Trivial Character Research Articles

Twisting of paramodular vectors

Let F be a non-Archimedean local field of characteristic zero, let (π, V) be an irreducible, admissible representation of GSp (4, F) with trivial central character, and let χ be a quadratic character of F× with conductor c(χ) > 1. We define a twisting operator Tχ from paramodular vectors for π of level n to paramodular vectors for χ ⊗ π of level max (n + 2c(χ), 4c(χ)), and prove that this operator has properties analogous to the well-known GL(2) twisting operator.

On the theta correspondence for (𝐺𝑆𝑝(4),𝐺𝑆𝑂(4,2)) and Shalika periods

We consider both local and global theta correspondences for G S p 4 \mathrm {GSp}_4 and G S O 4 , 2 \mathrm {GSO}_{4,2} . Because of the accidental isomorphism P G S O 4 , 2 ≃ P G U 2 , 2 \mathrm {PGSO}_{4,2} \simeq \mathrm {PGU}_{2,2} , these correspondences give rise to those between G S p 4 \mathrm {GSp}_4 and G U 2 , 2 \mathrm {GU}_{2,2} for representations with trivial central characters. In the global case, using this relation, we characterize representations with trivial central character, which have Shalika period on G U ( 2 , 2 ) \mathrm {GU}(2,2) by theta correspondences. Moreover, in the local case, we consider a similar relationship for irreducible admissible representations without an assumption on the central character.

On 1-cocycles induced by a positive definite function on a locally compact abelian group

For $\varphi$ a normalized positive definite function on a locally compact abelian group $G$, we consider on the one hand the unitary representation $\pi_\varphi$ associated to $\varphi$ by the GNS construction, on the other hand the probability measure $\mu_\varphi$ on the Pontryagin dual $\hat{G}$ provided by Bochner's theorem. We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^1(G,\pi_\varphi)$ and reduced 1-cohomology $\bar{H}^1(G,\pi_\varphi)$. For example, $\bar{H}^1(G,\pi_\varphi)=0$ if and only if either $Hom(G,\mathbb{C})=0$ or $\mu_\varphi(1_G)=0$, where $1_G$ is the trivial character of $G$.

Decomposing induced characters of the centralizer of ann-cycle in the symmetric group on 2nelements

We give explicit multiplicities and formulas for multiplicities of characters appearing in the decomposition of the induced character Ind^{S_{2n}}_{C_{S_{2n}}({\sigma})} 1_C, where {\sigma} is an n-cycle, C_{S_{2n}}({\sigma}) is the centralizer of {\sigma} in S_{2n}, and 1_C denotes the trivial character on C_{S_{2n}}({\sigma}).

Perturbation of Zeros of the Selberg Zeta Function for Γ0(4)

We study the asymptotic behavior of zeros of the Selberg zeta function for the congruence subgroup Γ0(4) as a function of a one-parameter family of characters tending to the trivial character. The motivation for the study comes from observations based on numerical computations. Some of the observed phenomena lead to precise theorems, which we prove and compare with the original numerical results. Supplementary material is available for this article. See the publisher's online edition of Experimental Mathematics for the following free supplementary resources: animations of the zeros of β↦Z(α, β) as a function of α.

