On the slopes of the $U_5$ operator acting on overconvergent modular forms

P-adic family of half-integral weight modular forms via overconvergent Shintani lifting

The goal of this paper is to construct the p-adic analytic family of overconvergent half-integral weight modular forms using Hecke-equivariant overconvergent Shintani lifting. The classical Shintani map is the Hecke-equivariant map from the space of cusp forms of integral weight to the space of cusp forms of half-integral weight. Glenn Stevens proved in [St1] that there is Lamda-adic lifting of this map to the Hida family of ordinary cusp forms of integral weight and consequently constructed Lamda-adic modular eigen form of half-integral weight. The natural thing to do is generalizing his resu…

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 …

Spectral expansions of overconvergent modular functions

The main result of this paper is an instance of the conjecture made by Gouvea and Mazur (Math. Res. Lett., 1995) which asserts that for certain values of r the space of r-overconvergent p-adic modular forms of tame level N and weight k should be spanned by the finite slope Hecke eigenforms. Using methods adapted from the work of Buzzard and Calegari I show that for p=2, k = 0, N = 1, this holds for all r in (5/12, 7/12). The proof relies on a certain factorisation of the U_p operator which is known in this case but I conjecture also holds for p = 3 and 5; this conjecture also implies exact for…

A Gluing Lemma And Overconvergent Modular Forms

We prove a gluing lemma for sections of line bundles on a rigid analytic variety. We apply the lemma, in conjunction with a result of Buzzard's, to give a proof of (a generalization) of Coleman's theorem which states that overconvergent modular forms of small slope are classical. The proof is "geometric" in nature, and is suitable for generalization to other PEL Shimura varieties.

Slope classicality via completed cohomology

We give a new proof of the slope classicality theorem in classical and higher Coleman theory for modular curves at arbitrary level using the completed cohomology classes attached to overconvergent modular forms. The latter give an embedding of the quotient of overconvergent modular forms by classical modular forms, which is the obstruction space for classicality in either cohomological degree, into a unitary representation of $\mathrm{GL}_2(\mathbb{Q}_p)$. The $U_p$ operator becomes a double-coset, and unitarity yields the slope vanishing.

Minimal slope conjecture of $F$-isocrystals

The minimal slope conjecture, which was proposed by K.Kedlaya, asserts that two irreducible overconvergent $F$-isocrystals on a smooth variety are isomorphic to each other if both minimal slope constitutions of slope filtrations are isomorphic to each other. We affirmatively solve the minimal slope conjecture for overconvergent $F$-isocrystals on curves and for overconvergent $\bar{\mathbb Q}_p$-$F$-isocrystals on smooth varieties over finite fields.

Slopes of modular forms and the ghost conjecture, II

In a previous article, we constructed an entire power series over $p$-adic weight space (the 'ghost series') and conjectured, in the $Γ_0(N)$-regular case, that this series encodes the slopes of overconvergent modular forms of any $p$-adic weight. In this paper, we construct 'abstract ghost series' which can be associated to various natural subspaces of overconvergent modular forms. This abstraction allows us to generalize our conjecture to, for example, the case of slopes of overconvergent modular forms with a fixed residual representation that is locally reducible at $p$. Ample numerical evi…

Computing overconvergent forms for small primes

In this note, we construct explicit bases for spaces of overconvergent $p$-adic modular forms when $p=2,3$ and study their stability under the Atkin operator. The resulting extension of the algorithms of Lauder is illustrated with computations of slope sequences of some $2$-adic eigencurves and the construction of Chow-Heegner points on elliptic curves via special values of Rankin triple product L-functions.

Fourier coefficients of the overconvergent generalized eigenform associated to a CM form

Let f be a modular form with complex multiplication. If f has critical slope, then Coleman's classicality theorem implies that there is a p-adic overconvergent generalized Hecke eigenform with the same Hecke eigenvalues as f. We give a formula for the Fourier coefficiets of this generalized Hecke eigenform. We also investigate the dimension of the generalized Hecke eigenspace of p-adic overconvergent forms containing f.

The 2-adic Eigencurve is Proper

For p=2 and tame level N=1 we prove that the map from the (Coleman-Mazur) Eigencurve to weight space satisfies the valuative criterion of properness. More informally, we show that the Eigencurve has no "holes"; given a punctured disc of finite slope overconvergent eigenforms over weight space, the center can be "filled in" with a finite slope overconvergent eigenform.

