Let $K$ be an algebraic number field, and $π=\otimesπ_{v}$ an irreducible, automorphic, cuspidal representation of $\GL_{m}(\mathbb{A}_{K})$ with analytic conductor $C(π)$. The theorem on analytic strong multiplicity one established in this note states, essentially, that there exists a positive constant $c$ depending on $\varepsilon>0, m,$ and $K$ only, such that $π$ can be decided completely by its local components $π_{v}$ with norm $N(v)<c\cdot C(π)^{2m+\varepsilon}.$
It is known that multiplicity one property holds for SL(2), while the strong multiplicity one property fails. However, in this paper, we show that if we require further that a pair of cuspidal representations $π$ and $π'$ of SL(2) have the same local components at archimedean places and the places above 2, and they are generic with respect to the same additive character, then they also satisfy the strong multiplicity one property. The proof is based on a local converse theorem for SL(2).
We show that signs of Fourier coefficients, on certain sub-families, determine the half-integral weight cuspidal eigenform uniquely, up to a positive constant. We also study sign change results for the product of the Fourier coefficients of two distinct half-integral weight eigenforms.
We give two applications of Arthur's multiplicity formula to Siegel modular forms. The one is a lifting theorem for vector valued Siegel modular forms, which contains Miyawaki's conjectures and Ibukiyama's conjectures. The other is the strong multiplicity one theorem for Siegel modular forms of scalar weights and level one.
Let $π=\otimesπ_{v}$ and $π^{\prime}=\otimesπ_{v}^{\prime}$ be two irreducible, automorphic, cuspidal representations of $GL_{m}(\mathbb{A}_{K}) >.$ Using the logarithmic zero-free region of Rankin-Selberg $L$-function, Moreno established the analytic strong multiplicity one theorem if at least one of them is self-contragredient, i.e. $π$ and $π^{\prime}$ will be equal if they have finitely many same local components $π_{v},π_{v}^{\prime},$ for which the norm of places are bounded polynomially by the analytic conductor of these cuspidal representations. Without the assumption of the self-contr…
Frobenius Traces for Rank-2 Drinfeld Modules, Higher-Dimensional Galois Representations, and a Strong Multiplicity One Theorem in Positive Characteristic
Following the method developed by Waldspurger and Beuzart-Plessis in their proof of the local Gan-Gross-Prasad conjecture, we are able to prove the multiplicity one theorem for the Ginzburg-Rallis model over the Vogan packets in the tempered case. In some cases, we can also relate the multiplicity to the epsilon factor. This is a sequel of our work \cite{Wan15} in which we consider the supercuspidal case.
We prove several multiplicity one theorems in this paper. For k a local field not of characteristic two, and V a symplectic space over k, any irreducible admissible representation of the symplectic similitude group GSp(V) decomposes with multiplicity one when restricted to the symplectic group Sp(V). We prove the analogous result for GO(V) and O(V), where V is an orthogonal space over k. When k is non-archimedean, we prove the uniqueness of Fourier-Jacobi models for representations of GSp(4), and the existence of such models for supercuspidal representations of GSp(4).
In this work we prove the local multiplicity at most one theorem underlying the definition and theory of local $γ$-, $ε$- and $L$-factors, defined by virtue of the generalized doubling method, over any local field of characteristic 0. We also present two applications: one to the existence of local factors for genuine representations of covering groups, the other to the global unfolding argument of the doubling integral.
We extend the strong multiplicity one theorem of Jacquet, Piatetski-Shapiro and Shalika. Let $π$ be a unitary, cuspidal, automorphic representation of $GL_n(\A_K)$. Let $S$ be a set of finite places of $K$, such that the sum $\sum_{v\in S}Nv^{-2/(n^2+1)}$ is convergent. Then $π$ is uniquely determined by the collection of the local components $\{π_v\mid v\not\in S, ~v \~\text{finite}\}$ of $π$. Combining this theorem with base change, it is possible to consider sets $S$ of positive density, having appropriate splitting behavior with respect to solvable extensions of $K$, and where $π$ is deter…
Zaremba's Conjecture concerns the formation of continued fractions with partial quotients restricted to a given alphabet. In order to answer the numerous questions that arrive from this conjecture, it is best to consider a semi-group, often denoted $Γ_{A}$, which arises naturally as a subset of $SL_2(\mathbb{Z})$ when considering finite continued fractions. To translate back from this semi-group into rational numbers, we select a projection mapping satisfying certain criteria to recover the numerator and denominator of the continued fractions in rational form. The central question of our resea…
In this paper we follow the approach of Bertrand-Beukers (and of later work of Bertrand), based on differential Galois theory, to prove a very general version of Shidlovsky's lemma that applies to Pad{é} approximation problems at several points, both at functional and numerical levels (i.e., before and after evaluating at a specific point). This allows us to obtain a new proof of the Ball-Rivoal theorem on irrationality of infinitely many values of Riemann zeta function at odd integers, inspired by the proof of the Siegel-Shidlovsky theorem on values of E-functions: Shidlovsky's lemma is used …