The analogue of the Gauss class number problem in motivic cohomology

The classical Gauss problem of determining all imaginary quadratic number fields of class number one has an analogue involving finite motivic cohomology groups attached to the ring of integers $$o_F$$ in a totally real number field $$F$$ . In the classical situation, the value of the zeta function of $$F$$ at $$0$$ can be written as a quotient of two—not necessarily coprime—integers, where the numerator is equal to the class number. For a totally real field $$F$$ and an even integer $$n \ge 2$$ , the value of the zeta-function of $$F$$ at $$1-n$$ can also be written as a quotient of two—not necessarily coprime—integers, where the numerator is equal to the order $$h_n(F)$$ of the motivic cohomology group $$H_\mathcal M ^2(o_F,\mathbb Z (n))$$ . This order is always divisible by $$2^d$$ , where $$d$$ is the degree of $$F$$ . We determine all totally real fields $$F (\ne \mathbb{Q })$$ and all even integers $$n \ge 2$$ , for which the quotient $$\frac{h_n(F)}{2^d}$$ is equal to 1. There are no fields for $$n \ge 6$$ , there is only the field $$\mathbb{Q }(\sqrt{5})$$ for $$n = 4$$ , and there are 11 fields for $$n=2$$ . The motivic cohomology groups $$H_\mathcal M ^2(o_F,\mathbb Z (n))$$ contain the canonical subgroups $$WK^\mathcal M _{2n-2}(F)$$ , called motivic wild kernels, which are analogous to Tate–Shafarevic groups. For even integers $$n \ge 4$$ , there is again only the case $$n = 4$$ and the field $$\mathbb{Q }(\sqrt{5})$$ , for which the motivic wild kernel $$WK^\mathcal M _{2n-2}(F)$$ vanishes. However, for $$n=2$$ , among the totally real fields $$F$$ of degrees between $$2$$ and $$9$$ , we found 21 of them, for which $$WK^\mathcal M _{2}(F)$$ vanishes, the largest degree being $$5$$ . A basic assumption in our approach is the validity of the $$2$$ -adic Main Conjecture in Iwasawa theory for the trivial character, which so far has only been proven for abelian number fields (by A. Wiles).

Factorial algebraic group actions and categorical quotients

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of a finitely generated algebra of invariants. As an application, we provide a combinatorial GIT-type construction of categorical quotients for actions of not necessarily reductive groups on, e.g. complete varieties with finitely generated Cox ring via lifting to the characteristic space.

Quiver varieties and the character ring of general linear groups over finite fields

Given a tuple (\mathcal X_1,\dots,\mathcal X_k) of irreducible characters of \mathrm{GL}_n(\mathbb F_q) we define a star-shaped quiver \Gamma together with a dimension vector \mathbf v . Assume that (\mathcal X_1,\dots,\mathcal X_k) is generic . Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product \mathcal X_1\otimes\cdots\otimes\mathcal X_k as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to (\Gamma,\mathbf v) . The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied, we prove our second result: The multiplicity \langle \mathcal X_1\otimes\cdots\otimes\mathcal X_k,1\rangle is non-zero if and only if \mathbf v is a root of the Kac–Moody algebra associated with \Gamma . This is somehow similar to the connection between Horn's problem and the representation theory of \mathrm{GL}_n(\mathbb C) .

Yoshida lifts and the Bloch–Kato conjecture for the convolution L -function

Let f1 (resp. f2) denote two (elliptic) newforms of prime level N, trivial character and weight 2 (resp. k+2, where k∈{8,12}). We provide evidence for the Bloch–Kato conjecture for the motive M=ρf1⊗ρf2(−k/2−1) by proving that under some assumptions the ℓ-valuation of the order of the Bloch–Kato Selmer group of M is bounded from below by the ℓ-valuation of the relevant L-value (a special value of the convolution L-function of f1 and f2). We achieve this by constructing congruences between the Yoshida lift Y(f1⊗f2) of f1 and f2 and Siegel modular forms whose ℓ-adic Galois representations are irreducible. Our result is conditional upon a conjectural formula for the Petersson norm of Y(f1⊗f2).

Determination of conjugacy class sizes from products of characters

Robinson showed that the character degrees are determined by knowing, for all $n$, the number of ways that the identity can be expressed as a product of $n$ commutators. Earlier, Strunkov showed that the existence of characters of $p$-defect 0 can be determined by counting solutions to certain equations involving commutators and conjugates. In this paper, we prove analogs to Robinson's and Strunkov's theorems by switching conjugacy classes and characters. We show that counting the multiplicity of the trivial character in certain products of characters determines the conjugacy class sizes and existence of conjugacy classes with $p$-defect 0.

The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey–Jarden Conjecture

AbstractFrey and Jarden asked if any abelian variety over a number field K has the infinite Mordell–Weil rank over the maximal abelian extension Kab. In this paper, we give an affirmative answer to their conjecture for the Jacobian variety of any smooth projective curve C over K such that #C(Kab) = ∞ and for any abelian variety of GL2-type with trivial character.

