Generic Cusp Research Articles

Whittaker functions for generalized principal series representations of GSp(2,R)

We study Whittaker functions for generalized principal series representations of GSp(2,R) induced from Siegel parabolic subgroup. We give Mellin–Barnes type integral representations of Whittaker functions belonging to certain K-types. As an application of our explicit formulas, we compute the archimedean parts of Novodvorsky's zeta integrals. As a consequence we can show the entireness of the spinor L-functions for generic cusp forms on GSp(2).

A multivariate integral representation on 𝐺𝐿₂×𝐺𝑆𝑝₄ inspired by the pullback formula

We give a two variable Rankin–Selberg integral inspired by consideration of Garrett’s pullback formula. For a globally generic cusp form on G L 2 × G S p 4 \mathrm {GL}_2\times \mathrm {GSp}_4 , the integral represents the product of the S t d × S p i n \mathrm {Std}\times \mathrm {Spin} and 1 × S t d \mathbf {1} \times \mathrm {Std} L L -functions. We prove a result concerning an Archimedean principal series representation in order to verify a case of Jiang’s first-term identity relating certain non-Siegel Eisenstein series on symplectic groups. Using it, we obtain a new proof of a known result concerning possible poles of these L L -functions.

On existence of generic cusp forms on semisimple algebraic groups

In this paper we discuss the existence of certain classes of cuspidal automorphic representations having non-zero Fourier coefficients for a general semisimple algebraic group $ G$ defined over a number field $ k$ such that its Archimedean group $ G_\infty $ is not compact. When $ G$ is quasi-split over $ k$, we obtain a result on existence of generic cuspidal automorphic representations which generalize results of Vigneras, Henniart, and Shahidi. We also discuss: (i) the existence of cuspidal automorphic forms with non-zero Fourier coefficients for congruence of subgroups of $ G_\infty $, and (ii) applications related to the work of Bushnell and Henniart on generalized Whittaker models.

Shock formation in the dispersionless Kadomtsev–Petviashvili equation

The dispersionless Kadomtsev–Petviashvili (dKP) equation is one of the simplest nonlinear wave equations describing two-dimensional shocks. To solve the dKP equation numerically we use a coordinate transformation inspired by the method of characteristics for the one-dimensional Hopf equation . We show numerically that the solutions to the transformed equation stays regular for longer times than the solution of the dKP equation. This permits us to extend the dKP solution as the graph of a multivalued function beyond the critical time when the gradients blow up. This overturned solution is multivalued in a lip shape region in the (x, y) plane, where the solution of the dKP equation exists in a weak sense only, and a shock front develops. A local expansion reveals the universal scaling structure of the shock, which after a suitable change of coordinates corresponds to a generic cusp catastrophe. We provide a heuristic derivation of the shock front position near the critical point for the solution of the dKP equation, and study the solution of the dKP equation when a small amount of dissipation is added. Using multiple-scale analysis, we show that in the limit of small dissipation and near the critical point of the dKP solution, the solution of the dissipative dKP equation converges to a Pearcey integral. We test and illustrate our results by detailed comparisons with numerical simulations of both the regularized equation, the dKP equation, and the asymptotic description given in terms of the Pearcey integral.

Large complex correlated Wishart matrices: the Pearcey kernel and expansion at the hard edge

We study the eigenvalue behaviour of large complex correlated Wishart matrices near an interior point of the limiting spectrum where the density vanishes (cusp point), and refine the existing results at the hard edge as well. More precisely, under mild assumptions for the population covariance matrix, we show that the limiting density vanishes at generic cusp points like a cube root, and that the local eigenvalue behaviour is described by means of the Pearcey kernel if an extra decay assumption is satisfied. As for the hard edge, we show that the density blows up like an inverse square root at the origin. Moreover, we provide an explicit formula for the $1/N$ correction term for the fluctuation of the smallest random eigenvalue.

Counting function of the embedded eigenvalues for some manifold with cusps, and magnetic Laplacian

We consider a non compact, complete manifold {\bf{M}} of finite area with cuspidal ends. The generic cusp is isomorphic to ${\bf{X}}\times ]1,+\infty [$ with metric $ds^2=(h+dy^2)/y^{2\delta}.$ {\bf{X}} is a compact manifold with nonzero first Betti number equipped with the metric $h.$ For a one-form $A$ on {\bf{M}} such that in each cusp $A$ is a non exact one-form on the boundary at infinity, we prove that the magnetic Laplacian $-\Delta_A=(id+A)^\star (id+A)$ satisfies the Weyl asymptotic formula with sharp remainder. We deduce an upper bound for the counting function of the embedded eigenvalues of the Laplace-Beltrami operator $-\Delta =-\Delta_0 .$

Spinor $L$-functions for generic cusp forms on $GSp(2)$ belonging to principal series representations

Let $\mathsf {G}=GSp(2)$ be the symplectic group with similitude of degree two, which is defined over $\mathbf {Q}$. For a generic cusp form $F$ on the adelized group $\mathsf {G}_{\mathbf {A}}$ whose archimedean type is a principal series representation, we show that its spinor $L$-function is continued to an entire function and satisfies the functional equation.

Entireness of the Spinor L -functions for certain generic cusp forms on GSp (2)

Let G = GSp (2) be the symplectic group with similitude of degree two, which is defined over Q . For a generic cusp form F on the adelized group G A whose archimedean type is either a (limit of) large discrete series representation or a certain principal series representation, we show that its spinor L -function is continued to an entire function and satisfies the functional equation.

