Rankin Selberg Type Research Articles

Fourier coefficients of Jacobi Poincaré series and applications

We define Jacobi Poincaré series over Cayley numbers and explicitly compute its Fourier coefficients. As an application, we obtain an estimate for the Fourier coefficients of a Jacobi cusp form. We also evaluate certain Petersson scalar products involving Jacobi cusp forms and Poincaré series. This evaluation yields certain special values of shifted convolution of Dirichlet series of Rankin-Selberg type associated to Jacobi cusp forms in consideration.

Special values of shifted Dirichlet series and quasi-modular forms

Multiplication by a given modular form can be viewed as a linear map on the space of modular forms. By computing its adjoint operator, one can obtain certain cusp forms whose Fourier coefficients are special values of Dirichlet series of Rankin-Selberg type associated to modular forms. We generalize this idea to the space of almost holomorphic modular forms with some cuspidal conditions. We prove that the generating function of special values of the Dirichlet series at certain points is a quasi-modular form.

An average of special values of Dirichlet series of Rankin–Selberg type

For an integer n and a Dirichlet character \(\xi \) modulo N, we denote by \(\mathcal {S}_n(N,\xi )\) the space of cusp forms of weight n with respect to \(\varGamma _0(N)\) and nebentypus \(\xi \). Here \(\varGamma _0(N)\) is the Hecke congruence subgroup. Let k, l be nonnegative integers with \(k-l \ge 2\) and \(\chi ,\psi \) Dirichlet characters modulo N. For a fixed \(g \in \mathcal {S}_l(N,\psi )\), we give an explicit expression for the average of special values of the Dirichlet series \(D(s,f \otimes g)\) of Rankin–Selberg type at each \(s=m \in {\mathbb {Z}}\) with \(\frac{k+l}{2}-1< m<k\) as f ranges over an orthogonal basis of \(\mathcal {S}_k(N,\chi )\).

Rankin–Cohen brackets on Jacobi forms of several variables and special values of certain Dirichlet series

We construct certain Jacobi cusp forms of several variables by computing the adjoint of linear map constructed using Rankin–Cohen-type differential operators with respect to the Petersson scalar product. We express the Fourier coefficients of the Jacobi cusp forms constructed, in terms of special values of the shifted convolution of Dirichlet series of Rankin–Selberg type. This is a generalization of an earlier work of the authors on Jacobi forms to the case of Jacobi forms of several variables.

RANKIN-SELBERG METHOD FOR JACOBI FORMS OF INTEGRAL WEIGHT AND OF HALF-INTEGRAL WEIGHT ON SYMPLECTIC GROUPS

In this article we show analytic properties of certain Rankin-Selberg type Dirichlet series for holomorphic Jacobi cusp forms of integral weight and of half-integral weight. The numerators of these Dirichlet series are the inner products of Fourier-Jacobi coefficients of two Jacobi cusp forms. The denominators and the range of summation of these Dirichlet series are like the ones of the Koecher-Maass series. The meromorphic continuations and functional equations of these Dirichlet series are obtained. Moreover, an identity between the Petersson norms of Jacobi forms with respect to linear isomorphism between Jacobi forms of integral weight and half-integral weight is also obtained.

A CERTAIN DIRICHLET SERIES OF RANKIN–SELBERG TYPE ASSOCIATED WITH THE IKEDA LIFT OF HALF‐INTEGRAL WEIGHT

In this article we obtain an explicit formula for certain Rankin-Selberg type Dirichlet series associated to certain Siegel cusp forms of half-integral weight. Here these Siegel cusp forms of half-integral weight are obtained from the composition of the Ikeda lift and the Eichler-Zagier-Ibukiyama correspondence. The integral weight version of the main theorem had been obtained by Katsurada and Kawamura. The result of the integral weight case is a product of $L$-function and Riemann zeta functions, while half-integral weight case is a infinite summation over negative fundamental discriminants with certain infinite products. To calculate explicit formula of such Rankin-Selberg type Dirichlet series, we use a generalized Maass relation and adjoint maps of index-shift maps of Jacobi forms.

