The classical Waldspurger formula, which computes periods of quaternionic automorphic forms in maximal torus, has been used in a wide variety of arithmetic applications, such as the Birch and Swinnerton-Dyer conjecture in rank 0 situations. This is why this formula is considered the rank 0 analogue of the celebrated Gross-Zagier formula.
On the other hand, Eichler-Shimura correspondence allows us to interpret this quaternionic automorphic form as a cocycle in higher cohomology spaces of certain arithmetic groups. In this way we can realize the corresponding automorphic representation in the etale cohomology of certain Shimura varieties. In this work we find a formula, analogous to that of Waldspurger, which relates cap-products of this cocycle and fundamental classes associated with maximal torus with special values of Rankin-Selberg L-functions.
We describe a construction of preimages for the Shimura map on Hilbert modular forms using generalized theta series, and give an explicit Waldspurger type formula relating their Fourier coefficients to central values of twisted $L$-functions. This formula extends our previous work, allowing to compute these central values when the main central value vanishes.
We describe a construction of preimages for the Shimura map on Hilbert modular forms, and give an explicit Waldspurger type formula relating their Fourier coefficients to central values of twisted L-functions. Our construction is inspired by that of Gross and applies to any nontrivial level and arbitrary base field, subject to certain conditions on the Atkin-Lehner eigenvalues and on the weight.
In this paper, we derive a function field version of the Waldspurger formula for the central critical values of the Rankin-Selberg L-functions. This formula states that the central critical L-values in question can be expressed as the "ratio" of the global toric period integral to the product of the local toric period integrals. Consequently, this result provides a necessary and sufficient criterion for the non-vanishing of these central critical L-values, and supports the Gross-Prasad conjecture for $SO(3)$ over function fields.
We reprove a Waldspurger's formula which relates the toric periods and the central values of L-functions of GL2. Our technique, different from the original theta-correspondence approach and the more recent relative trace formula, relies on the exploit of distributions defined on prehomogeneous vector spaces.
In this article, we study $p$-adic torus periods for certain $p$-adic valued functions on Shimura curves coming from classical origin. We prove a $p$-adic Waldspurger formula for these periods, generalizing the recent work of Bertolini, Darmon, and Prasanna. In pursuing such a formula, we construct a new anti-cyclotomic $p$-adic $L$-function of Rankin-Selberg type. At a character of positive weight, the $p$-adic $L$-function interpolates the central critical value of the complex Rankin-Selberg $L$-function. Its value at a Dirichlet character, which is outside the range of interpolation, essent…
In this paper we give quantitative local test vectors for Waldspurger's period integral (i.e., a toric period on $\text{GL}_2$) in new cases with joint ramifications. The construction involves minimal vectors, rather than newforms and their variants. This paper gives a uniform treatment for the matrix algebra and division algebra cases under mild assumptions, and establishes an explicit relation between the size of the local integral and the finite conductor $C(π\timesπ_{χ^{-1}})$. As an application, we combine the test vector results with the relative trace formula, and prove a hybrid type su…
In this paper, by inputting the Bessel identities over the complex field in previous work of the authors, the Waldspurger formula of Baruch and Mao is extended from totally real fields to arbitrary number fields. This is applied to give a non-trivial bound towards the Ramanujan conjecture for automorphic forms of the metaplectic group $\widetilde{\mathrm{SL}}_2$ for the first time in the generality of arbitrary number fields.
Through combining the work of Jean-Loup Waldspurger (\cite{waldspurger10} and \cite{waldspurgertemperedggp}) and Raphaël Beuzart-Plessis (\cite{beuzart2015local}), we give a proof for the tempered part of the local Gan-Gross-Prasad conjecture (\cite{ggporiginal}) for special orthogonal groups over any local fields of characteristic zero, which was already proved by Waldspurger over $p$-adic fields.
In this paper we will study certain models of irreducible admissible representations of the split special orthogonal group $SO(2n+1)$ over a nonarchimedean local field. If $n=1$, these models were considered by Waldspurger. If $n=2$, they were considered by Novodvorsky and Piatetski-Shapiro \cite{NP}, who called them {\it Bessel models}. They arise from a variety of Rankin-Selberg integrals nad the resultos of this paper will naturally have applications to the study of L-functions. They also arise in the study of the theta correspondence between $SO(2n+1)$ and the double cover of $Sp(2n)$, and…
We determine local test vectors for Waldspurger functionals for GL(2), in the case where both the representation of GL(2) and the character of the degree two extension are ramified, with certain restrictions. We use this to obtain an explicit version of Waldspurger's formula relating twisted central L-values of automorphic representations on GL(2) with certain toric period integrals. As a consequence, we generalize an average value formula of Feigon and Whitehouse, and obtain some nonvanishing results.
We prove a generalization to the totally real field case of the Waldspurger's formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger's formula in terms of equality between global distributions. As applications we generalize the Kohnen-Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindelof hypothesis for integral weight forms. We also study the Kohnen space in t…
We generalize our method for subconvex bounds for $\mathrm{GL}_2 \times \mathrm{GL}_1$ to the setting of the Waldspurger's formula for compact torical integrals. We address the two major difficulties: one is the lack of split places with small norm, the other is the test vector problem. The final bound is valid with arbitrary high probability and is better than the known bounds for a non-empty interval.