The half-integral weight eigencurve

In this paper we define Banach spaces of overconvergent half-integral weight $p$-adic modular forms and Banach modules of families of overconvergent half-integral weight $p$-adic modular forms over admissible open subsets of weight space. Both spaces are equipped with a continuous Hecke action for which $U_{p^2}$ is moreover compact. The modules of families of forms are used to construct an eigencurve parameterizing all finite-slope systems of eigenvalues of Hecke operators acting on these spaces. We also prove an analog of Coleman's theorem stating that overconvergent eigenforms of suitably l…

Overconvergent quaternionic forms and anticyclotomic p-adic L-functions

We reinterpret the explicit construction of Gross points given by Chida-Hsieh as a non-Archimedian analogue of the standard geodesic cycle from zero to the infinity on the Poincare upper half plane. This analogy allows us to consider certain distributions, which can be regarded as anticyclotomic p-adic L-functions for modular forms of non-critical slope following the overconvergent strategy a la Stevens. We also give a geometric interpretation of their Gross points for the case of weight two forms. Our construction generalizes those of Bertolini-Darmon, Bertolini-Darmon-Iovita-Spiess, and Chid…

Iwasawa theory of overconvergent modular forms, I: Critical $p$-adic $L$-functions

We construct an Euler system of $p$-adic zeta elements over the eigencurve which interpolates Kato's zeta elements over all classical points. Applying a big regulator map gives rise to a purely algebraic construction of a two-variable $p$-adic $L$-function over the eigencurve. As a first application of these ideas, we prove the equality of the $p$-adic $L$-functions associated with a critical-slope refinement of a modular form by the works of Bellaïche/Pollack-Stevens and Kato/Perrin-Riou.

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…

Triple product p-adic L-functions associated to finite slope p-adic families of modular forms, with an appendix by Eric Urban

We p-adically interpolate the relative de Rham cohomology of the universal elliptic curve over strict neighbourhoods of the ordinary locus of modular curves, together with the Hodge filtration and Gauss-Manin connection. Sections of these sheaves provide the so called nearly overconvergent modular forms. This extends previous work of Andreatta, Iovita and Pilloni where we p-adically interpolate powers of the Hodge bundle and in that case the sections coincide with Coleman overconvergent modular forms. We also show that, under suitable assumptions, one can p-adically interpolate the Gauss-Manin…

Overconvergent algebraic automorphic forms

I present a general theory of overconvergent p-adic automorphic forms and eigenvarieties for connected reductive algebraic groups G whose real points are compact modulo centre, extending earlier constructions due to Buzzard, Chenevier and Yamagami for forms of GL(n). This leads to some new phenomena, including the appearance of intermediate spaces of "semi-classical" automorphic forms; this gives a hierarchy of interpolation spaces (eigenvarieties) interpolating classical automorphic forms satisfying different finite slope conditions (corresponding to a choice of parabolic subgroup of G at p).…

对数迪乌多理论与过收敛扩展

Logarithmic Dieudonné theory and overconvergent extensions

本文通过引入对数迪乌多理论，为特定类别的$F$-isocrystals提供了抛物性猜想的另一种证明，并建立了潜在半稳定$p$-divisible群与过收敛$F$-isocrystals之间的等价关系。

$2$-adic slopes of Hilbert modular forms over $\mathbb{Q}(\sqrt{5})$

We show that for arithmetic weights with a fixed finite order character, the slopes of $U_p$ (for $p=2$) acting on overconvergent Hilbert modular forms of level $U_0(4)$ are independent of the (algebraic part of the) weight and can be obtained by a simple recipe from the classical slopes in parallel weight $3$.

Partial Classicality of Hilbert Modular Forms

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.

Slopes of modular forms and the ghost conjecture

We formulate a conjecture on slopes of overconvergent p-adic cuspforms of any p-adic weight in the Gamma_0(N)-regular case. This conjecture unifies a conjecture of Buzzard on classical slopes and more recent conjectures on slopes "at the boundary of weight space".