Each orthogonal group $\OO(n)$ has a nontrivial $\GL(1)$-extension, which we call $\GPin(n)$. The identity component of $\GPin(n)$ is the more familiar $\GSpin(n)$, the general Spin group. We prove that the restriction to $\GPin(n-1)$ of an irreducible admissible representation of $\GPin(n)$ over a nonarchimedean local field of characteristic zero is multiplicity free and also prove the analogous theorem for $\GSpin(n)$. Our proof uses the method of Aizenbud, Gourevitch, Rallis and Schiffman, who proved the analogous theorem for $\OO(n)$, and Waldspurger, who proved that for $\SO(n)$. We also …
Let $π_1$ be a standard representation of $\mathrm{GL}_{n+1}(F)$ and let $π_2$ be the smooth dual of a standard representation of $\mathrm{GL}_n(F)$. When $F$ is non-Archimedean, we prove that $\mathrm{Ext}^i_{\mathrm{GL}_n(F)}(π_1, π_2)$ is $\cong \mathbb C$ when $i=0$ and vanishes when $i \geq 1$. The main tool of the proof is a notion of left and right Bernstein-Zelevinsky filtrations. An immediate consequence of the result is to give a new proof on the multiplicity at most one theorem. Along the way, we also study an application of an Euler-Poincaré pairing formula of D. Prasad on the coef…
Mazur's fundamental work on Eisenstein ideals of prime level has a variety of arithmetic applications. In this article, we generalize some of his work to square-free level. More specifically, we attempt to compute the index of an Eisenstein ideal and the dimension of the m-torsion of the modular Jacobian variety, where m is an Eisenstein maximal ideal. In many cases, the dimension of the m-torsion is 2, in other words, a multiplicity one theorem holds.
We obtain a sharp refinement of the strong multiplicity one theorem for the case of unitary non-dihedral cuspidal automorphic representations for GL(2). Given two unitary cuspidal automorphic representations for GL(2) that are not twist-equivalent, we also find sharp lower bounds for the number of places where the Hecke eigenvalues are not equal, for both the general and non-dihedral cases. We then construct examples to demonstrate that these results are sharp.
We prove a new general multiplicity estimate applicable to sets of functions without any assumption on algebraic independence. The multiplicity estimates are commonly used in determining measures of algebraic independence of values of functions, for instance within the context of Mahler's method. For this reason, our result provides an important tool for the proofs of algebraic independence of complex numbers. At the same time, these estimates can be considered as a measure of algebraic independence of functions themselves. Hence our result provides, under some conditions, the measure of algeb…
We establish zero-free regions tapering as an inverse power of the analytic conductor for Rankin-Selberg L-functions on GL(n) x GL(n'). Such zero-free regions are equivalent to commensurate lower bounds on the edge of the critical strip, and in the case of $L(s,f x f~), on the residue at s=1. As an application we show that a cuspidal automorphic representation on GL(n) is determined by a finite number of its Dirichlet series coefficients, and that this number grows at most polynomially in the analytic conductor.
Inertial multiplicity bounds for two dimensional projective representations and bounds for the number of $\operatorname{PSL}_2(\mathbb{F}_q)$ and $\operatorname{PGL}_2(\mathbb{F}_q)$ number fields
We consider the problem of estimating the multiplicity of a polynomial when restricted to the smooth analytic trajectory of a (possibly singular) polynomial vector field at a given point or points, under an assumption known as the D-property. Nesterenko has developed an elimination theoretic approach to this problem which has been widely used in transcendental number theory. We propose an alternative approach to this problem based on more local analytic considerations. In particular we obtain simpler proofs to many of the best known estimates, and give more general formulations in terms of New…
We prove a strong multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces i.e. if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length.
We show that the number of partitions of n with alternating sum k such that the multiplicity of each part is bounded by 2m+1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is bounded by m. The first proof relies on two formulas with two parameters that are related to the four-parameter formulas of Boulet. We also give a combinatorial proof of this result by using Sylvester's bijection, which implies a stronger partition theorem. For m=0, our result reduces to Bessenrodt's refinement of Euler's Theorem. If the alternating sum and the number of…
In this paper we show a local Jacquet-Langlands correspondence for all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and strong multiplicity one theorems for inner forms of GL(n) as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by Moeglin-Waldspurger and Jacquet-Shalika for GL(n).
We prove that automorphic representations whose local components are certain small representations have multiplicity one. The proof is based on the multiplicity-one theorem for certain functionals of small representations, also proved in this paper.
Publications on automorphic representations of the group U(3) assumed the validity of multiplicity one theorem since I claimed it in 1982. But the argument, published 1988, was based on a misinterpretation of a claim of Gelbart and Piatetski-Shapiro, SLN 1041 (1984), Prop. 2.4(i): ``L^2_{0,1} has multiplicity 1'', as meaning that each irreducible in the space L^2_{0,1} of generic cusp forms has multiplicity one in the space L^2_0 of cusp forms. The statement meant in SLN 1041, ``each irreducible in L^2_{0,1} has multiplicity 1 in L^2_{0,1}'' is too weak to be useful for nongeneric representati…
