Modular representations of p-adic groups

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…

关于Igusa塔上p进积分的理论

On the p-adic integration over Igusa towers of Siegel modular varieties

本文提出了一种针对Igusa塔的显式p进积分理论，用于研究模Siegel流形的互反律。

The p-adic Shintani modular symbol and evil Eisenstein series

We compute the p-adic L-function (in the sense of Pollack-Stevens and Bellaiche) of the "evil twin" Eisenstein series. We use Solomon's Shintani cocycle to construct a p-adic Shintani modular symbol, and show how to specialize this modular symbol to overconvergent modular symbols with the same p-adic L-function as evil/critical-slope Eisenstein series. This gives a new proof of recent results of Bellaiche and Dasgupta.

p-adic Abel-Jacobi map and p-adic Gross-Zagier formula for Hilbert modular forms

We compute the p-adic Abel-Jacobi map of the product of a Hilbert modular surface and a modular curve at a null-homologous (modified) embedding of the modular curve in this product, evaluated on differentials associated to a Hilbert cuspidal form f of weight (2,2) and a cuspidal form of weight 2. We generalize this computation to suitable null-homological cycles in the fibre products of the universal families on the surface and the curve, evaluated at differentials associated to f and g of higher weights. We express the values of the p-adic Abel-Jacobi map at these weights in terms of a p-adic…

A short note on p-adic families of Hilbert Modular Forms

We extend previous work of the author using an idea of Buzzard and give an elementary construction of non-ordinary $p$-adic families of Hilbert Modular Eigenforms.

Traces on Hecke algebras and families of p-adic modular forms

In this preprint we prove that any finite slope modular form fits into a p-adic family of modular forms which is indexed by the weight. Here, the term p-adic family means that p-adic congruences between weights entail certain p-adic congruences between the corresponding modular forms. We also show that the dimension of the slope subspace of the space of modular forms does not depend on the weight. Both statements are predicted by the Mazur-Gouvea Conjecture, which has been proven by Coleman using methods from rigid analytic geometry. In contrast our proof is based on a comparison of (topologic…

On the $p$-adic periods of the modular curve $X(Γ_0(p) \cap Γ(2))$

We prove a variant of Oesterlé's conjecture about $p$-adic periods of the modular curve $X_0(p)$, with an additional $Γ(2)$-structure. We use de Shalit's techniques and $p$-adic uniformization of curves whose reduction is semi-stable.

Modular forms and $p$-adic numbers (in Russian)

Let $p$ be a prime. We discuss $p$-adic properties of various arithmetical functions related to the coefficients of modular form and generating functions. Modular forms are considered as a tool of solving arithmetical problems. Examples of congruences between modular forms modulo $p$ and modulo $p^n$ are given, and the use of a computer for the study of modular forms is illistrated.

近过收敛模形式上的高斯–马宁联络的p进插值与p进L函数

p-adic interpolation of Gauss--Manin connections on nearly overconvergent modular forms and p-adic L-functions

本文通过新的几何定义研究近过收敛模形式，并对高斯–马宁联络进行p进插值，改进Andreatta–Iovita的结果，进而应用于Rankin–Selberg和三重积p进L函数的构造。

$p$-Adic hypergeometric functions in the connections with certain twisted Kloosterman sheaf sum and modular forms

In this paper we establish certain identities connecting $p$-adic hypergeometric functions with 4-th twisted Kloosterman sheaf sum. To prove these identities we express certain character sum over finite field in terms of special values of $p$-adic hypergeometric functions. One conjecture of Evans behaves as a bridge to connect $p$-adic hypergeometric functions with Kloosterman sheaf sum. We also connect $p$-adic hypergeometric functions with Fourier coefficients of certain modular forms.

Hecke algebras associated to $Λ$-adic modular forms

We show that if an Eisenstein component of the $p$-adic Hecke algebra associated to modular forms is Gorenstein, then it is necessary that the plus-part of a certain ideal class group is trivial. We also show that this condition is sufficient whenever a conjecture of Sharifi holds.

Slopes of overconvergent 2-adic modular forms

The slope of a p-adic overconvergent eigenform of weight k is the p-adic valuation of its U_p eigenvalue. We find the slope of all 2-adic finite slope overconvergent eigenforms of tame level 1 and weight 0. As a consequence we prove that any finite slope 2-adic overconvergent eigenform of tame level 1 and weight 0 has coefficients in Q_2. These results provide evidence towards conjectures of the first author that predict the slopes of classical p-adic modular forms under a mild ``Gamma_0(N)''-regular hypothesis on p.

