Our main theorem describes the degree 0 cohomology of Igusa varieties in terms of one-dimensional automorphic representations in the setup of mod p Hodge-type Shimura varieties with hyperspecial level at p, mirroring the well known analogue for complex Shimura varieties. As an application, we obtain a completely new approach to two geometric questions. (See Sect. 1.5 for a comparison with independent results by van Hoften and Xiao via a different approach.) Firstly, we verify the discrete part of the Hecke orbit conjecture, which amounts to irreducibility of central leaves, generalizing preceding works by Chai, Oort, Yu, et al. Secondly, we deduce irreducibility of Igusa towers and its generalization to non-basic Igusa varieties in the same generality, extending previous results by Igusa, Ribet, Faltings--Chai, Hida, and others. Our proof is based on a Langlands--Kottwitz-type formula for Igusa varieties due to Mack-Crane, an asymptotic study of the trace formula, and an estimate for unitary representations and their Jacquet modules in representation theory of $p$-adic groups due to Howe--Moore and Casselman.
We describe the connected components of Igusa's varieties of second species defined par Harris and Taylor in their book. There are in bijection with the characters of the inversible group of the maximal order of the division algebra attached to them.
Igusa varieties are algebraic varieties that arise in the study of special fibers of Shimura varieties, and have demonstrated many applications in the Langlands program via a Langlands-Kottwitz style point-counting formula due to Shin in the case of PEL type. In this paper we formulate and prove an analogue of the Langlands-Rapoport conjecture for Igusa varieties of Hodge type, building off the work of Kisin in the case of Shimura varieties. We then use this description of the points to derive a point-counting formula for Igusa varieties of Hodge type, generalizing the fomula in PEL type of Sh…
We establish asymptotic formulae for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying variety. In the adelic setting, this involves the construction of general Tamagawa measures.
We formulate and prove certain vanishing theorems for p-adic automorphic forms on unitary groups of arbitrary signature. The p-adic q-expansion principle for p-adic modular forms on the Igusa tower says that if the coefficients of (sufficiently many of) the q-expansions of a p-adic modular form f are zero, then f vanishes everywhere on the Igusa tower. There is no p-adic q-expansion principle for unitary groups of arbitrary signature in the literature. By replacing q-expansions with Serre-Tate expansions (expansions in terms of Serre-Tate deformation coordinates) and replacing modular forms wi…
Let O be a maximal order in a totally indefinite quaternion algebra over a totally real number field. In this note we study the locus Q_O of quaternionic multiplication by O in the moduli space A_g of principally polarized abelian varieties of even dimension g with particular emphasis in the two-dimensional case. We describe Q_O as a union of Atkin-Lehner quotients of Shimura varieties and we compute the number of irreducible components of Q_O in terms of class numbers of CM-fields.
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.
For several Hodge-type Shimura varieties of good reduction in characteristic $p$, we show that the cone of weights of automorphic forms is encoded by the stack of $G$-zips of Pink-Wedhorn-Ziegler. This establishes several instances of a general conjecture formulated in previous papers by the authors. Furthermore, we prove in these cases that any mod $p$ automorphic form whose weight lies in a specific region of the weight space is divisible by a partial Hasse invariant. This generalizes to other Shimura varieties previous results of Diamond--Kassaei on Hilbert modular forms.
We establish vanishing results for spaces of automorphic forms in characteristic $0$ and characteristic $p$. We prove that for Hodge-type Shimura varieties, the weight of any nonzero automorphic form in characteristic $0$ satisfies the Griffiths-Schmid conditions, by purely algebraic, characteristic $p$ methods. We state a conjecture for general Hodge-type Shimura varieties regarding the vanishing of the space of automorphic forms in characteristic $p$ in terms of the weight. We verify this conjecture for unitary PEL Shimura varieties of signature $(n-1,1)$ at a split prime.
The goal of this paper is to show that the cohomology of compact unitary Shimura varieties is concentrated in the middle degree and torsion-free, after localizing at a maximal ideal of the Hecke algebra satisfying a suitable genericity assumption. Along the way, we establish various foundational results on the geometry of the Hodge-Tate period map. In particular, we compare the fibres of the Hodge-Tate period map with Igusa varieties.
We present a pairing of automorphic distributions that applies in situations where a Lie group acts with an open orbit on a product of generalized flag varieties. The pairing gives meaning to an integral of products of automorphic distributions on these varieties. This generalizes classical integral representations or "Rankin-Selberg integrals" of L-functions, and gives new constructions and analytic continuations of automorphic L-functions. Keywords: Automorphic forms, invariant pairings, automorphic distributions, L-functions, analytic continuation, rapid decay.
We propose a new method to construct rigid $G$-automorphic representations and rigid $\widehat{G}$-local systems for reductive groups $G$. The construction involves the notion of euphotic representations, and the proof for rigidity involves the geometry of certain Hessenberg varieties.
We define and construct integral canonical models for automorphic vector bundles over Shimura varieties of abelian type. More precisely, we first build on Kisin's work to construct integral canonical models over rings of integers of number fields with finitely many primes inverted for Shimura varieties of abelian type with hyperspecial level at all primes we do not invert, compatible with Kisin's construction. We then define a notion of an integral canonical model for the standard principal bundles lying over Shimura varieties and proceed to construct them in the abelian type case. With these …
We study the rigid cohomology of the ordinary locus in some compact PEL Shimura varieties of type C with values in automorphic local systems and use it to prove a small slope criterion for classicality of overconvergent Hecke eigenforms. This generalises the work of Coleman, and is a first step in an ongoing project to extend the cohomological approach to classicality to higher-dimensional 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.
