Searcharxiv⌕ Search

arXiv subjects

David Savitt

Publications and source records attributed to David Savitt.

At least 19 recordsLinked to original sources
arXiv2026

Inclusions between p-bounded crystalline loci in dimension two

Let p be an odd prime and K/Qp a finite unramified extension of degree f > 1. Let Z(r) be the reduced special fiber of the Emerton-Gee stack of two-dimensional crystalline representations of Hodge type r of the absolute Galois group of K. We study the collection of stacks Z(r) as r varies over p-bounded Hodge types, as a set partially ordered under inclusion. We prove that aside from two degenerate cases, simple inclusions can be classified in terms of three operations on Hodge types, two of which have standard automorphic interpretations. We also prove, with one exception, that inclusions can be detected at the level of an inclusion of closed points (equivalently, semisimple mod p Galois representations). GPT-5.5 Pro was used extensively in the course of this work.

math.NT↗
arXiv2025

An elementary proof of Newman's eta-quotient theorem

Let eta(z) be the Dedekind eta function. Newman studied the modularity of eta-quotients, giving necessary and sufficient conditions for a function of the form \prod_{0 < m | N} eta(mz)^{r_m} to be a (weakly) holomorphic modular form of level N. We explain a proof of Newman's theorem, developed while teaching a class for talented high school students at Canada/USA Mathcamp. The key observation is that although Gamma_1(N) is not generated by its upper triangular and lower triangular subgroups, it is generated by those subgroups together with any congruence subgroup. Modularity with respect to some congruence subgroup is established using one simple identity involving the multiplier system of eta(z), whose proof is elementary in the sense that it avoids the use of Dedekind sums.

math.NT↗
arXiv2023

Irregular loci in the Emerton-Gee stack for GL_2

Let K/Q_p be unramified. Inside the Emerton-Gee stack X_2, one can consider the locus of two-dimensional mod p representations of the absolute Galois group of K having a crystalline lift with specified Hodge-Tate weights. We study the case where the Hodge-Tate weights are irregular, which is an analogue for Galois representations of the partial weight one condition for Hilbert modular forms. We prove that if the gap between each pair of weights is bounded by p (the irregular analogue of a Serre weight), then this locus is irreducible. We also establish various inclusion relations between these loci.

math.NT↗
arXiv2022

Local geometry of moduli stacks of two-dimensional Galois representations

We construct moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field, as well as their resolutions by moduli stacks of two-dimensional Breuil-Kisin modules with tame descent data. We study the local geometry of these moduli stacks by comparing them with local models of Shimura varieties at hyperspecial and Iwahori level.

math.NT↗
arXiv2022

Components of moduli stacks of two-dimensional Galois representations

In a previous article we introduced various moduli stacks of two-dimensional tamely potentially Barsotti-Tate representations of the absolute Galois group of a p-adic local field, as well as related moduli stacks of Breuil-Kisin modules with descent data. We study the irreducible components of these stacks, establishing in particular that the components of the former are naturally indexed by certain Serre weights.

math.NT↗
arXiv2017

Potentially crystalline lifts of certain prescribed types

We prove several results concerning the existence of potentially crystalline lifts with prescribed Hodge-Tate weights and inertial types of a given n-dimensional mod p representation of the absolute Galois group of K, where K/Q_p is a finite extension. Some of these results are proved by purely local methods, and are expected to be useful in the application of automorphy lifting theorems. The proofs of the other results are global, making use of automorphy lifting theorems.

math.NT↗
arXiv2015

General Serre weight conjectures

We formulate a number of related generalisations of the weight part of Serre's conjecture to the case of GL(n) over an arbitrary number field, motivated by the formalism of the Breuil-M\'ezard conjecture. We give evidence for these conjectures, and discuss their relationship to previous work. We generalise one of these conjectures to the case of connected reductive groups which are unramified over Q_p, and we also generalise the second author's previous conjecture for GL(n)/Q to this setting, and show that the two conjectures are generically in agreement.

math.NT↗
arXiv2014

The weight part of Serre's conjecture for GL(2)

Let p > 2 be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call "pseudo-Barsotti-Tate representations", over arbitrary finite extensions of the p-adics. As a consequence, we establish (under the usual Taylor-Wiles hypothesis) the weight part of Serre's conjecture for GL(2) over arbitrary totally real fields.