Hypergeometric type identities in the $p$-adic setting and modular forms

We prove hypergeometric type identities for a function defined in terms of quotients of the $p$-adic gamma function. We use these identities to prove a supercongruence conjecture of Rodriguez-Villegas between a truncated $_4F_3$ hypergeometric series and the Fourier coefficients of certain weight four modular form.

Geometric and $p$-adic modular forms of half-integral weight

We introduce a geometric formalism for studying modular forms of half-integral weight and explore some of its basic properties. Geometric Hecke operators are constructed and some basic spaces of $p$-adic forms are introduced. The $p$-adic theory is greatly expanded in subsequent papers, making that part of this paper largely obsolete.

Moduli-friendly Eisenstein series over the p-adics and the computation of modular Galois representations

We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of "moduli-friendly" Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic of modular curves points and dispenses us of the need for equations of modular curves and for q-expansion computations. The resulting algorithm compares very favourably to the complex-analytic method.

Hilbert modular forms and p-adic Hodge theory

We consider the p-adic Galois representation associated to a Hilbert modular form. We show the compatibility with the local Langlands correspondence at a place divising p under a certain assumption. We also prove the monodromy-weight conjecture. The prime-to-p case is established by Carayol.

$p$-adic properties of certain half-integral weight modular forms

In this paper, we study the parallel cases of Zagier's and Folsom-Ono's grids of weakly holomorphic (resp. weakly holomorphic and mock modular) forms of weights 3/2 and 1/2, investigating their $p$-adic properties under the action of Hecke operators.

Modularity of nearly ordinary 2-adic residually dihedral Galois representations

We prove modularity of some two dimensional, 2-adic Galois representations over totally real fields that are nearly ordinary and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles, using Hida families, together with the 2-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above 2.

Overconvergent modular symboles and p-adic L-functions

We come back to the construction of p-adic L-functions attached to cusp forms of even weight k in the spirit of G. Stevens, R. Pollack [7] and M. Greenberg [3] with a new unified presentation including the non-ordinary case. This construction is based on Stevens's modular symbols rather than q-developments. We review the proofs in order to obtain an effective algorithm guaranteeing a given p-adic accuracy.

Modular symbols for reductive groups and p-adic Rankin-Selberg convolutions over number fields

We give a construction of a wide class of modular symbols attached to reductive groups. As an application we construct a p-adic distribution interpolating the special values of the twisted Rankin-Selberg L-function attached to cuspidal automorphic representations of GL(n) and GL(n-1) over number fields. If the representations are ordinary at p, our distribution is bounded and yields analyticity of the associated p-adic L-function.

A geometric perspective on p-adic properties of mock modular forms

Bringmann, Guerzhoy and Kane have shown how to correct mock modular forms by a certain linear combination of the Eichler integral of their shadows in order to obtain p-adic modular forms in the sense of Serre. In this paper, we give a new proof of their results (for good primes p) by employing the geometric theory of harmonic Maass forms developed by the first author and the theory of overconvergent modular forms due to Katz and Coleman.

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 interpolation of square roots of central $L$-values of modular forms

We construct a meromorphic function on the eigencurve that interpolates a square root of the ratio of the central values of two quadratic twists of the $L$-function at classical points.

On the $U_p$ operator acting on $p$-adic overconvergent modular forms when $X_0(p)$ has genus 1

In this article we will show how to compute $U_p$ acting on spaces of overconvergent $p$-adic modular forms when $X_0(p)$ has genus 1. We first give a construction of Banach bases for spaces of overconvergent $p$-adic modular forms, and then give an algorithm to approximate both the characteristic power series of the $U_p$ operator and eigenvectors of finite slope for $U_p$, and present some explicit examples. We will also relate this to the conjectures of Clay on the slopes of overconvergent modular forms.

Theta operators, Goss polynomials, and v-adic modular forms

We investigate hyperderivatives of Drinfeld modular forms and determine formulas for these derivatives in terms of Goss polynomials for the kernel of the Carlitz exponential. As a consequence we prove that v-adic modular forms in the sense of Serre, as defined by Goss and Vincent, are preserved under hyperdifferentiation. Moreover, upon multiplication by a Carlitz factorial, hyperdifferentiation preserves v-integrality.

$p$-adic properties of coefficients of weakly holomorphic modular forms

We examine the Fourier coefficients of modular forms in a canonical basis for the spaces of weakly holomorphic modular forms of weights 4, 6, 8, 10, and 14, and show that these coefficients are often highly divisible by the primes 2, 3, and 5.