We construct p-adic L-functions associated to cuspidal Hilbert modular eigenforms of parallel weight two in certain dihedral or anticyclotomic extensions via the Jacquet-Langlands correspondence, generalizing works of Bertolini-Darmon, Vatsal and others. The construction given here is adelic, which allows us to deduce a precise interpolation formula from a Waldspurger type formula, as well as a formula for the dihedral mu-invariant. We also make a note of Howard's nonvanishing criterion for these p-adic L-functions, which can be used to reduce the associated Iwasawa main conjecture to a certai…
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…
Waldspurger's formula gives an identity between the norm of a torus period and an L-function of the twist of an automorphic representation on GL(2). For any two Hecke characters of a fixed quadratic extension, one can consider the two torus periods coming from integrating one character against the automorphic induction of the other. Because the corresponding L-functions agree, (the norms of) these periods---which occur on different quaternion algebras---are closely related. In this paper, we give a direct proof of an explicit identity between the torus periods themselves.
Let $\mathbb{E}$ be a quadratic extension of a number field $\mathbb{F}$. Let $E(g, s)$ be an Eisenstein series on $GL_2(\mathbb{E})$, and let $F$ be a cuspidal automorphic form on $GL_2(\mathbb{F})$. We will consider in this paper the following automorphic integral: $$\int_{Z_{A}GL_{2}(\mathbb{F})\backslash GL_{2}(\mathbb{A}_{\mathbb{F}})} F(g)E(g,s) dg.$$ This is in some sense the complementary case to the well-known Rankin-Selberg integral and the triple product formula. We will approach this integral by Waldspurger's formula. We will discuss when the integral is automatically zero, and oth…
Assuming everywhere good reduction we generalize the class number formula of Taelman to Drinfeld modules over arbitrary coefficient rings. In order to prove this formula we develop a theory of shtukas and their cohomology.
We give a new formula for the special value at s=2 of the Hasse-Weil zeta function for smooth projective fourfolds under some assumptions (the Tate and Beilinson conjecture, finiteness of some cohomology groups, etc.). Our formula may be considered as a generalization of the Artin-Tate(-Milne) formula for smooth surfaces, and expresses the special zeta value almost exclusively in terms of inner geometric invariants such as higher Chow groups (motivic cohomology groups). Moreover we compare our formula with Geisser's formula for the same zeta value in terms of Weil-étale motivic cohomology grou…
We compute the cohomology with compact supports of a Picard modular surface as a virtual module over the product of the appropriate Galois group and the appropriate Hecke algebra. We use the method developed by Ihara, Langlands, and Kottwitz: comparison of the Grothendieck-Lefschetz formula and the Arthur-Selberg trace formula. Our implementation of this method takes as its starting point the works of Laumon and Morel.
In this article, we investigate the alternating sum of the l-adic cohomology of the Lubin-Tate tower by the Lefschetz trace formula. Our method gives slightly stronger results than in the preceding work of Strauch.
Deligne's conjecture is the Lefschetz trace formula for correspondences defined over a finite field. In this paper, we prove an analogous statement of Deligne's conjecture with respect to $p^n$-torsion étale cohomology under certain conditions, where $p$ is the characteristic of the base field.
We present a strategy to use the stable trace formula to compute growth in the cohomology of cocompact arithmetic lattices in a reductive group $G$. We then implement it in the case where $G = U(2,1)$, using results of Rogwaski. This recovers a result of Marshall.
We investigate the alternating sum of the l-adic cohomology of the Rapoport-Zink tower for GSp(4) by the Lefschetz trace formula. Under some assumptions on L-packets of GSp(4) and its inner form, we observe that the local Jacquet-Langlands correspondence appears in the cohomology.
We give a formula for the Eisenstein cohomology of local systems on the partial compactification of the moduli of principally polarized abelian varieties given by rank 1 degenerations. For genus 2 we give a formula for the full Eisenstein cohomology.
We formulate a detailed conjectural Eichler-Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes $\mathrm{GU}(2,1,\mathbb{Q}(\sqrt{-3}))$. The formula is based on counting points over finite fields on curves of genus three which are cyclic triple covers of the projective line. Assuming the conjecture we are able to calculate traces of Hecke operators on spaces of Picard modular forms. We provide ample evidence for the conjectural formula. Along the way we prove new results on characteristic polynomials of Frobenius …
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…
Let p be a prime and F a field containing a primitive pth root of unity. Then for n in N, the cohomological dimension of the maximal pro-p-quotient G of the absolute Galois group of F is <=n if and only if the corestriction maps H^n(H,Fp) -> H^n(G,Fp) are surjective for all open subgroups H of index p. Using this result we derive a surprising generalization to dim_Fp H^n(H,Fp) of Schreier's formula for dim_Fp H^1(H,Fp).