Critical slope p-adic L-functions of CM modular forms

For ordinary modular forms, there are two constructions of a p-adic L-function attached to the non-unit root of the Hecke polynomial, which are conjectured but not known to coincide. We prove this conjecture for modular forms of CM type, by calculating the the critical-slope L-function arising from Kato's Euler system and comparing this with results of Bellaiche on the critical-slope L-function defined using overconvergent modular symbols.

p-adic Eichler-Shimura maps for the modular curve

We give a new proof of Faltings's p-adic Eichler-Shimura decomposition of the modular curves via BGG methods and the Hodge-Tate period map. The key property is the relation between the Tate module and the Faltings extension, which was already used in the original proof. Then, we construct overconvergent Eichler-Shimura (ES) maps for the modular curves providing ''the second half'' of the overconvergent ES map of Andreatta-Iovita-Stevens. We use higher Coleman theory on the modular curve developed by Boxer-Pilloni to show that the small slope part of the ES maps interpolates the classical p-adi…

Lubin-Tate theory and overconvergent Hilbert modular forms of low weight

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $Γ$ be the Galois group of the cyclotomic extension of $K$. Fontaine's theory gives a classification of $p$-adic representations of $\mathrm{Gal}\left(\overline{K}/K\right)$ in terms of $(φ,Γ)$-modules. A useful aspect of this classification is Berger's dictionary which expresses invariants coming from $p$-adic Hodge theory in terms of these $\left(φ,Γ\right)$-modules. In this paper, we use the theory of locally analytic vectors to generalize this dictionary to the setting where $Γ$ is the Galois group of a Lubin-Tate extension of $K$. …

p-adic Cohomology and classicality of overconvergent Hilbert modular forms

Let $F$ be a totally real field in which $p$ is unramified. We prove that, if a cuspidal overconvergent Hilbert cuspidal form has small slopes under $U_p$-operators, then it is classical. Our method follows the original cohomological approach of Coleman. The key ingredient of the proof is giving an explicit description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety. A byproduct of the proof is to show that, at least when $p$ is inert, of the rigid cohomology of the ordinary locus has the same image as the classical forms in the Grothendieck group of Hecke …

Slopes of modular forms and reducible Galois representations: an oversight in the ghost conjecture

The ghost conjecture, formulated by this article's authors, predicts the list of p-adic valuations of the non-zero p-th eigenvalues ("slopes") for overconvergent p-adic modular eigenforms in terms of the Newton polygon of an easy-to-describe power series (the "ghost series"). The prediction is restricted to eigenforms whose Galois representation modulo p is reducible on a decomposition group at p. It has been discovered, however, that the conjecture is not formulated correctly. Here we explain the issue and propose a salvage.

Slopes of modular forms and the ghost conjecture (unabridged version)

We formulate a conjecture on slopes of overconvergent p-adic cuspforms of any p-adic weight in the Gamma_0(N)-regular case. This conjecture unifies a conjecture of Buzzard on classical slopes and more recent conjectures on slopes "at the boundary of weight space".

$p$-adic families of automorphic forms over some unitary Shimura varieties

We construct some $n$-dimensional eigenvarieties for finite slope overconvergent eigenforms over some unitary Shimura varieties with signature $(1,n-1)\times(0,n)\times\cdots\times(0,n)$ by adapting Andreatta-Iovita-Pilloni's method. We also show that there are some Galois pseudo-characters over our eigenvarieties by studying analytic continuation of finite slope eigenforms over these Shimura varieties.

Quaternionic modular forms of any weight

In this work we construct an eigencurve for p-adic modular forms attached to an indefinite quaternion algebra over Q. Our theory includes the definition, both as rules on test objects and sections of line bundle, of p-adic modular forms, convergent and overconvergent, of any p-adic weight. We prove that our modular forms can be put in analytic families over the weight space and we introduce the Hecke operators U and T_l, that can also be put in families. We show that the U-operator acts compactly on the space of overconvergent modular forms. We finally construct the eigencurve, a rigid analyti…

Familles de formes modulaires de Drinfeld pour le groupe général linéaire

Let $F$ be a function field over $\mathbb{F}_q$, $A$ its ring of regular functions outside a place $\infty$ and $\mathfrak{p}$ a prime ideal of $A$. First, we develop Hida theory for Drinfeld modular forms of rank $r$ which are of slope zero for a suitably defined Hecke operator $\mathrm{U}_{\mathfrak{p}}$. Second, we show the existence in the finite slope case of families of Drinfeld modular forms varying continuously with respect to the weight. Finally, we show a classicity result: an overconvergent Drinfeld modular form of sufficiently small slope with respect to the weight is a classical D…

