Shimura's conjectire (1963) concerns the rationality of the generating series for Hecke operators for the symplectic group of genus g. This conjecture wes proved by Andrianov for arbitrary genus g. For genus g=4, we explicify the rational fraction in this conjecture. Using formulas for images of double cosets, we first compute the sum of the generating series under the Satake spherical map, which is a rational fraction with polynomial coefficients. Then we recover the coefficients of this fraction as elements of the Hecke algebra using polynomial representation of basic Hecke operators under spherical map. Numerical examples of these fractions for special choice of Satake parameters are given.
We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain $p$-adic estimates of Satake parameters of certain non-regular self dual automorphic representations of $\mathrm{GL}_n$.
Let $X$ be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on $X$ in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An interesting by-product of our analysis is the evaluation $$_2F_1(1/24,7/24,5/6, -\frac{2^{10}\cdot3^3\cdot5}{11^4})=\sqrt6 \sqrt[6]{\frac{11}{5^5}} $$ and other similar identities.
In \cite{Shimura}, Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm's bounds give generators of the Hecke algebra as a module. We also have well-known recursion formulae for the operators $T_{p^\ell}$ with $p$ prime. It is the purpose of this paper to prove analogous results in the half-integral weight setting. We also give an explicit formula for how operators $T_{p^{\ell}}$ commute with the Shimura correspondence.
The Hecke orbit conjecture asserts that every prime-to-$p$ Hecke orbit in a Shimura variety is dense in the central leaf containing it. In this paper, we prove the conjecture for certain irreducible components of Newton strata in Shimura varieties of PEL type A and C, when $p$ is an unramified prime of good reduction. Our approach generalizes Chai and Oort's method for Siegel modular varieties.
We define and study a collection of special cycles on certain non-PEL Shimura varieties for $U(2,1) \times U(1,1)$ that appear naturally in the context of the recent conjectures of Gan, Gross and Prasad on restrictions of automorphic forms for unitary groups and conjectural generalizations of the Gross--Zagier formula. We prove a combinatorial formula expressing the Galois action in terms of the distance function on the Bruhat--Tits buildings for these groups. In addition, we calculate explicitly the Hecke polynomial appearing in the congruence relation conjectured by Blasius and Rogawski. Usi…
We study the local factor at p of the semi-simple zeta function of a Shimura variety of Drinfeld type for a level structure given at p by the pro-unipotent radical of an Iwahori subgroup. Our method is an adaptation to this case of the Langlands-Kottwitz counting method. We use Hecke algebra isomorphisms to determine the test functions at p.
This paper proves that the $\mathbb{U}$-operator attached to a cocharacter is a right root of the corresponding Hecke polynomial. This result is an important ingredient in the proof of (i) the horizontal norm relations in the context of Gross--Gan--Prasad cycles and of (ii) the generalization of Eichler--Shimura relations.
We introduce a method for producing congruences between Hecke eigenclasses, possibly torsion, in the coherent cohomology of automorphic vector bundles on certain good reduction Shimura varieties. The congruences are produced using some "generalized Hasse invariants" adapted to the Ekedahl-Oort stratification of the special fiber.
We continue our study of the Wiles defect of deformation rings $R$ and Hecke rings $T$ (at a newform $f$) acting on the cohomology of Shimura curves. The Wiles defect at an augmentation $λ_f:T \to O$ measures the failure of $R,T$ to be complete intersections locally at $λ_f$. In situations we study here the Taylor-Wiles-Kisin patching method gives an isomorphism $R=T$ without the rings being complete intersections. Using novel arguments in commutative algebra and patching, we generalize significantly and give different proofs of our earlier results that compute the Wiles defect at $λ_f: R=T \t…
We find a different method to compute the generating series in Shimura's conjecture for Sp3, proved by Andrianov in 1967. Formulas for the Satake spherical map for Sp3 are used.
We utilize effective algorithms for computing in the cohomology of a Shimura curve together with the Jacquet-Langlands correspondence to compute systems of Hecke eigenvalues associated to Hilbert modular forms over a totally real field.
We prove an equidistribution result for Hecke operators acting on the basic stratum of certain Shimura varieties. We relate the rate of convergence to the bounds from the Ramanujan conjecture of certain cuspidal automorphic representations on Gl_n for which this conjecture is known, and therefore we obtain optimal estimates on the rate of convergence.
We obtain an explicit formula for the spherical image of the polynomial fraction conjectured by Shimura in 1963 for generating Hecke series in particular case of genus 4. As in our previous work we used the Satake spherical map for ${\rm Sp}_4$ and formulas by Andrianov. Key words: Hecke operators, Dirichlet series, Euler factorization, spherical map.
We compute by a different method the generating series in Shimura's conjecture for Sp3, proved by Andrianov in 1967. We develop formulas for the Satake spherical maps for Sp_n and Gl_n.