math.NT↗
arXiv2014

Lattices in the cohomology of Shimura curves

We prove conjectures of Breuil and Breuil-Dembele (C. Breuil, "Sur un probleme de compatibilite local-global modulo p pour GL(2)"), including a generalisation from the principal series to the cuspidal case, subject to a mild global hypothesis that we make in order to apply certain R=T theorems. More precisely, we prove a multiplicity one result for the mod p cohomology of a Shimura curve at Iwahori level, and we show that certain apparently globally defined lattices in the cohomology of Shimura curves are determined by the corresponding local p-adic Galois representations. We also indicate a new proof of the Buzzard-Diamond-Jarvis conjecture in generic cases. Our main tools are the geometric Breuil-Mezard philosophy developed by two of the authors, and a new and more functorial perspective on the Taylor-Wiles-Kisin patching method. Along the way, we determine the tamely potentially Barsotti-Tate deformation rings of generic two-dimensional mod p representations, generalising a result of Breuil-Mezard in the principal series case.

math.NT↗
arXiv2014

Serre weights for locally reducible two-dimensional Galois representations

Let F be a totally real field, and v a place of F dividing an odd prime p. We study the weight part of Serre's conjecture for continuous, totally odd, two-dimensional mod p representations rhobar of the absolute Galois group of F that are reducible locally at v. Let W be the set of predicted Serre weights for the semisimplification of rhobar restricted to the decomposition group at v. We prove that when the local representation is generic, the Serre weights in W for which rhobar is modular are exactly the ones that are predicted (assuming that rhobar is modular). We also determine precisely which subsets of W arise as predicted weights when the local representation varies with fixed generic semisimplification.

math.NT↗
arXiv2013

The Buzzard-Diamond-Jarvis conjecture for unitary groups

Let p > 2 be prime. We prove the weight part of Serre's conjecture for rank two unitary groups for mod p representations in the unramified case (that is, the Buzzard-Diamond-Jarvis conjecture for unitary groups), by proving that any Serre weight which occurs is a predicted weight. This completes the analysis begun in [BLGG11], which proved that all predicted Serre weights occur. Our methods are purely local, using the theory of (phi,Ghat)-modules to determine the possible reductions of certain two-dimensional crystalline representations.

math.NT↗
arXiv2011

Crystalline extensions and the weight part of Serre's conjecture

Let p>2 be prime. We complete the proof of the weight part of Serre's conjecture for rank two unitary groups for mod p representations in the totally ramified case, by proving that any weight which occurs is a predicted weight. Our methods are a mixture of local and global techniques, and in the course of the proof we establish some purely local results on crystalline extension classes.

math.NT↗
arXiv2010

Serre weights for mod p Hilbert modular forms: the totally ramified case

We study the possible weights of an irreducible 2-dimensional modular mod p representation of the absolute Galois group of F, where F is a totally real field which is totally ramified at p, and the representation is tamely ramified at the prime above p. In most cases we determine the precise list of possible weights; in the remaining cases we determine the possible weights up to a short and explicit list of exceptions.

math.NT↗
arXiv2010

On a Conjecture of Conrad, Diamond, and Taylor

We prove a conjecture of Conrad, Diamond, and Taylor on the size of certain deformation rings parametrizing potentially Barsotti-Tate Galois representations. To achieve this, we extend results of Breuil and Mezard (classifying Galois lattices in semistable representations in terms of "strongly divisible modules") to the potentially crystalline case in Hodge-Tate weights (0,1). We then use these strongly divisible modules to compute the desired deformation rings. As a corollary, we obtain new results on the modularity of potentially Barsotti-Tate representations.

math.NT↗
arXiv2010

Serre weights for quaternion algebras

We study the possible weights of an irreducible two-dimensional mod p representation of the absolute Galois group of F which is modular in the sense of that it comes from an automorphic form on a definite quaternion algebra with centre F which is ramified at all places dividing p, where F is a totally real field. In most cases we determine the precise list of possible weights; in the remaining cases we determine the possible weights up to a short and explicit list of exceptions.

math.NT↗
arXiv2008

Polygones de Hodge, de Newton et de l'inertie modérée des représentations semi-stables

Let k be a perfect field, and K be a totally ramified extension of K_0 = Frac W(k) of degree e. To a semi-stable p-adic representation of G_K (the absolute Galois group of K), one can classicaly associate two polygons : the Hodge polygon et the Newton polygon. It is well known that the former lies below the latter, and that they have same endpoints. In this note, we introduce a third polygon gotten from the semi-simplification of the representation mod p, and, under some conditions on Hodge-Tate weights, we prove that it lies above the Hodge polygon again with same endpoint. We finally examine one exemple associated to a crystalline representation.

math.NT↗