A converse theorem for metaplectic Eisenstein series on the double and triple covers of SL2(ℚ(−D))

In this work, we prove a converse theorem for metaplectic Eisenstein series on the [Formula: see text]th metaplectic cover of the group [Formula: see text], where [Formula: see text] is an imaginary quadratic number field containing the [Formula: see text]th roots of unity. This is an analog to previous converse theorems relating certain double Dirichlet series to the Mellin transforms of Eisenstein series of half-integer weight. We also propose a way to generalize this result to any number field.

Metaplectic Tensor Products for Automorphic Representation of (r)

Let M = GLr1 ✗ … × GLrk ⊆ GLr be a Levi subgroup of GLr, where r = r1 + … +rk, and its metaplectic preimage in the n-fold metaplectic cover r1 of GLr. For automorphic representations π1, …, πk of r1 (), … ,rk (), we construct (under a certain technical assumption that is always satisfied when n = 2) an automorphic representation π of () that can be considered as the “tensor product” of the representations π1, … , πk. This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place v, πv is equivalent to the local metaplectic tensor product of π1,v, … , πk,v defined by Mezo. Then we show that if all of the πi are cuspidal (resp. square-integrable modulo center), then the metaplectic tensor product is cuspidal (resp. square-integrable modulo center). We also show that (both locally and globally) the metaplectic tensor product behaves in the expected way under the action of a Weyl group element and show the compatibility with parabolic inductions.

Short incomplete Gauss sums and rational points on metaplectic horocycles

In this paper, we investigate the limiting behavior of short incomplete Gauss sums at random argument as the number of terms goes to infinity. We prove that the limit distribution is given by the distribution of theta sums and differs from the limit law for long Gauss sums studied by the author and Marklof. The key ingredient in the proof is an equidistribution theorem for rational points on horocycles in the metaplectic cover of SL(2, ℝ).

EXPLICIT DOUBLING INTEGRALS FOR Sp2(F) AND $\widetilde{{{\rm Sp}}_{2}}(F)$ USING "GOOD TEST VECTORS"

In this paper, we offer some explicit computations of a formulation of the doubling method of Piatetski-Shapiro and Rallis for the groups Sp 2(F) (the rank 2 symplectic group) and its metaplectic cover [Formula: see text] for F a finite extension of ℚp with p ≠ 2. We determine a set of "good test vectors" for the irreducible constituents of unramified principal series representations for these groups as well as a set of "good theta test sections" in a family of degenerate principal series representations of Sp 4(F) and [Formula: see text]. Determining "good test data" that produces a non-vanishing doubling integral should indicate the existence of a non-vanishing theta lifts for dual pairs of the type ( Sp 2(F), O (V)) (respectively [Formula: see text]) where V is a quadratic space of an even (respectively odd) dimension.

Lifting of characters on p-adic orthogonal and metaplectic groups

Let F be a p-adic field. Consider a dual pair $({\rm SO}(2n+1)_+, \widetilde {{\rm Sp}}(2n)),$ where SO(2n+1)+ is the split orthogonal group and $\widetilde {{\rm Sp}}(2n)$ is the metaplectic cover of the symplectic group Sp(2n) over F. We study lifting of characters between orthogonal and metaplectic groups. We say that a representation of SO(2n+1)+ lifts to a representation of $\widetilde {{\rm Sp}}(2n)$ if their characters on corresponding conjugacy classes are equal up to a transfer factor. We study properties of this transfer factor, which is essentially the character of the difference of the two halves of the oscillator representation. We show that the lifting commutes with parabolic induction. These results were motivated by the paper ‘Lifting of characters on orthogonal and metaplectic groups’ by Adams who considered the case F=ℝ.

Tori in metaplectic covers of GL2 and applications to a formula of Loxton–Matthews

In this paper we have two objectives. The first is to investigate the restriction of a metaplectic cover to an arbitrary torus in GL2. This will be explained at greater length below, and the main results are Theorems 1 and 2. The second is an application of the same ideas to introduce the arithmetic function P, which has already appeared in a special case in [9], and to prove the fundamental property given by Theorem 3. These theorems will be proved in §§ 2 and 3. In §§ 4 and 5 we remark on the appearence of the function P in the formula of Loxton and Matthews [5], [6] for the biquadratic Gauss sum and discuss the structure of this formula.

Some Remarks on Eisenstein Series for Metaplectic Coverings

In recent years the harmonic analysis of n-fold (n > 2) metaplectic coverings of GL2 has played an increasingly important role in certain aspects of algebraic number theory. In large part this has been inspired by the pioneering work of Kubota (see [3] for example); as an application one could cite the solution by Heath-Brown and Patterson [3] to a question of Kummer's on the distribution of the arguments of cubic Gauss sums. In that paper, Eisenstein series on the 3-fold metaplectic cover of GL2(A) play a crucial role.The object of this note is to point out that the theory of Eisenstein series can be made to work for a wide class of finite central coverings. Indeed, once the assumptions are made, the usual theory carries over readily, and one obtains a spectral decomposition of the appropriate L2-space of functions; this is done in Section 2 of this paper.