $l$ of the types $σ(τ)$ defined by Schneider and Zink.

math.NT↗