Let f be a newform, as specified by its Hecke eigenvalues, on a Shimura curve X. We describe a method for evaluating f. The most interesting case is when X arises as a compact quotient of the hyperbolic plane, so that classical q-expansions are not available. The method takes the form of an explicit, rapidly-convergent formula that is well-suited for numerical computation. We apply it to the problem of computing modular parametrizations of elliptic curves, and illustrate with some numerical examples.
We study the action of the derived Hecke algebra on the space of weight one forms. By analogy with the topological case, we formulate a conjecture relating this to a certain Stark unit. We verify the truth of the conjecture numerically, for the weight one forms of level $23$ and $31$, and many derived Hecke operators at primes less than $200$. Our computation depends in an essential way on Merel's evaluation of the pairing between the Shimura and cuspidal subgroups of $J_0(q)$.
We study "quadratic periods" on quaternionic Shimura varieties and formulate an integral refinement of Shimura's conjecture regarding Petersson inner products of automorphic forms that are related by the Jacquet-Langlands correspondence. The main result is that this integral refinement is implied by another conjecture (Conjecture D below) regarding integrality of theta lifts between certain quaternionic unitary groups.
We express the Frobenius-Hecke traces on the compactly supported cohomology of a Shimura variety of abelian type in terms of elliptic parts of stable Arthur-Selberg trace formulas for the endoscopic groups. This confirms predictions of Langlands and Kottwitz at primes where the level is hyperspecial.
We compare the conjecture of Buzzard-Gee on the association of Galois representations to C-algebraic automorphic representations with the conjectural description of the cohomology of Shimura varieties due to Kottwitz, and the reciprocity law at infinity due to Arthur. This is done by extending Langlands's representation of the L-group associated with a Shimura datum to a representation of the C-group of Buzzard-Gee. The approach offers an explanation of the explicit Tate twist appearing in Kottwitz's description.
Let $F$ be a totally real field in which a fixed prime $p$ is inert, and let $E$ be a CM extension of $F$ in which $p$ splits. We fix two positive integers $r,s \in \mathbb N$. We investigate the Tate conjecture on the special fiber of $G(U(r,s) \times U(s,r))$-Shimura variety. We construct cycles which we conjecture to generate the Tate classes and verify our conjecture in the case of $G(U(1,s) \times U(s,1))$. We also discuss the general conjecture regarding special cycles on the special fibers of unitary Shimura varieties.
In this paper, we will study various connections between double zeta values relative to $μ_N$, Hecke operators $T_N$, and newforms of level $Γ_0(N)$ for $N=2,3$. Those various connections generalize the well-known of Baumard and Schneps. We also give a conjecture about the Eichler-Shimura-Manin correspondence between $\mathcal{S}_k^{\textrm{new}}(Γ_0(2))^{\pm}$, $\mathcal{S}_k^{\textrm{new}}(Γ_0(3))^{\pm}$ and some period polynomial spaces.
We explore a natural action of Hecke operators acting on formal sums of optimal embeddings of real quadratic orders into Eichler orders. By associating an optimal embedding to its root geodesic on the corresponding Shimura curve, we can consider the signed intersection number of pairs of embeddings. Using the Hecke operators and the intersection pairing, we construct a generating series that is demonstrated to be a classical modular form of weight two.
The goal of this paper is to calculate the trace of the composition of a Hecke correspondence and a (high enough) power of the Frobenius at a good place on the intersection cohomology of the Satake-Baily-Borel compactification of certain Shimura varieties, to stabilize the result for Shimura varieties associated to unitary groups over $\mathbb{Q}$ and to give applications of this calculations using base change from these unitary groups to $GL_n$. ----- Le but de ce texte est de calculer la trace d'une correspondance de Hecke composee avec une puissance (assez grande) du Frobenius en une bonne …
The present paper studies Hecke rings derived by the automorphism groups of certain algebras $L_p$ over the ring of $p$-adic integers. Our previous work considered the case where $L_p$ is the Heisenberg Lie algebra (of dimension 3) over the ring of $p$-adic integers. Although this Hecke ring is noncommutative, we showed that a formal power series with coefficients in this Hecke ring satisfies an identity similar to the rationality of the Hecke series for $GL_2$ due to E.~Hecke. In the present paper, we establish an analogous result in the case of the Heisenberg Lie algebra of dimension 5 over …
In this paper we introduce the notion of Shimura's period symbols over function fields in positive characteristic and establish their fundamental properties. We further formulate and prove a function field analogue of Shimura's conjecture on the algebraic independence of period symbols. Our results enable us to verify the algebraic independence of the coordinates of any nonzero period vector of an abelian t-module with complex multiplication whose CM type is non-degenerate and defined over an algebraic function field. This is an extension of Yu's work on Hilbert-Blumenthal t-modules.