We formulate an analogue of the archimedean motivic action conjecture of Prasanna--Venkatesh for irregular cohomological automorphic forms on Shimura varieties, which appear on multiple degrees of coherent cohomology of Shimura varieties. Such multiple appearances are due to many infinity types in a single $L$-packet with equal minimal $K$-types. Accordingly, we formulate the conjecture comparing periods of forms of different automorphic representations. We provide evidences for the conjecture by showing its compatibility with existing conjectures on periods of automorphic forms. The conjectur…
We construct Eigenvarieties for PEL Shimura varieties which interpolate cuspidal, finite slope automorphic forms for PEL Shimura varieties appearing as global sections of (coherent) automorphic sheaves, under the hypothesis that the primes are unramified in the Shimura datum, but allowing the ordinary locus to be empty. We then use this construction to extend previous results on the Bloch-Kato conjecture for Hecke characters of a quadratic imaginary field.
We construct the compatible system of $l$-adic representations associated to a regular algebraic cuspidal automorphic representation of $GL_n$ over a CM (or totally real) field and check local-global compatibility for the $l$-adic representation away from $l$ and finite number of rational primes above which the CM field or the automorphic representation ramify. The main innovation is that we impose no self-duality hypothesis on the automorphic representation.
In this article, we improve our main results from \emph{Chow groups and $L$-derivatives of automorphic motives for unitary groups} in two direction: First, we allow ramified places in the CM extension $E/F$ at which we consider representations that are spherical with respect to a certain special maximal compact subgroup, by formulating and proving an analogue of the Kudla--Rapoport conjecture for exotic smooth Rapoport--Zink spaces. Second, we lift the restriction on the components at split places of the automorphic representation, by proving a more general vanishing result on certain cohomolo…
We prove under certain assumptions a formula for the cohomology of PEL-type Rapoport--Zink spaces that "averages" over the Kottwitz set. Our formula generalizes that of Shin's beyond the EL-type case and is proven by combining Mantovan's formula with descriptions of the cohomology of Shimura and Igusa varieties. We then use this averaging formula to derive a conjectural description of the cohomology of Rapoport--Zink spaces, generalizing our earlier work for EL-type spaces. Along the way, we give a description of the cohomology of Igusa varieties in terms of the Langlands correspondence, gener…
We prove vanishing results for the $0$th cohomology of automorphic line bundles on the Ekedahl--Oort strata of the special fiber of Hilbert--Blumenthal Shimura varieties. This work vastly extends to all Ekedahl--Oort strata previous results of Diamond--Kassaei and Goldring--Koskivirta. Furthermore, our results also cover the case of unitary Shimura varieties of rank $\leq 2$ attached to an arbitrary CM extension. We also prove a conjecture of Goldring--Koskivirta on the set of possible weights of automorphic forms.
In this paper, we give a geometric construction of the Jacquet-Langlands transfer for automorphic forms of higher weights by studying the geometry of the mod $p$ fibres of different Hodge type Shimura varieties which satisfy a mild assumption and the cohomological correspondences between them.
A particular case of Bergeron-Venkatesh's conjecture predicts that torsion classes in the cohomology of Shimura varieties are rather rare. According to this and for Kottwitz-Harris-Taylor type of Shimura varieties, we first associate to each such torsion class an infinity of irreducible automorphic representations in characteristic zero, which are pairwise non isomorphic and weakly congruent. Then, using completed cohomology, we construct torsion classes in regular weight and then deduce explicit examples of such automorphic congruences.
We show how to deduce the standard sign conjecture (a weakening of the Künneth standard conjecture) for Shimura varieties from some statements about discrete automorphic representations (Arthur's conjectures plus a bit more). We also indicate what is known (to us) about these statements.
In this article we study a differential algebra of modular-type functions attached to the periods of a one parameter family of Calabi-Yau varieties which is mirror dual to the universal family of quintic threefolds. Such an algebra is generated by seven functions satisfying functional and differential equations in parallel to the modular functional equations of classical Eisenstein series and the Ramanujan differential equation. Our result is the first example of automorphic-type functions attached to varieties whose period domain is not Hermitian symmetric. It is a reformulation and realizati…
We discuss generalizations of the Langlands program, from reductive groups to the local and automorphic spectra of spherical varieties, and to more general representations arising as "quantizations" of suitable Hamiltonian spaces. To a spherical $G$-variety $X$, one associates a dual group ${^LG_X}$ and an $L$-value (encoded in a representation of ${^LG_X}$), which conjecturally describe the local and automorphic spectra of the variety. This sets up a problem of functoriality, for any morphism ${^LG_X}\to {^LG_Y}$ of dual groups. We review, and generalize, Langlands' "beyond endoscopy" approac…
We determine the behavior of automorphic Green functions along the boundary components of toroidal compactifications of orthogonal Shimura varieties. We use this analysis to define boundary components of special divisors and prove that the generating series of the resulting special divisors on a toroidal compactification is modular.
Generalizing work of Gross--Zagier and Schofer on singular moduli, we study the CM values of regularized theta lifts of harmonic Whittaker forms. We compute the archimedian part of the height pairing of arithmetic special divisors and CM cycles on Shimura varieties associated to certain quadratic spaces over an arbitrary totally real base field. As a special case, we obtain an explicit formula for the norms of the CM values of those meromorphic modular forms arising as regularized theta lifts of holomorphic Whittaker forms.
We first give a relative flexible process to construct torsion cohomology classes for Shimura varieties of Kottwitz-Harris-Taylor type with coefficient in a non too regular local system. We then prove that associated to each torsion cohomology class, there exists a infinity of irreducible automorphic representations in characteristic zero, which are pairwise non isomorphic and weakly congruent.
