The overconvergent Shimura lifting

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…

关于 $\mathrm{GSp}_4$ 的超收敛 Eichler-Shimura 态射

Overconvergent Eichler-Shimura morphisms for $\mathrm{GSp}_4$

本文为亏格二的超收敛 Siegel 模形式族构造显式 Eichler-Shimura 态射，实现 Faltings-Chai 分解的 p-进插值，关键工具为 Boxer-Pilloni 高阶 Coleman 理论与带支集 pro-Kummer 平展上同调。

Analytic continuation on Shimura varieties with $μ$-ordinary locus

We study the geometry of unitary Shimura varieties without assuming the existence of an ordinary locus. We prove, by a simple argument, the existence of canonical subgroups on a strict neighborhood of the $μ$-ordinary locus (with an explicit bound). We then define the overconvergent modular forms (of classical weight), as well as the relevant Hecke operators. Finally, we show how an analytic continuation argument can be adapted to this case to prove a classicality theorem, namely that an overconvergent modular form which is an eigenform for the Hecke operators is classical under certain assump…

A trace formula approach to control theorems for overconvergent automorphic forms

We present an approach to proving control theorems for overconvergent automorphic forms on some Harris-Taylor unitary Shimura varieties based on a comparison between the rigid coho- mology of the multiplicative ordinary locus and the rigid cohomology of the overlying Igusa tower, the latter which may be computed using the Harris-Taylor version of the Langlands-Kottwitz method. We also prove a higher level version, generalizing work of Coleman.

完美体单位ary Shimura 变种与 $p$-adic Eichler-Shimura 映射 I

Perfectoid unitary Shimura varieties and $p$-adic Eichler-Shimura map I

本文通过完美体构造和收敛Eichler-Shimura映射，研究单位ary群上的$p$-adic自动机形与特征种几何及$p$-adic $L$-函数的关系。

p-adic modular forms of non-integral weight over Shimura curves

In this work, we set up a theory of p-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of k-th invariant differentials over the Shimura curves we are interested in, for any p-adic character. In this way, we are able to introduce the notion of overconvergent modular form of any p-adic weight. Moreover, our sheaves can be put in p-adic families over a suitable rigid-analytic space, that parametrizes the weights. Finally, we define Hecke operators, including the U operator, th…

$p$-adic families of modular forms for Hodge type Shimura varieties with non-empty ordinary locus

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.

Formes modulaires surconvergentes, ramification et classicité

We prove in this paper a classicality result for overconvergent modular forms on PEL Shimura varieties of type (A) or (C), without any ramification hypothesis. We use an analytic continuation method, which generalizes previous results in the unramified setting. We work with the rational model of the Shimura variety, and use an embedding into the Siegel variety to define the integral structures on the rigid space.

The Lambda-adic Shimura-Shintani-Waldspurger Correspondence

We generalize the $Λ$-adic Shintani lifting for $GL_2(Q)$ to indefinite quaternion algebras over $Q$.

Clifford algebras and Shimura's lift for Theta-series

In this paper we use automorph class theory formalism to construct a lifting of similitudes of quadratic Z-modules of arbitrary ternary nondegenerate quadratic forms to morphisms between certain subrings of associated Clifford algebras. The construction explains and generalizes Shimura's correspondence in the case of theta-series of positive-definite quadratic forms. Relations between associated zeta-functions are considered.

Classicité de formes modulaires surconvergentes

We prove in this paper a classicality result for overconvergent modular forms on PEL Shimura varieties of type (A) or (C) associated to an unramified reductive group on $\mathbb{Q}_p$. To get this result, we use the analytic continuation method, first used by Buzzard and Kassaei.

Higher Coleman Theory

We develop local cohomology techniques to study the finite slope part of the coherent cohomology of Shimura varieties. The local cohomology groups we consider are a generalization of overconvergent modular forms, and they are defined by using a stratification on the Shimura variety obtained from the Bruhat stratification on a flag variety via the Hodge-Tate period map. We construct a spectral sequence from the local cohomologies to the classical cohomology and use it to obtain classicality and vanishing results. We also develop a theory of p-adic families and construct eigenvarieties. As an ap…

Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz $(1,1)$ theorem

In this article we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, stating that a rational (logarithmic) line bundle on the special fibre of a semistable scheme over $k [\![ t ]\!]$ lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham-Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter example to show that the variational Tate conject…

Modularity lifting results in parallel weight one and applications to the Artin conjecture: the tamely ramified case

We extend the modularity lifting result of the arXiv:1111.2804 to allow Galois representations with some ramification at p. We also prove modularity mod 2 and 5 of certain Galois representations. We use these results to prove many new cases of the strong Artin conjecture over totally real fields in which 5 is unramified. As an ingredient of the proof, we provide a general result on the automatic analytic continuation of overconvergent p-adic Hilbert modular forms of finite slope which substantially generalizes a similar result in arXiv:1111.2804.

例外 Shimura 簇的积分典范模型

Integral canonical models of exceptional Shimura varieties

本文证明例外 Shimura 簇在足够大的素数处存在积分典范模型，并推广至周期映射像，进而得到若干算术推论。

Modularity Lifting beyond the Taylor-Wiles Method

We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions -- one must be in a Shimura variety setting and the automorphic forms must be of regular weight at infinity. In this paper, we essentially show how to remove these restrictions. Our most general result is a modularity lifting theorem which, on…

Periodicity properties of coefficients of half integral weight modular forms

In this paper we prove a theorem about the coefficients in a block of a half integral weight modular form. We show that the result of Serre and Stark for weight 1/2 forms does not generalize to higher higher weights. Let f be a half integral weight cusp form of weight at leat 3/2. We consider blocks of coefficients of f and prove that, under some weak assumptions on f, if such a coefficient block is periodic when considered as a function of in the square root of its index, then it must vanish completely. The proof is analytic in nature and uses Shimura's lifting theorem together with estimates…