A p-adic Waldspurger formula

In this article, we study p-adic torus periods for certain p-adic-valued functions on Shimura curves of classical origin. We prove a p-adic Waldspurger formula for these periods as a generalization of 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 finite-order character, which is outside the range of interpolation, essentially computes the corresponding p-adic torus period.

A Dirichlet series attached to three Siegel modular forms

For three Siegel modular forms of degrees \(n_1, n_2\) and \(n_1+n_2-1\) respectively, we attach a certain Dirichlet series which has a meromorphic continuation to the whole complex plane and satisfies a functional equation. We also show some algebraicity property of its special values. For the proof we use a Rankin-Selberg type integral involving a pullback of Siegel-Eisenstein series of degree \(2n_1+2n_2-1\). In some special cases our series coincides with known Dirichlet series associated with automorphic L-functions.

On epsilon factors attached to supercuspidal representations of unramified 𝑈(2,1)

Let G G be the unramified unitary group in three variables defined over a p p -adic field F F with p ≠ 2 p \neq 2 . Gelbart, Piatetski-Shapiro and Baruch attached zeta integrals of Rankin-Selberg type to irreducible generic representations of G G . In this paper, we formulate a conjecture on L L - and ε \varepsilon -factors defined through zeta integrals in terms of newforms for G G , which is an analogue of the result by Casselman and Deligne for G L ( 2 ) \mathrm {GL}(2) . We prove our conjecture for the generic supercuspidal representations of G G .

Special Values of Green Functions at Big CM Points

We give a formula for the values of automorphic Green functions on the special rational 0-cycles (big CM points) attached to certain maximal tori in the Shimura varieties associated to rational quadratic spaces of signature (2d, 2). Our approach depends on the fact that the Green functions in question are constructed as regularized theta lifts of harmonic weak Maass forms, and it involves the Siegel-Weil formula and the central derivatives of incoherent Eisenstein series for totally real fields. In the case of a weakly holomorphic form, the formula is an explicit combination of quantities obtained from the Fourier coefficients of the central derivative of the incoherent Eisenstein series. In the case of a general harmonic weak Maass form, there is an additional term given by the central derivative of a Rankin-Selberg type convolution.

POWER SERIES EXPANSIONS OF MODULAR FORMS AND THEIR INTERPOLATION PROPERTIES

We define a power series expansion of an holomorphic modular form f in the p-adic neighborhood of a CM point x of type K for a split good prime p. The modularity group can be either a classical conguence group or a group of norm 1 elements in an order of an indefinite quaternion algebra. The expansion coefficients are shown to be closely related to the classical Maass operators and give p-adic information on the ring of definition of f. By letting the CM point x vary in its Galois orbit, the rth coefficients define a p-adic K×-modular form in the sense of Hida. By coupling this form with the p-adic avatars of algebraic Hecke characters belonging to a suitable family and using a Rankin–Selberg type formula due to Harris and Kudla along with some explicit computations of Watson and of Prasanna, we obtain in the even weight case a p-adic measure whose moments are essentially the square roots of a family of twisted special values of the automorphic L-function associated with the base change of f to K.

Distinguished Generic Representations of GL(n) over p-adic Fields

Let K/F be a quadratic extension of p-adic fields, and n a positive integer. A smooth irreducible representation of the group GL(n, K) is said to be distinguished, if it admits on its space a nonzero GL(n, F)-invariant linear form. In the present work, we classify distinguished generic representations of the group GL(n, K) in terms of inducing quasi-square-integrable representations. This has, as a consequence, the truth of the expected equality between the Rankin–Selberg type Asai L-function of a generic representation and the Asai L-function of its Langlands parameter.