Archimedean zeta integrals on GL n × GL m and SO2n+1 × GL m

In this paper, we evaluate archimedean zeta integrals for automorphic L-functions on GL n × GL n-1+l and on SO2n+1 × GL n+l , for l = −1, 0, and 1. In each of these cases, the zeta integrals in question may be expressed as Mellin transforms of products of class one Whittaker functions. Here, we obtain explicit expressions for these Mellin transforms in terms of Gamma functions and Barnes integrals. When l = 0 or l = 1, the archimedean zeta integrals amount to integrals over the full torus. We show that, as has been predicted by Bump for such domains of integration, these zeta integrals are equal to the corresponding local L-factors—which are simple rational combinations of Gamma functions. (In the cases of GL n × GL n-1 and GL n × GL n this has, in large part, been shown previously by the second author of the present work, though the results here are more general in that they do not require the assumption of trivial central characters. Our techniques here are also quite different. New formulas for GL(n, R) Whittaker functions, obtained recently by the authors of this work, allow for substantially simplified computations). In the case l = −1, we express our archimedean zeta integrals explicitly in terms of Gamma functions and certain Barnes-type integrals. These evaluations rely on new recursive formulas, derived herein, for GL(n, R) Whittaker functions. Finally, we indicate an approach to certain unramified calculations, on SO2n+1 × GL n and SO2n+1 × GL n+1, that parallels our method herein for the corresponding archimedean situation. While the unramified theory has already been treated using more direct methods, we hope that the connections evoked herein might facilitate future archimedean computations.

Symmetries of the transfer operator for Γ0(N) and a character deformation of the Selberg zeta function for Γ0(4)

The transfer operator for Γ0(N) and trivial character χ0 possesses a finite group of symmetries generated by permutation matrices P with P2= id. Every such symmetry leads to a factorization of the Selberg zeta function in terms of Fredholm determinants of a reduced transfer operator. These symmetries are related to the group of automorphisms in GL(2,ℤ) of the Maass wave forms of Γ0(N). For the group Γ0(4) and Selberg’s character χα there exists just one nontrivial symmetry operator P. The eigenfunctions of the corresponding reduced transfer operator with eigenvalue λ=±1 are related to Maass forms that are even or odd, respectively, under a corresponding automorphism. It then follows from a result of Sarnak and Phillips that the zeros of the Selberg function determined by the eigenvalue λ=−1 of the reduced transfer operator stay on the critical line under deformation of the character. From numerical results we expect that, on the other hand, all the zeros corresponding to the eigenvalue λ=+1 are off this line for a nontrivial character χα.

Cohomology algebra of plane curves, weak combinatorial type, and formality

We determine an explicit presentation by generators and relations of the cohomology algebra H ∗ ( P 2 ∖ C , C ) H^*(\mathbb {P}^2\setminus \mathcal {C},\mathbb {C}) of the complement to an algebraic curve C \mathcal {C} in the complex projective plane P 2 \mathbb {P}^2 via the study of log-resolution logarithmic forms on P 2 \mathbb {P}^2 . As a first consequence, we derive that H ∗ ( P 2 ∖ C , C ) H^*(\mathbb {P}^2\setminus \mathcal {C},\mathbb {C}) depends only on the following finite pieces of data: the number of irreducible components of C \mathcal {C} together with their degrees and genera, the number of local branches of each component at each singular point, and the intersection numbers of every two distinct local branches at each singular point of C \mathcal {C} . This finite set of data is referred to as the weak combinatorial type of C \mathcal {C} . A further corollary is that the twisted cohomology jumping loci of H ∗ ( P 2 ∖ C , C ) H^*(\mathbb {P}^2\setminus \mathcal {C},\mathbb {C}) containing the trivial character also depend on the weak combinatorial type of C \mathcal {C} . Finally, the explicit construction of the generators and relations allows us to prove that complements of plane projective curves are formal spaces in the sense of Sullivan.

Base change and the Birch and Swinnerton-Dyer conjecture

In this paper we prove that if the Birch and Swinnerton-Dyer conjecture holds for products of abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character, then the Birch and Swinnerton-Dyer conjecture holds for products of abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character regarded over arbitrary totally real number fields.