$P$-adic $L$-functions of Bianchi modular forms

The theory of overconvergent modular symbols, developed by Rob Pollack and Glenn Stevens, gives a beautiful and effective construction of the $p$-adic $L$-function of a modular form. In this paper, we give an analogue of their results for Bianchi modular forms, that is, modular forms over imaginary quadratic fields. In particular, we prove control theorems that say that the canonical specialisation map from overconvergent to classical Bianchi modular symbols is an isomorphism on small slope eigenspaces of suitable Hecke operators. We also give an explicit link between the classical modular sym…

临界Λ-adic模形式与双普通复形

Critical Lambda-adic modular forms and bi-ordinary complexes

本文构造了Λ-adic临界斜率过收敛模形式的平坦Λ-模，实现Hida型插值，并插值双普通复形，证明相关Hecke代数的R=T定理，且产生度移位Hecke作用及Stark单位作用。

作用于 $\mathrm{GL}_3$ 调和上循环的 $U$-算子及其斜率

$U$-Operators Acting on Harmonic Cocycles for $\mathrm{GL}_3$ and Their Slopes

本文在正特征设定下，计算研究了作用于 $\mathrm{GL}_3$ 调和上循环空间的两个自然 $U$-算子，实现了算子公式并分析了 Hecke 作用的斜率数据，揭示了若干模式与现象。

一个对称的 $p$-adic 符号，用于三个模形式的三重积

A symmetric $p$-adic symbol for triples of modular forms

本文基于 Garrett-Rankin 三重积 p-adic L-函数引入新的 p-adic 三重符号，证明其在置换三个模形式时满足对称性，并扩展 Lauder 算法以支持普通投影，提供计算示例及高效计算高权重 Poincaré 配对的方法。

有限斜率族的 $p$-进 Asai 与扭曲三重积 $L$-函数

$p$-adic Asai and twisted triple product $L$-functions for finite slope families

本文利用Hilbert模曲面的近过度收敛高阶Coleman理论，定义了有限斜率Hilbert模形式族的二元$p$-进Asai及扭曲三重积$L$-函数，推广了已有构造并拓展了经典$L$-值的插值范围。

Reductions of Galois representations and the theta operator

Let $p\ge 5$ be a prime, and let $f$ be a cuspidal eigenform of weight at least $2$ and level coprime to $p$ of finite slope $α$. Let $\barρ_f$ denote the mod $p$ Galois representation associated with $f$ and $ω$ the mod $p$ cyclotomic character. Under an assumption on the weight of $f$, we prove that there exists a cuspidal eigenform $g$ of weight at least $2$ and level coprime to $p$ of slope $α+1$ such that $$\barρ_f \otimes ω\simeq \barρ_g,$$ up to semisimplification. The proof uses Hida-Coleman families and the theta operator acting on overconvergent forms. The structure of the reductions…

Galois level and congruence ideal for $p$-adic families of finite slope Siegel modular forms

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…

On $p$-adic $L$-functions for Hilbert modular forms

We construct $p$-adic $L$-functions associated with $p$-refined cohomological cuspidal Hilbert modular forms over any totally real field under a mild hypothesis. Our construction is canonical, varies naturally in $p$-adic families, and does not require any small slope or non-criticality assumptions on the $p$-refinement. The main new ingredients are an adelic definition of a canonical map from overconvergent cohomology to a space of locally analytic distributions on the relevant Galois group and a smoothness theorem for certain eigenvarieties at critically refined points.

Hilbert模形式的有限斜率族的扭曲三重积$p$-adic $L$-函数

Twisted Triple Product $p$-adic $L$-function for Finite Slope Families of Hilbert Modular Forms

本文构造了Hilbert模簇上相对de Rham上同调类的过收敛族，证明其Gauss-Manin联络的$p$-adic迭代收敛性，并据此构造了Hilbert模形式有限斜率族的$p$-adic扭曲三重积$L$-函数。

一个p进上同调方法用于解析模形式的同余

A $p$-adic cohomological approach to congruences of meromorphic modular forms

本文提出了一种基于p进上同调的方法，用于研究解析模形式的同余关系，通过刚性上同调与晶格结构的相互作用，建立了弗罗贝尼乌斯作用与U_p算子之间的联系。

Families of Picard modular forms and an application to the Bloch-Kato conjecture

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…

Arithmetic properties of Fredholm series for p-adic modular forms

We study the relationship between recent conjectures on slopes of overconvergent p-adic modular forms "near the boundary" of p-adic weight space. We also prove in tame level 1 that the coefficients of the Fredholm series of the U_p operator never vanish modulo p, a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo p using a modular interpretation of the mod p reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.

