For each subset of primes in a totally real field above a rational prime $p$, there is the notion of partially classical Hilbert modular forms, where the empty set recovers the overconvergent forms and the full set of primes above $p$ yields classical forms. Given such a set, we $p$-adically interpolate the classical modular sheaves to construct families of partially classical Hilbert modular forms with weights varying in appropriate weight spaces and construct the corresponding eigenvariety, generalizing the construction of Andreatta, Iovita, Pilloni, and Stevens.
Let $F$ be a totally real number field and let $f$ be a classical cuspidal $p$-regular Hilbert modular eigenform over $F$ of parallel weight $1$. Let $x$ be the point on the $p$-adic Hilbert eigenvariety $\mathcal E$ corresponding to an ordinary $p$-stabilization of $f$. We show that if the $p$-adic Schanuel Conjecture is true, then $\mathcal E$ is smooth at $x$ if $f$ has CM. If we additionally assume that $F/\mathbb Q$ is Galois, we show that the weight map is étale at $x$ if $f$ has either CM or exotic projective image (which is the case for almost all cuspidal Hilbert modular eigenforms of…
In this article we construct a $p$-adic three dimensional Eigenvariety for the group $U(2,1)(E)$, where $E$ is a quadratic imaginary field and $p$ is inert in $E$. The Eigenvariety parametrizes Hecke eigensystems on the space of overconvergent, locally analytic, cuspidal Picard modular forms of finite slope. The method generalized the one developed in Andreatta-Iovita-Pilloni by interpolating the coherent automorphic sheaves when the ordinary locus is empty. As an application of this construction, we reprove a particular case of the Bloch-Kato conjecture for some Galois characters of $E$, exte…
Generalising the recent method of Andreatta, Iovita, and Pilloni for cuspidal forms, we construct an eigenvariety for symplectic and unitary groups that parametrises systems of eigenvalues of overconvergent and locally analytic $p$-adic automorphic forms. This is achieved by gluing some intermediates eigenvarieties of a fixed 'degree of cuspidality'. The dimension of these eigenvarieties is explicit and depends on the degree of cuspidality, it is maximal for cuspidal forms and it is $1$ for forms that are 'not cuspidal at all'. Under mild assumption, we are able to prove a conjecture of Urban …
Let $F$ be a totally real field and $p$ be an odd prime which splits completely in $F$. We prove that the eigenvariety associated to a definite quaternion algebra over $F$ satisfies the following property: over a boundary annulus of the weight space, the eigenvariety is a disjoint union of countably infinitely many connected components which are finite over the weight space; on each fixed connected component, the ratios between the $U_\mathfrak{p}$-slopes of points and the $p$-adic valuations of the $\mathfrak{p}$-parameters are bounded by explicit numbers, for all primes $\mathfrak{p}$ of $F$…
We consider $p$-adic families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zer…
Andreatta-Iovita-Pilloni constructed eigenvarieties for cuspidal Hilbert modular forms. The eigenvariety has a natural map to the weight space, called the weight map. At a classical point, we compute a lower bound of the dimension of the tangent space of the fiber of the weight map using Galois deformation theory. Along with the classicality theorem due to Tian-Xiao, this enables us to characterize the classical points of the eigenvariety which are ramified over the weight space, in terms of the local splitting behavior of the associated Galois representation.
Let $p>2$ be a prime number, and $L$ be a finite extension of $\mathbb{Q}_p$, we prove Breuil's locally analytic socle conjecture for $\mathrm{GL}_2(L)$, showing the existence of all the companion points on the definite (patched) eigenvariety. This work relies on infinitesimal "R=T" results for the patched eigenvariety and the comparison of (partially) de Rham families and (partially) Hodge-Tate families. This method allows in particular to find companion points of non-classical points.
We generalize some of the results of Andreatta, Iovita, and Pilloni and the author to Hodge type Shimura varieties having non-empty ordinary locus. For any $p$-adic weight $κ$, we give a geometric definition of the space of overconvergent modular forms of weight $κ$ in terms of sections of a sheaf. We show that our sheaves live in analytic families, interpolating the classical sheaves for integral weights. We define an action of the Hecke algebra, including a completely continuous operator at $p$. In some simple cases, we also build the eigenvariety.
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…
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.
Let F be a totally real field of degree d and let p be an odd prime which is totally split in F. We define and study one-dimensional partial eigenvarieties interpolating Hilbert modular forms over F with weight varying only at a single place v above p. For these eigenvarieties, we show that methods developed by Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight space of sufficiently small radius, the partial eigenvarieties decompose as a disjoint union of components which are finite over weight space. We apply this result to prove the parity version of the Bloch--Kato c…
Let F be a totally real field and p a rational prime unramified in F. We prove a partial classicality theorem for overconvergent Hilbert modular forms: when the slope is small compared to certain but not all weights, an overconvergent form is partially classical. We use the method of analytic continuation.
We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms $\bmod \ell$, partially answering a question of Bellaïche-Soundararajan concerning the asymptotic formula for the count of the number of Fourier coefficients upto $x$ which do not vanish $\bmod \ell$. We also propose a precise conjecture as a possible answer to this question. Further, we prove several results related to the nonvanishing of arithmetically interesting (e.g., primitive or fundamental) Fourier coefficients $\bmod \ell $ of a Siegel modular form with integral algebraic Fourier coefficients provi…
The existence and construction of vector-valued modular forms (vvmf) for any arbitrary Fuchsian group $\mathrm{G}$, for any representation $ρ:\mathrm{G} \longrightarrow \mathrm{GL}_{d}(\mathbb{C})$ of finite image can be established by lifting scalar-valued modular forms of the finite index subgroup $Ker(ρ)$ of $\mathrm{G}$. In this article vvmf are explicitly constructed for any admissible multiplier (representation) $ρ$, see Section 3 for the definition of admissible multiplier. In other words, the following question has been partially answered: For which representations $ρ$ of a given $\mat…
For each prime $\ell$, let $|\cdot|_\ell$ be an extension to $\bar \Q$ of the usual $\ell$-adic absolute value on $\Q$. Suppose $g(z) = \sum_{n=0}^\infty c(n)q^n \in M_{k+\half}(N)$ is an eigenform whose Fourier coefficients are algebraic integers. Under a mild condition, for all but finitely many primes $\ell$ there are infinitely many square-free integers $m$ for which $|c(m)|_\ell = 1$. Consequently we obtain indivisibility results for ``algebraic parts'' of central critical values of modular $L$-functions and class numbers of imaginary quadratic fields. These results partially answer a con…
Mock modular forms have found applications in numerous branches of mathematical sciences since they were first introduced by Ramanujan nearly a century ago. In this proceeding we highlight a new area where mock modular forms start to play an important role, namely the study of three-manifold invariants. For a certain class of Seifert three-manifolds, we describe a conjecture on the mock modular properties of a recently proposed quantum invariant. As an illustration, we include concrete computations for a specific three-manifold, the Brieskorn sphere $Σ(2,3,7)$. This note is partially based on …