Anticyclotomic Iwasawa’s Main Conjecture for Hilbert modular forms

Let F/\mathbb{Q} be a totally real extension and f an Hilbert modular cusp form of level \mathfrak{n} , with trivial central character and parallel weight 2, which is an eigenform for the action of the Hecke algebra. Fix a prime \wp | \mathfrak{n} of F of residual characteristic p . Let K/F be a quadratic totally imaginary extension and K_{\wp^\infty} be the \wp -anticyclotomic \mathbb{Z}_p -extension of K . The main result of this paper, generalizing the analogous result [5] of Bertolini and Darmon, states that, under suitable arithmetic assumptions and some technical restrictions, the characteristic power series of the Pontryagin dual of the Selmer group attached to (f,K_{\wp^\infty}) divides the p -adic L -function attached to (f,K_{\wp^\infty}) , thus proving one direction of the Anticyclotomic Main Conjecture for Hilbert modular forms. Arithmetic applications are given.

THE TWISTED TENSOR L-FUNCTION OFGSp4

The author gives an integral representation for the twisted tensor L-function of a cuspidal, globally generic automorphic representation of GSp4over a quadratic extension E of a number field F with trivial central character. He proves the Euler product factorization of the global integral; computes the unramified L-factor via explicit branching from GL4to Sp4and shows it is equal to the normalized unramified local integral; and proves the absolute convergence and nonvanishing of all local integrals.

Scott modules in finite groups with cyclic Sylow p-subgroups

We present a simple method for determining Scott modules of finite groups using just the Brauer tree of the principal block and the values of ordinary characters at non-identity p-elements. Previously, determining Scott modules often involved long calculations. The method is applicable to the case where finite groups have cyclic Sylow p-subgroups. The structure of a Scott module is closely related to the length of the shortest path from the vertex labeled by the trivial ordinary character to the exceptional vertex in the Brauer tree. We classify the structures of the Scott modules and the ordinary characters they afford, by the length of the associated path in the Brauer tree.

Siegel modular forms of degree two attached to Hilbert modular forms

Let E / Q be a real quadratic field and π 0 a cuspidal, irreducible, automorphic representation of GL ( 2 , A E ) with trivial central character and infinity type ( 2 , 2 n + 2 ) for some non-negative integer n . We show that there exists a non-zero Siegel paramodular newform F : H 2 → C with weight, level, Hecke eigenvalues, epsilon factor and L -function determined explicitly by π 0 . We tabulate these invariants in terms of those of π 0 for every prime p of Q .

Quasi‐convexly dense and suitable sets in the arc component of a compact group

AbstractLet G be an abelian topological group. The symbol \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\widehat{G}$\end{document} denotes the group of all continuous characters \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\chi :G\rightarrow {\mathbb T}$\end{document} endowed with the compact open topology. A subset E of G is said to be qc‐dense in G provided that χ(E)⊆φ([− 1/4, 1/4]) holds only for the trivial character \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\chi \in \widehat{G}$\end{document}, where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\varphi : {\mathbb R}\rightarrow {\mathbb T}={\mathbb R}/{\mathbb Z}$\end{document} is the canonical homomorphism. A super‐sequence is a non‐empty compact Hausdorff space S with at most one non‐isolated point (to which S converges). We prove that an infinite compact abelian group G is connected if and only if its arc component Ga contains a super‐sequence converging to 0 that is qc‐dense in G. This gives as a corollary a recent theorem of Außenhofer: For a connected locally compact abelian group G, the restriction homomorphism \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$r:\widehat{G}\rightarrow \widehat{G}_a$\end{document} defined by \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$r(\chi )=\chi \upharpoonright _{G_a}$\end{document} for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\chi \in \widehat{G}$\end{document}, is a topological isomorphism. We show that an infinite compact group G is connected if and only if its arc component Ga contains a super‐sequence converging to the identity that is qc‐dense in G and generates a dense subgroup of G. We also offer a short alternative proof of the result of Hofmann and Morris on the existence of suitable sets of minimal size in the arc component of a compact connected group.