arXiv subjects

Daniel Barrera Salazar

Publications and source records attributed to Daniel Barrera Salazar.

17 recordsLinked to original sources

arXiv2026

The non-abelian Leopoldt conjecture and equalities of $\mathcal{L}$-invariants

Let $G$ be a reductive group quasi-split at $p$. Using arguments of Hansen--Thorne, we show that under the non-abelian Leopoldt conjecture (NALC), Hansen's $p$-adic overconvergent cohomology eigenvariety for $G$ is étale over its image in weight space at any non-critical classical tempered cuspidal point of `cohomological multiplicity one'. This applies to all non-critical classical cuspidal points if $G = \mathrm{Res}_{F/\mathbb{Q}}\mathrm{GL}_n$. We then let $π$ be a $p$-ordinary regular algebraic cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ such that $π_p$ is Steinberg. Combining the above étaleness result for the classical point attached to $π$, and a local-global compatibility result from our earlier work, we deduce -- under a tangent vector hypothesis that is true for at least half the simple roots -- the equality of Fontaine--Mazur and automorphic $\mathcal{L}$-invariants for $π$. Where this assumption is satisfied, we deduce the NALC implies a conjecture of Gehrmann: that automorphic $\mathcal{L}$-invariants are independent of cohomological degree. Our approach is inspired by (and generalises) previous work of Gehrmann--Rosso. When $π= \operatorname{Sym}^{n-1} π_f$ is the symmetric power lift of a modular form, we verify all assumptions other than the NALC, and deduce a functoriality result for the automorphic $\mathcal{L}$-invariants.

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arXiv2026

On periods of Elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$ having split multiplicative reduction at a prime number $p$. We describe the tame part of the $\mathcal{L}$-invariant of $E$ at $p$ in terms of automorphic $p$-adic periods introduced in the work of Darmon. More precisely, we prove an equality of refined $\mathcal{L}$-invariants using twisted versions of refined exceptional zero conjectures. When the conductor of the elliptic curve is exactly $p$ and the automorphic period is attached to an optimal embedding of conductor $1$ then we prove this equality unconditionally by using the work of de-Shalit.

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arXiv2026

On $p$-adic $L$-functions for $GL_{2n}$ in finite slope Shalika families

In this paper, we propose and explore a new connection in the study of $p$-adic $L$-functions and eigenvarieties. We use it to prove results on the geometry of the cuspidal eigenvariety for $\mathrm{GL}_{2n}$ over a totally real number field $F$ at classical points admitting Shalika models. We also construct $p$-adic $L$-functions over the eigenvariety around these points. Our proofs proceed in the opposite direction to established methods: rather than using the geometry of eigenvarieties to deduce results about $p$-adic $L$-functions, we instead show that non-vanishing of a (standard) $p$-adic $L$-function implies smoothness of the eigenvariety at such points. Key to our methods are a family of distribution-valued functionals on (parahoric) overconvergent cohomology groups, which we construct via $p$-adic interpolation of classical representation-theoretic branching laws for $\mathrm{GL}_n \times \mathrm{GL}_n \subset \mathrm{GL}_{2n}$. More precisely, we use our functionals to attach a $p$-adic $L$-function to a non-critical refinement $\tildeπ$ of a regular algebraic cuspidal automorphic representation $π$ of $\mathrm{GL}_{2n}/F$ which is spherical at $p$ and admits a Shalika model. Our new parahoric distribution coefficients allow us to obtain optimal non-critical slope and growth bounds for this construction. When $π$ has regular weight and the corresponding $p$-adic Galois representation is irreducible, we exploit non-vanishing of our functionals to show that the parabolic eigenvariety for $\mathrm{GL}_{2n}/F$ is étale at $\tildeπ$ over an $([F:\mathbb{Q}]+1)$-dimensional weight space and contains a dense set of classical points admitting Shalika models. Under a hypothesis on the local Shalika models at bad places which is empty for $π$ of level 1, we construct a $p$-adic $L$-function for the family.

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arXiv2025

Local-global compatibility and the exceptional zero conjecture for GL(3)

We prove exceptional zero conjectures for $p$-ordinary regular algebraic cuspidal automorphic representations of $\mathrm{GL}_3(\mathbb{A})$ which are Steinberg at $p$. We make no self-duality assumptions. The paper has two parts. In Part 1, we use $p$-arithmetic cohomology to unconditionally prove an automorphic exceptional zero conjecture in this setting, using Gehrmann's automorphic $\mathcal{L}$-invariant. In Part 2 we prove, under mild assumptions that are expected to always hold, the equality of automorphic and Fontaine--Mazur $\mathcal{L}$-invariants, and thus deduce cases of the full Greenberg--Benois exceptional zero conjecture. As one of the key ingredients for this, we establish local-global compatibility at $\ell = p$ for Galois representations attached to $p$-ordinary torsion classes for $\mathrm{GL}_n$, confirming a conjecture of Hansen in this setting. We prove this for all $n$ following the strategy in the "10-author paper", and use the $n=3$ case to deduce the desired equality of $\mathcal{L}$-invariants.