Integral Representation for U3 × GL2

Abstract Gelbart and Piatetskii-Shapiro constructed various integral representations of Rankin–Selberg type for groups G×GLn, where G is of split rank n. Here we show that their method can equally well be applied to the product U3 × GL2, where U3 denotes the quasisplit unitary group in three variables. As an application, we describe which cuspidal automorphic representations of U3 occur in the Siegel induced residual spectrum of the quasisplit U4.

On the asymptotics of Whittaker functions

We study the asymptotics of Whittaker functions on split groups and relate them to the cuspidal exponents of the representation.

A certain Dirichlet series of Rankin–Selberg type associated with the Ikeda lifting

We consider a certain Dirichlet series of Rankin–Selberg type associated with two Siegel cusp forms of the same integral weight with respect to Sp n ( Z ) . In particular, we give an explicit formula for the Dirichlet series associated with the Ikeda lifting of cuspidal Hecke eigenforms with respect to SL 2 ( Z ) . We also comment on a contribution to the Ikeda's conjecture on the period of the lifting.

Special values of L-functions by a Siegel–Weil–Kudla–Rallis formula

We study the arithmeticity of special values of L-functions attached to cuspforms which are Hecke eigenfunctions on hermitian quaternion groups Sp ∗ ( m , 0 ) which form a reductive dual pair with G = O ∗ ( 4 n ) . For f 1 and f 2 two cuspforms on H, consider their theta liftings θ f 1 and θ f 2 on G. Then we compute a Rankin–Selberg type integral and obtain an integral representation of the standard L-function: 〈 θ f 1 ⋅ E s , θ f 2 〉 G = 〈 f 1 , f 2 〉 H ⋅ L std ( f 1 , s ) . Also a short proof the Siegel–Weil–Kudla–Rallis formula is given. This implies that at the critical point s = s 0 = m − n + 1 2 Eisenstein series E s have rational Fourier coefficients. Via the natural embedding G × G ↪ G ˜ = O ∗ ( 8 n ) we restrict the holomorphic Siegel-type Eisenstein series E ˜ on G and decompose as a sum over an orthogonal basis for holomorphic cusp forms of fixed type. As a consequence we prove that the space of holomorphic cuspforms for O ∗ ( 4 n ) of given type is spanned by cuspforms so that the finite-prime parts of Fourier coefficients are rational and obtain special value results for the L-functions.

On Petersson products of not necessarily cuspidal modular forms

We show that Petersson products of any pair of elliptic modular forms of weight 1 2 and of any pair of Siegel modular forms of degree 2 and weight 1 have an expression as the residue of a Dirichlet series of Rankin–Selberg type. This generalizes a well-known property of cuspidal Petersson products.

Jacobi-Hecke algebras and a rationality theorem for a formal Hecke series

It is well known that the L-function associated to a Siegel eigenform f is equal to a Rankin-Selberg type zeta-integral involving f and a restricted Eisenstein series ([3], [14]). At some point in the proof one has to show the equality of a certain Dirichlet series and the L-function, which follows from a rationality theorem for a certain formal power series over the Hecke algebra. The main purpose of this paper is to develop a Hecke theory for the Jacobi group and to prove such a rationality theorem.

On certain L-series of Rankin-Selberg type associated to Siegel modular forms of degree 2

We study certain L-series of Rankin-Selberg type associated to Siegel modular forms of degree 2: twists by Dirichlet characters, relation to zeta functions of spinor type, construction of Siegel cusp forms whose Fourier coefficients involve special values of such L-series, etc.

OnL-functions and intertwining operators for unitary groups

The ingredients of an “L-function machine” for the quasi-split groupU n, n +1 × Res GL n are treated here, following similar theories of P. Shapiro and S. Gelbart. We start with a known Rankin-Selberg type integral having an Euler product. In section 2 we compute the local integral to get a localL function. This is done by working with an “L group” related to L G and the relative root system. All computations are carried out for the split and the non-split case. In section 3 we address the problem of analytic continuation of the Eisenstein series. This involves computation of poles of intertwining operators.