In this paper we formulate some conjectures about algebraic flows on Shimura varieties. In the first part of the paper we prove the `logarithmic Ax-Lindemann theorem'. We then prove a result concerning the topological closure of the images of totally geodesic subvarieties of symmetric spaces uniformising Shimura varieties. This is a special case of our conjectures.
Let S_{w+2} be the vector space of cusp forms of weight w+2 on the full modular group, and let S_{w+2}^* denote its dual space. Periods of cusp forms can be regarded as elements of S_{w+2}^*. The Eichler-Shimura isomorphism theorem asserts that odd (or even) periods span S_{w+2}^*. However, periods are not linearly independent; in fact, they satisfy the Eichler-Shimura relations. This leads to a natural question: which periods would form a basis of S_{w+2}^*. First we give an answer to this question. Passing to the dual space S_{w+2}, we will determine a new basis for S_{w+2}. The even period …
In this paper, we generalize a conjecture due to Darmon and Logan in an adelic setting. We study the relation between our construction and Kudla's works on cycles on orthogonal Shimura varieties. This relation allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's points.
We develop a general framework to study Szpiro's conjecture and the $abc$ conjecture by means of Shimura curves and their maps to elliptic curves, introducing new techniques that allow us to obtain several unconditional results for these conjectures. We first prove various general results about modular and Shimura curves, including bounds for the Manin constant in the case of additive reduction, a detailed study of maps from Shimura curves to elliptic curves and comparisons between their degrees, and lower bounds for the Petersson norm of integral modular forms on Shimura curves. Our main appl…
We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks $G$-Zip$^μ$. Restricting to zip data of Hodge type, we obtain a group-theoretical Hasse invariant on every Ekedahl-Oort stratum closure of a general Hodge-type Shimura variety. A key tool is the construction of a stack of zip flags $G$-ZipFlag$^μ$, fibered in flag varieties over $G$-Zip$^μ$. It provides a simultaneous generalization of the "classical case" homogeneous complex manifolds studied by Griffiths-Schmid and the "flag space" for Siegel varieties studied by Ekedahl-van der Geer. Four …
We state an improved version of the conjecture of Langlands and Rapoport, and we prove the conjecture for a large class of Shimura varieties. In particular, we obtain the first proof of the (original) conjecture for Shimura varieties of PEL-type.
We study the prime-to-$p$ Hecke action on the projective limit of the sets of connected components of Shimura varieties with fixed parahoric or Bruhat--Tits level at $p$. In particular, we construct infinitely many Shimura varieties for CM unitary groups in odd variables for which the considering actions are not transitive. We prove this result by giving negative examples on the question of Bruhat--Colliot-Thélène--Sansuc--Tits or its variant, which is related to the weak approximation on tori over $\mathbb{Q}$.
A conjecture by Yves Andre and Frans Oort says that closed subvarieties of Shimura varieties that contain a Zariski dense subset of special points are subvarieties of Hodge type. We prove this in the case where the subvariety is a curve that contains infinitely many special points that lie in one Hecke orbit. Via work of Juergen Wolfart, Paula Cohen and Gisbert Wuestholz, this result has an application to algebraicity of values of hypergeometric functions.
It is well-known for an elliptic curve with complex multiplication that the existence of a $\mathbb{Q}$-rational model is equivalent to its field of moduli being equal to $\mathbb{Q}$, or its endomorphism ring being the ring of integers of 9 possible fields ($\ast$). Murabayashi and Umegaki proved analogous results for abelian surfaces. For three dimensional CM abelian varieties with rational fields of moduli, Chun narrowed down to a list of 37 possible CM fields. In this paper, we show that his list is exact. By constructing certain Hecke characters that satisfy a theorem of Shimura, we prove…
Xue proved an equational refinement of the unitary Shimura curve case of the arithmetic Gan-Gross-Prasad conjecture via the Gross-Zagier formula for quaternionic Shimura curves. On the other hand, Rapoport, Smithling and Zhang posed a variant of the conjecture, using modified PEL type Shimura varieties, which we call RSZ Shimura varieties. We reinterpret the result of Xue in terms of the modified Shimura curves. We then use the reinterpretation to prove a case of the variant of the conjecture. Our result combined with the work of Xue establishes a connection between the variant and the Gross-Z…
We prove the Mumford--Tate conjecture for those abelian varieties over number fields whose extensions to C have attached adjoint Shimura varieties that are products of simple, adjoint Shimura varieties of certain Shimura types. In particular, we prove the conjecture for the orthogonal case (i.e., for the $B_n$ and $D_n^R$ Shimura types). As a main tool, we construct embeddings of Shimura varieties (whose adjoints are) of prescribed abelian type into unitary Shimura varieties of PEL type. These constructions implicitly classify the adjoints of Shimura varieties of PEL type.