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arXiv2025

Higher Hida theory for Drinfeld modular curves

Inspired by the construction of Higher Hida theory of Boxer and Pilloni, we develop Higher Hida theory for the cohomology of the line bundles of Drinfeld modular forms on the Drinfeld modular curve. We also interpolate Serre duality.

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arXiv2025

On $p$-refined Friedberg-Jacquet integrals and the classical symplectic locus in the $\mathrm{GL}_{2n}$ eigenvariety

Friedberg--Jacquet proved that if $π$ is a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A})$, then $π$ is a functorial transfer from $\mathrm{GSpin}_{2n+1}$ if and only if a global zeta integral $Z_H$ over $H = \mathrm{GL}_n \times \mathrm{GL}_n$ is non-vanishing on $π$. We conjecture a $p$-refined analogue: that any $P$-parahoric $p$-refinement $\tildeπ^P$ is a functorial transfer from $\mathrm{GSpin}_{2n+1}$ if and only if a $P$-twisted version of $Z_H$ is non-vanishing on the $\tildeπ^P$-eigenspace in $π$. This twisted $Z_H$ appears in all constructions of $p$-adic $L$-functions via Shalika models. We connect our conjecture to the study of classical symplectic families in the $\mathrm{GL}_{2n}$ eigenvariety, and -- by proving upper bounds on the dimensions of such families -- obtain various results towards the conjecture.

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arXiv2025

On the GL(2n) eigenvariety: branching laws, Shalika families and $p$-adic $L$-functions

In this paper, we prove that a $\mathrm{GL}(2n)$-eigenvariety is étale over the (pure) weight space at non-critical Shalika points, and construct multi-variable $p$-adic $L$-functions varying over the resulting Shalika components. Our constructions hold in tame level 1 and Iwahori level at $p$, and give $p$-adic variation of $L$-values (of regular algebraic cuspidal automorphic representations of $\mathrm{GL}(2n)$ admitting Shalika models) over the whole pure weight space. In the case of $\mathrm{GL}(4)$, these results have been used by Loeffler and Zerbes to prove cases of the Bloch--Kato conjecture for $\mathrm{GSp}(4)$. Our main innovations are: (a) the introduction and systematic study of `Shalika refinements' of local representations of $\mathrm{GL}(2n)$, and evaluation of their attached local twisted zeta integrals; and (b) the $p$-adic interpolation of representation-theoretic branching laws for $\mathrm{GL}(n) \times \mathrm{GL}(n)$ inside $\mathrm{GL}(2n)$. Using (b), we give a construction of multi-variable $p$-adic functionals on the overconvergent cohomology groups for $\mathrm{GL}(2n)$, interpolating the zeta integrals of (a). We exploit the resulting non-vanishing of these functionals to prove our main arithmetic applications.

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arXiv2024

Geometry of the Bianchi eigenvariety around non-cuspidal points and strong multiplicity-one results

Let $K$ be an imaginary quadratic field. In this article, we study the local geometry of the Bianchi eigenvariety around non-cuspidal classical points, in particular, ordinary non-cuspidal base change points. To perform this study we introduce Bianchi Eisenstein eigensystems and prove strong multiplicity-one results on the cohomology of the corresponding Bianchi threefolds. We believe these results are of independent interest.

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arXiv2023

On the $2$-Selmer group of Jacobians of hyperelliptic curves

Let $\mathcal{C}$ be a hyperelliptic curve $y^2 = p(x)$ defined over a number field $K$ with $p(x)$ integral of odd degree. The purpose of the present article is to prove lower and upper bounds for the $2$-Selmer group of the Jacobian of $\mathcal{C}$ in terms of the class group of the $K$-algebra $K[x]/(p(x))$. Our main result is a formula relating these two quantities under some mild hypothesis. We provide some examples that prove that our lower and upper bounds are as sharp as possible. As a first application, we study the rank distribution of the $2$-Selmer group in families of quadratic twists. Under some extra hypothesis we prove that among prime quadratic twists, a positive proportion has fixed $2$-Selmer group. As a second application, we study the family of octic twists of the genus $2$ curve $y^2 = x^5 + x$.

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arXiv2021

Overconvergent cohomology, $p$-adic $L$-functions and families for $\mathrm{GL}(2)$ over CM fields

The use of overconvergent cohomology in constructing $p$-adic $L$-functions, initiated by Stevens and Pollack--Stevens in the setting of classical modular forms, has now been established in a number of settings. The method is compatible with constructions of eigenvarieties by Ash--Stevens, Urban and Hansen, and is thus well-adapted to non-ordinary situations and variation in $p$-adic families. In this note, we give an exposition of the ideas behind the construction of $p$-adic $L$-functions via overconvergent cohomology. Conditional on the non-abelian Leopoldt conjecture, we illustrate them by constructing $p$-adic $L$-functions attached to families of base-change automorphic representations for $\mathrm{GL}(2)$ over CM fields. As a corollary, we prove a $p$-adic Artin formalism result for base-change $p$-adic $L$-functions.