A 2-adic control theorem for modular curves

We study the behaviour of ordinary parts of the homology modules of modular curves, associated to a decreasing sequence of congruence subgroups $Γ_1(N2^r)$ for $r \geq 2$, and prove a control theorem for these homology modules.

关于四元数普通模形式族和 $p$-adic $L$-函数的家族

On quaternionic ordinary families of modular forms and $p$-adic $L$-functions

本文利用Serre-Tate展开构造四元数普通模形式族的幂级数，并关联到插值大 $p$-adic $L$-函数，扩展了Ohta的部分结果。

On p-adic haromonic Maass functions

Modular and mock modular forms possess many striking $p$-adic properties, as studied by Bringmann, Guerzhoy, Kane, Kent, Ono, and others. Candelori developed a geometric theory of harmonic Maass forms arising from the de Rham cohomology of modular curves. In the setting of over-convergent $p$-adic modular forms, Candelori and Castella showed this leads to $p$-adic analogs of harmonic Maass forms. In this paper we take an analytic approach to construct $p$-adic analogs of harmonic Maass forms of weight $0$ %and $1/2$ with square free level. Although our approaches differ, where the two theories…

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.

Bianchi 模形式的 $p$-进 $L$-函数的函数方程

Functional equation of the $p$-adic $L$-function of Bianchi modular forms

本文研究类数为 1 的虚二次域上 Bianchi 模形式的 $p$-进 $L$-函数，证明小斜率 $p$-稳定化情形的函数方程，并借助 $p$-进族推广至 $\Sigma$-光滑基变换形式。

A remark on non-integral p-adic slopes for modular forms

We give a sufficient condition, namely "Buzzard irregularity", for there to exist a cuspidal eigenform which does not have integral p-adic slope.

关于四元数模形式的 p-进测度

On $p$-adic Measures for Quaternionic Modular Forms

本文利用加倍方法研究四元数模形式的标准 L-函数特殊值，通过建立带特征扭曲的积分表示，构造了插值特殊 L-值的 p-进测度。

$P$-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…

On special zeros of $p$-adic $L$-functions of Hilbert modular forms

Let $E$ be a modular elliptic curve over a totally real number field $F$. We prove the weak exceptional zero conjecture which links a (higher) derivative of the $p$-adic $L$-function attached to $E$ to certain $p$-adic periods attached to the corresponding Hilbert modular form at the places above $p$ where $E$ has split multiplicative reduction. Under some mild restrictions on $p$ and the conductor of $E$ we deduce the exceptional zero conjecture in the strong form (i.e.\ where the automorphic $p$-adic periods are replaced by the $\cL$-invariants of $E$ defined in terms of Tate periods) from a…

非尖点 Bianchi 模形式与 Katz $p$-进 $L$-函数

Non-cuspidal Bianchi modular forms and Katz $p$-adic $L$-functions

本文引入 $C$-尖点性质与部分 Bianchi 模符号，构造非尖点 Bianchi 模形式的 $p$-进 $L$-函数，并在 $p$ 分裂情形证明基变换形式的相应函数分解为两个 Katz $p$-进 $L$-函数之积。

Eisenstein series, p-adic modular functions, and overconvergence

Let $p$ be a prime $\ge 5$. We establish explicit rates of overconvergence for members of the "Eisenstein family", notably for the $p$-adic modular function $V(E_{(1,0)}^{\ast})/E_{(1,0)}^{\ast}$ ($V$ the $p$-adic Frobenius operator) that plays a pi\-votal role in Coleman's theory of $p$-adic families of modular forms. The proof goes via an in-depth analysis of rates of overconvergence of $p$-adic modular functions of form $V(E_k)/E_k$ where $E_k$ is the classical Eisenstein series of level $1$ and weight $k$ divisible by $p-1$. Under certain conditions, we extend the latter result to a vast g…

Siegel模形式的非尖点Hida理论与p进L-函数的平凡零点

Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of $p$-adic $L$-functions

本文通过发展非尖点Siegel模形式的Hida理论，利用Greenberg-Stevens方法构造改进的p进L-函数，研究了P-普通Siegel模形式的标准p进L-函数在半稳定平凡零点处的导数，从而部分证明了Greenberg关于p进L-函数平凡零点的阶数与首项系数的猜想。

On certain constructions of p-adic families of Siegel modular forms of even genus

Suppose that p > 5 is a rational prime. Starting from a well-known p-adic analytic family of ordinary elliptic cusp forms of level p due to Hida, we construct a certain p-adic analytic family of holomorphic Siegel cusp forms of arbitrary even genus and of level p associated with Hida's p-adic analytic family via the Duke-Imamoglu lifting provided by Ikeda. Moreover, we also give a similar results for the Siegel Eisenstein series of even genus with trivial Nebentypus.