Syntomic regulators of Asai--Flach classes

In this paper, we derive a formula for the p-adic syntomic regulators of Asai--Flach classes. These are cohomology classes forming an Euler system associated to a Hilbert modular form over a quadratic field, introduced in an earlier paper (arXiv:1607.07813) by Antonio Lei and the first and third authors. The formula we develop here is expressed in terms of differential operators acting on overconvergent Hilbert modular forms; it is analogous to existing formulae for the regulators of Beilinson--Flach classes, but a novel feature is the appearance of a projection operator associated to a critic…

A control theorem for $p$-adic automorphic forms and Teitelbaum's $\mathcal{L}$-invariant

In this article, we describe an efficient method for computing Teitelbaum's $p$-adic $\mathcal{L}$-invariant. These invariants are realized as the eigenvalues of the $\mathcal{L}$-operator acting on a space of harmonic cocycles on the Bruhat-Tits tree $\mathcal{T}$, which is computable by the methods of Franc and Masdeu described in [FM14]. The main difficulty in computing the $\mathcal{L}$-operator is the efficient computation of the $p$-adic Coleman integrals in its definition. To solve this problem, we use overconvergent methods, first developed by Darmon, Greenberg, Pollack and Stevens. In…

Approximation to points in the plane by SL(2,Z)-orbits

Let x be a point in R^2 with irrational slope and let Γdenote the lattice SL(2,Z) acting linearly on R^2. Then, the orbit Γx is dense in R^2. We give efective results on the approximation of a point y in R^2 by points of the form γx, where γbelongs to Γ, in terms of the size of γ.

$\ell$-刚性空间的$\ell$-进上同调群的独立性、斜率与权重

$\ell$-Independence, slopes, and weights of $\ell$-adic cohomology groups of rigid spaces

针对不同素数ℓ和p的局部域上的刚性空间，研究ℓ进上同调群的相关性质，证明Weil群正部作用的迹交错和与ℓ无关且为整数，给出几何Frobenius提升特征值p进赋值的最优界，推广相关结果并证明Lu-Zheng猜想。

On the Atkin $U_t$-operator for $Γ_0(t)$-invariant Drinfeld cusp forms

We study the diagonalizability of the Atkin $U_t$-operator acting on Drinfeld cusp forms for $Γ_0(t)$: starting with the slopes of eigenvalues and then moving to the space of cusp forms for $Γ_1(t)$ to use Teitelbaum's interpretation as harmonic cocycles which makes computations more explicit. We prove $U_t$ is diagonalizable in odd characteristic for (relatively) small weights and explicitly compute the eigenvalues. In even characteristic we show that it is not diagonalizable when the weight is odd (except for the trivial cases) and prove some cases of non diagonalizability in even weight as …

Triple products of Coleman's families

We discuss modular forms as objects of computer algebra and as elements of certain p-adic Banach modules. Problem-solving approach in number theory is discussed which is based on the use of generating functions and their links with modular forms. In particular, the critical values of various L-functions of modular forms produce non-trivial but computable solutions of arithmetical problems. Namely, for a prime number $p\ge 5$, we consider three classical cusp eigenforms $$f_j(z)=\sum_{n=1}^\infty a_{n,j}e(nz)\in \Sr_{k_j}(N_j, ψ_j),\ (j=1, 2,3)$$ of weights k_1, k_2, k_3, of conductors N_1, N_2…

p-adic Banach modules of arithmetical modular forms and triple products of Coleman's families

For a prime number $p\ge 5$, we consider three classical cusp eigenforms $f_j(z)$ of weights $k_1, k_2, k_3$, of conductors $N_1, N_2, N_3$, and of nebentypus characters $ψ_j \bmod N_j$. According to H.Hida and R.Coleman, one can include each $f_j$ into a {$p$-adic analytic family} $k_j \mapsto \{f_{j,k_j}\}$ of cusp eigenforms $f_{j,k_j}$ of weights $k_j$ in such a way that $f_{j,k_j}=f_j$, and that all their Fourier coefficients $a_n(f_{j, k_j})$ are given by certain $p$-adic analytic functions $k_j{}\mapsto a_{n, j}(k_j{})$. The purpose of this paper is to describe a four variable $p$-adic …