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arXiv2021

Families of Bianchi modular symbols: critical base-change p-adic L-functions and p-adic Artin formalism

Let $K$ be an imaginary quadratic field. In this article, we study the eigenvariety for $\mathrm{GL}_2/K$, proving an étaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical points. We give three main applications of this; let $f$ be a $p$-stabilised newform of weight $k \geq 2$ without CM by $K$. Suppose $f$ has finite slope at $p$ and its base-change $f_{/K}$ to $K$ is $p$-regular. Then: (1) We construct a two-variable $p$-adic $L$-function attached to $f_{/K}$ under assumptions on $f$ that conjecturally always hold, in particular with no non-critical assumption on $f/K$. (2) We construct three-variable $p$-adic $L$-functions over the eigenvariety interpolating the $p$-adic $L$-functions of classical base-change Bianchi cusp forms. (3) We prove that these base-change $p$-adic $L$-functions satisfy a $p$-adic Artin formalism result, that is, they factorise in the same way as the classical $L$-function under Artin formalism. In an appendix, Carl Wang-Erickson describes a base-change deformation functor and gives a characterisation of its Zariski tangent space.

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arXiv2021

Parabolic eigenvarieties via overconvergent cohomology

Let $G'$ be a connected reductive group over $\mathbb{Q}$ such that $G = G'/\mathbb{Q}_p$ is quasi-split, and let $Q \subset G$ be a parabolic subgroup. We introduce parahoric overconvergent cohomology groups with respect to $Q$, and prove a classicality theorem showing that the small slope parts of these groups coincide with those of classical cohomology. This allows the use of overconvergent cohomology at parahoric, rather than Iwahoric, level, and provides flexible lifting theorems that appear to be particularly well-adapted to arithmetic applications. When $Q$ is the Borel, we recover the usual theory of overconvergent cohomology, and our classicality theorem gives a stronger slope bound than in the existing literature. We use our theory to construct $Q$-parabolic eigenvarieties, which parametrise $p$-adic families of systems of Hecke eigenvalues that are finite slope at $Q$, but that allow infinite slope away from $Q$.

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arXiv2020

On $2$-Selmer groups and quadratic twists of elliptic curves

Let $K$ be a number field and $E/K$ be an elliptic curve with no $2$-torsion points. In the present article we give lower and upper bounds for the $2$-Selmer rank of $E$ in terms of the $2$-torsion of a narrow class group of a certain cubic extension of $K$ attached to $E$. As an application, we prove (under mild hypotheses) that a positive proportion of prime conductor quadratic twists of $E$ have the same $2$-Selmer group.

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arXiv2019

Triple product p-adic L-functions for Shimura curves over totally real number fields

Let F be a totally real number field. Using a recent geometric approach developed by Andreatta and Iovita we construct several variables p-adic families of finite slope quaternionic automorphic forms over F. It is achieved by interpolating the modular sheaves defined over some auxiliary unitary Shimura curves. Secondly, we attach p-adic L-functions to triples of ordinary p-adic families of quaternionic automorphic eigenforms. This is done by relating trilinear periods to some trilinear products over unitary Shimura curves which can be interpolated adapting the work of Liu-Zhang-Zhang to our families.

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arXiv2017

P-adic L-functions for GL(2)

Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct p-adic L-functions for non-critical slope rational modular forms, the theory has been extended to construct p-adic L-functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then there is a canonical way of constructing a ray class distribution from the corresponding overconvergent symbol, that moreover interpolates critical values of the L-function of the eigenform. We thus define its p-adic L-function to be this distribution.

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arXiv2017

$\mathcal{L}$-invariants and exceptional zeros of Bianchi modular forms

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F. In this paper, we prove an exceptional zero conjecture in the case where f is new at a prime above p. More precisely, for each prime $\mathfrak{p}$ of F above p we prove the existence of an L-invariant $\mathcal{L}_{\mathfrak{p}}$, depending only on $\mathfrak{p}$ and f, such that when the p-adic L-function of f has an exceptional zero at $\mathfrak{p}$, its derivative can be related to the classical L-value multiplied by $\mathcal{L}_{\mathfrak{p}}$. The proof uses cohomological methods of Darmon and Orton, who proved similar results for GL(2) over the rationals. When p is not split and f is the base-change of a classical modular form F, we relate $\mathcal{L}_{\mathfrak{p}}$ to the L-invariant of F, resolving a conjecture of Trifković in this case.

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arXiv2016

On critical values of L-functions of potentially automorphic motives

In this paper we prove a version of Deligne's conjecture for potentially automorphic motives, twisted by certain algebraic Hecke characters. The Hecke characters are chosen in such a way that we can use automorphic methods in the context of totally definite unitary groups.

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