[Paper Review] Les Variétés de Hecke-Hilbert aux points classiques de poids $1$

Adel Betina|arXiv (Cornell University)|Jun 28, 2016

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper studies the local geometry of Hilbert eigenvarieties at classical weight one Hilbert modular forms using Galois deformation theory and p-adic cohomology. It proves that the eigenvariety is smooth at such points under Leopoldt and regularity conditions, computes the tangent space dimension of the fiber, and constructs non-classical overconvergent eigenforms whose Fourier coefficients are expressed via p-adic logarithms of algebraic numbers.

ABSTRACT

We show that the Eigenvariety attached to Hilbert modular forms over a totally real field $F$ is smooth at the points corresponding to certain classical weight one theta series and we give a precise criterion for etaleness over the weight space at those points. In the case where the theta series has real multiplication, we construct a non-classical overconvergent generalised eigenform and compute its Fourier coefficients in terms of $p$-adic logarithms of algebraic numbers. Our approach uses deformations and pseudo-deformations of Galois representations.

Motivation & Objective

• To describe the local structure of Hilbert eigenvarieties at classical weight one Hilbert modular forms.
• To determine the smoothness and tangent space dimension of the eigenvariety at such points.
• To construct non-classical overconvergent eigenforms and relate their Fourier coefficients to p-adic logarithms.
• To establish conditions under which the weight morphism is étale at weight one points.
• To generalize results from the elliptic case to totally real fields using deformation theory and class field theory.

Proposed method

• Uses Galois deformation theory of Mazur and pseudo-deformations to study local geometry of the eigenvariety.
• Applies cohomological tools from class field theory to analyze the structure of the eigenvariety and its fibers.
• Introduces p-stabilized weight one forms as points on the quasi-ordinary locus of the eigenvariety.
• Computes the tangent space of the fiber of the weight morphism using ramification and decomposition data at primes above p.
• Constructs overconvergent eigenforms via a surjection from the local ring of the eigenvariety to a truncated power series ring.
• Expresses Fourier coefficients of non-classical forms using p-adic logarithms of algebraic integers under Leopoldt's conjecture.

Experimental results

Research questions

• RQ1 Under what conditions is the Hilbert eigenvariety smooth at a classical weight one Hilbert modular form?
• RQ2 What is the dimension of the tangent space to the fiber of the weight morphism at such a point?
• RQ3 When is the weight morphism étale at a classical weight one point?
• RQ4 Can non-classical overconvergent eigenforms be constructed at weight one points, and how are their Fourier coefficients characterized?
• RQ5 How do p-adic logarithms of algebraic numbers relate to Fourier coefficients of overconvergent Hilbert modular forms of weight one?

Key findings

• The Hilbert eigenvariety is smooth at classical weight one Hilbert modular forms under the assumption of p-regularity and Leopoldt's conjecture for the associated quadratic extension.
• The dimension of the tangent space of the fiber of the weight morphism at such a point is equal to ∑_{p_i ∈ S_p} e_i f_i, where e_i and f_i are the ramification and inertia degrees at primes above p.
• The tangent space dimension of the quasi-ordinary locus is max{1, ∑_{p_i ∈ S_p} e_i f_i}, reflecting the presence of ordinary structures.
• When S_p is non-empty, there exists a surjection from the local ring of the eigenvariety to a truncated power series ring over Q_p, yielding a 2-dimensional space of overconvergent eigenforms.
• Among these, a non-classical eigenform exists, and its Fourier coefficients are expressed as p-adic logarithms of algebraic numbers.
• The construction relies on the validity of Leopoldt's conjecture for the quadratic extension M associated to the theta series.