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。

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…

On certain constructions of p-adic families of Siegel modular forms of even genus

Suppose that p > 5 is a rational prime. Starting from a well-known p-adic analytic family of ordinary elliptic cusp forms of level p due to Hida, we construct a certain p-adic analytic family of holomorphic Siegel cusp forms of arbitrary even genus and of level p associated with Hida's p-adic analytic family via the Duke-Imamoglu lifting which has been established by Ikeda. Moreover, we also give a similar results for the Siegel Eisenstein series of even genus with trivial Nebentypus.

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…

$Λ$-adic Families of Jacobi Forms

We show that Hida's families of $p$-adic elliptic modular forms generalize to $p$-adic families of Jacobi forms. We also construct $p$-adic versions of theta lifts from elliptic modular forms to Jacobi forms. Our results extend to Jacobi forms previous works by Hida and Stevens on the related case of half-integral weight modular forms.

Bianchi 模符号与 $p$-进 $L$-函数

Bianchi modular symbols and $p$-adic $L$-functions

研究通过特殊 Bianchi 模符号刻画第一同调群并构造 Bianchi 模形式的 $p$-进 $L$-函数，证明正比例普通素数下相关 $\mu$-不变量消失及积分 $L$-值剩余非零。

$p$-adic Limit of Weakly Holomorphic Modular Forms of Half Integral Weight

Serre obtained the p-adic limit of the integral Fourier coefficient of modular forms on $SL_2(\mathbb{Z})$ for $p=2,3,5,7$. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on $Γ_{0}(4N)$ for $N=1,2,4$. A proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications we obtain congruences of Borcherds exponents, congruences of quotient of Eisentein series and congruences of values of $L$-functions at a certain point are also studied. Furthermore, the congruences of the Fourier coeff…

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.

On $p$-adic quaternionic Eisenstein series

We show that certain $p$-adic Eisenstein series for quaternionic modular groups of degree 2 become "real" modular forms of level $p$ in almost all cases. To prove this, we introduce a $U(p)$ type operator. We also show that there exists a $p$-adic Eisenstein series of the above type that has transcendental coefficients. Former examples of $p$-adic Eisenstein series for Siegel and Hermitian modular groups are both rational (i.e., algebraic).

$2$-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 $3$.

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.

P-adic Asai L-functions of Bianchi modular forms

The Asai (or twisted tensor) $L$-function of a Bianchi modular form $Ψ$ is the $L$-function attached to the tensor induction to $\mathbb{Q}$ of its associated Galois representation. In this paper, when $Ψ$ is ordinary at $p$ we construct a $p$-adic analogue of this $L$-function: that is, a $p$-adic measure on $\mathbb{Z}_p^\times$ that interpolates the critical values of the Asai $L$-function twisted by Dirichlet characters of $p$-power conductor. The construction uses techniques analogous to those used by Lei, Zerbes and the first author in order to construct an Euler system attached to the A…

Big image of Galois representations associated with finite slope $p$-adic families of modular forms

We consider the Galois representation associated with a finite slope $p$-adic family of modular forms. We prove that the Lie algebra of its image contains a congruence Lie subalgebra of a non-trivial level. We describe the largest such level in terms of the congruences of the family with $p$-adic CM forms.

A finiteness result for $p$-adic families of Bianchi modular forms

We study $p$-adic families of cohomological automorphic forms for ${\mathrm{GL}}(2)$ over imaginary quadratic fields and prove that families interpolating a Zariski-dense set of classical cuspidal automorphic forms only occur under very restrictive conditions. We show how to computationally determine when this is not the case and establish concrete examples of families interpolating only finitely many Bianchi modular forms.

$p$-adic properties of modular shifted convolution Dirichlet series

Hoffstein and Hulse recently introduced the notion of shifted convolution Dirichlet series for pairs of modular forms $f_1$ and $f_2$. The second two authors investigated certain special values of symmetrized sums of such functions, numbers which are generally expected to be mysterious transcendental numbers. They proved that the generating functions of these values in $h$-aspect are linear combinations of mixed mock modular forms and quasimodular forms. Here we examine the special cases when $f_1=f_2$ where, in addition, there is a prime $p$ for which $p^2$ divides the level. We prove that th…