On a correspondence between cuspidal representations of 𝐺𝐿_{2𝑛} and 𝑆𝑝̃_{2𝑛}

Let η \eta be an irreducible, automorphic, self-dual, cuspidal representation of GL 2 n ⁡ ( A ) \operatorname {GL}_{2n}(\mathbb A) , where A \mathbb A is the adele ring of a number field K K . Assume that L S ( η , Λ 2 , s ) L^S(\eta ,\Lambda ^2,s) has a pole at s = 1 s=1 and that L ( η , 1 2 ) ≠ 0 L(\eta , \frac 12)\neq 0 . Given a nontrivial character ψ \psi of K ∖ A K\backslash \mathbb A , we construct a nontrivial space of genuine and globally ψ − 1 \psi ^{-1} -generic cusp forms V σ ψ ( η ) V_{\sigma _{\psi }(\eta )} on Sp ~ 2 n ( A ) \widetilde {\operatorname {Sp}}_{2n}(\mathbb A) —the metaplectic cover of Sp 2 n ( A ) {\operatorname {Sp}}_{2n}(\mathbb A) . V σ ψ ( η ) V_{\sigma _{\psi }(\eta )} is invariant under right translations, and it contains all irreducible, automorphic, cuspidal (genuine) and ψ − 1 \psi ^{-1} -generic representations of Sp ~ 2 n ( A ) \widetilde {\operatorname {Sp}}_{2n}(\mathbb A) , which lift (“functorially, with respect to ψ \psi ") to η \eta . We also present a local counterpart. Let τ \tau be an irreducible, self-dual, supercuspidal representation of GL 2 n ⁡ ( F ) \operatorname {GL}_{2n}(F) , where F F is a p p -adic field. Assume that L ( τ , Λ 2 , s ) L(\tau ,\Lambda ^2,s) has a pole at s = 0 s=0 . Given a nontrivial character ψ \psi of F F , we construct an irreducible, supercuspidal (genuine) ψ − 1 \psi ^{-1} -generic representation σ ψ ( τ ) \sigma _\psi (\tau ) of Sp ~ 2 n ( F ) \widetilde {\operatorname {Sp}}_{2n}(F) , such that γ ( σ ψ ( τ ) ⊗ τ , s , ψ ) \gamma (\sigma _\psi (\tau )\otimes \tau ,s,\psi ) has a pole at s = 1 s=1 , and we prove that σ ψ ( τ ) \sigma _\psi (\tau ) is the unique representation of Sp ~ 2 n ( F ) \widetilde {\operatorname {Sp}}_{2n}(F) satisfying these properties.

Form of cosmic string cusps

We classify the possible shapes of cosmic string cusps and how they transform under Lorentz boosts. A generic cusp can be brought into a form in which the motion of the cusp tip lies in the plane of the cusp. The cusp whose motion is perpendicular to this plane, considered by some authors, is a special case and not the generic situation. We redo the calculation of the energy in the region where the string overlaps itself near a cusp, which is the maximum energy that can be released in radiation. We take into account the motion of a generic cusp and the resulting Lorentz contraction of the string core. The result is that the energy scales as $\sqrt {rL}$ instead of the usual value of $r^{1/3} L^{2/3}$, where $r$ is the string radius and $L$ and is the typical length scale of the string. Since $r << L $ for cosmological strings, the radiation is strongly suppressed and could not be observed.

A Rankin-Selberg integral for the adjointL-function of Sp4

In this paper we study the AdjointL-function for Sp4. For generic cusp forms of Sp4(A) we construct a global Rankin-Selberg integral which represents thisL-function.

On the asymptotic formula of Kuznetsov and existence of generic cusp forms

On the asymptotic formula of Kuznetsov and existence of generic cusp forms

On the Ramanujan Conjecture and Finiteness of Poles for Certain L- Functions

In this paper we prove a number of general results about automorphic L-functions which appear in the constant terms of Eisenstein series for an arbitrary quasi-split connected reductive algebraic group over a number field. These L-functions were first introduced by Langlands for Chevalley groups [20] and through them he was led to make some of his deep conjectures [21]. One significance of these L-functions is that all the automorphic L-functions studied so far are among them. There are three general results. First, we obtain a uniform line of absolute convergence for all of these L-functions (Theorem 5.1). A uniform estimate for the Hecke eigenvalues of generic cusp forms on many absolutely simple quasi-split connected reductive algebraic groups over number fields (Corollary 5.4) and an improvement on the best available estimate for the Fourier coefficients of Maass wave forms (Corollary 5.5) also follow. Next, we establish a meromorphic continuation and functional equation for each of these L-functions (Theorem 6.1). Finally, in Theorem 6.2, we prove finiteness of poles on the whole complex plane for an important class of these L-functions (Corollaries 6.6 through 6.10). More precisely, let G be a quasi-split connected reductive algebraic group over a number field F. Set G = G(A F)' where A F is the ring of adeles of F. Fix a Borel subgroup B of G over F and let U be its unipotent radical. Let P be a maximal F-parabolic subgroup of G with P D B. In the context of the problems studied here, nothing new will be obtained if one drops the maximality condition on P. Write P = MN, a Levi decomposition, N C U. and let B, U, P, M, and N be the corresponding groups of adelic points. For every place v of F. let GV = G(FJ). Similarly we have Bv, Us, Pv, Me, and Nv. Let X = ?~ Xv be a generic character of U(F) U (cf. Section 3). Then each Xv is generic. Let r = Ov 7Tv be a cusp form on M. We shall say 7T is