There has been a recent coming together of the Converse Theorem for $\gln$ and the Langlands-Shahidi method of controlling the analytic properties of automorphic $L$-functions which has allowed us to establish a number of new cases of functoriality, or the lifting of automorphic forms. In this article we would like to present the current state of the Converse Theorem and outline the method one uses to apply the Converse Theorem to obtain liftings. We will then turn to an exposition of the new liftings and some of their applications.
This is a report on the global aspects of the Langlands-Shahidi method which in conjunction with converse theorems of Cogdell and Piatetski-Shapiro has recently been instrumental in establishing a significant number of new and surprising cases of Langlands Functoriality Conjecture over number fields. They have led to striking new estimates towards Ramanujan and Selberg conjectures.
We show that in Weil's converse theorem the functional equations of multiplicative twists for at least the first $\sqrt{\frac{p-24}{3}}$ moduli are needed in order to prove the modularity for $Γ_0(p)$.
We study some tempered endoscopic cases of Langlands functoriality on the $n$-variable unitary groups via the simple stable trace formula. This extends previous work of Rogawski and Clozel-Harris-Labesse. Ramakrishnan and Kim-Shahidi have proved the existence of the automorphic tensor product for $GL_2 \times GL_2$ and $GL_2 \times GL_3$. We apply our endoscopic results to, in a sense, "lift" these theorems and deduce the existence of the automorphic tensor product for the quasi-split unitary groups $U^*_2 \times U^*_2$ and $U^*_2 \times U^*_3$. We then formulate the notion of overconvergent L…
We answer a challenge posed in (Math. Ann. 363 (2015), no. 1-2, 423-454) by proving a version of Weil's converse theorem that assumes a functional equation for character twists but allows their root numbers to vary arbitrarily.
Given a prime p and a positive integer m satisfying a certain inequality, the converse of Wolstenholme's Theorem is shown to hold for the product mp^k where k is any positive integer, generalizing a result by Helou and Terjanian.
The purpose of this work is to produce a converse theorem for adelic Eisenstein series on the double metaplectic cover of the group $SL_2(\mathbb{A})$. We show that the double Dirichlet series, which satisfy the natural functional equations required in our theorem, are in fact Mellin transforms at infinity of metaplectic Eisenstein series.
We extend Venkatesh's proof of the converse theorem for classical holomorphic modular forms to arbitrary level and character. The method of proof, via the Petersson trace formula, allows us to treat arbitrary degree 2 gamma factors of Selberg class type.
This is an expanded version of the author's lecture at the XX Congresso U.M.I., held in Siena in September 2015. After a brief review of L-functions, we turn to the classical converse theorems of H.Hamburger, E.Hecke and A.Weil, and to some later developments. Finally we present several results on converse theorems in the framework of the Selberg class of L-functions.
We prove an analogue of Jacquet's conjecture on the local converse theorem for \ell-adic families of co-Whittaker representations of GL_n(F), where F is a finite extension of Q_p and \ell does not equal p. We also prove an analogue of Jacquet's conjecture for a descent theorem, which asks for the smallest collection of gamma factors determining the subring of definition of an \ell-adic family. These two theorems are closely related to the local Langlands correspondence in \ell-adic families.
Let $F$ be a $p$-adic field and $E/F$ be a quadratic extension. In this paper, we prove the local converse theorem for generic representations of $\textrm{U}_{E/F}(2,2)$ if $E/F$ is unramified or the residue characteristic of $F$ is odd. Our method is purely local and analytic, and the same method also gives the local converse theorem for $\textrm{Sp}_4(F)$ and $\widetilde {\textrm{Sp}}_4(F)$ if the residue characteristic of $F$ is odd.
In this paper, we define a $γ$-factor for generic representations of $\RU(1,1)\times \Res_{E/F}(\GL_1)$ and prove a local converse theorem for $\RU(1,1)$ using the $γ$-factor we defined. We also give a new proof of the local converse theorem for $\GL_2$ using a $γ$-factor of $\GL_2\times \GL_2$ type which was originally defined by Jacquet in \cite{J}.
By Zeckendorf's Theorem, every positive integer is uniquely written as a sum of non-adjacent terms of the Fibonacci sequence, and its converse states that if a sequence in the positive integers has this property, it must be the Fibonacci sequence. If we instead consider the problem of finding a monotone sequence with such a property, we call it the weak converse of Zeckendorf's theorem. In this paper, we first introduce a generalization of Zeckendorf conditions, and subsequently, Zeckendorf's theorems and their weak converses for the general Zeckendorf conditions. We also extend the generaliza…
We present an approximate converse theorem which measures how close a given set of irreducible admissible unramified unitary generic local representations of GL(n) is to a genuine cuspidal representation. To get a formula for the measure, we introduce a quasi-Maass form on the generalized upper half plane for a given set of local representations. We also construct an annihilating operator which enables us to write down an explicit cuspidal automorphic function.
Let $A/\mathbb{Q}$ be an elliptic curve having split multiplicative reduction at an odd prime $p$. Under some mild technical assumptions, we prove the statement: $$rank_{\mathbb{Z}}A(\mathbb{Q})=1 \ \ and\ \ \ \#\Big(\underline{III}(A/\mathbb{Q})[p^\infty]\Big)<\infty\ \ \implies\ \ \mathrm{ord}_{s=1}L(A/\mathbb{Q},s)=1,$$ thus providing a "$p$-converse" to a celebrated theorem of Kolyvagin-Gross-Zagier.
We prove a GL(n)xGL(n-1) local converse theorem for l-adic families of smooth representations of GL(n,F) where F is a finite extension of Q_p and l is different from p. To do so, we also extend the theory of Rankin-Selberg integrals, first introduced by Jacquet, Piatetski-Shapiro, and Shalika, to the setting of families, continuing previous work of the author.
We prove that a Dirichlet series with a functional equation and Euler product of a particular form can only arise from a holomorphic cusp form on the Hecke congruence group $Γ_0(13)$. The proof does not assume a functional equation for the twists of the Dirichlet series. The main new ingredient is a generalization of the familiar Weil's lemma that played a prominent role in previous converse theorems.
We investigate the relations for $L$-functions satisfying certain functional equation, summationa formulas of Voronoi-Ferrar type and Maass forms of integral and half-integral weight. Summation formulas of Voronoi-Ferrar type can be viewed as an automorphic property of distribution vectors of non-unitary principal series representations of the double covering group of $SL(2)$. Our goal is converse theorems for automorphic distributions and Maass forms of level $N$ characterizing them by analytic properties of the associated $L$-functions. As an application of our converse theorems, we construc…
We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main idea is to linearize the Euler product, replacing it by twists by Ramanujan sums. We provide evidence for the conjecture, including proofs of some special cases and under various additional hypotheses.
We construct a family of harmonic Maass forms of polynomial growth of any level corresponding to any cusp whose shadows are Eisenstein series of integral weight. We further consider Dirichlet series attached to a harmonic Maass form of polynomial growth, study its analytic properties, and prove an analogue of Weil's converse theorem.
We prove a converse theorem for a family of L functions of degree 2 with gamma factor coming from a holomorphic cuspform. We show these L functions coincide with either those coming from a newform or a product of L functions arising from Dirichlet characters. We require some analytic data on the Euler factors, but don't require anything on the shape. We also suppose that the twisted L functions satisfy expected functional equations. We allow the non-trivial twists to have arbitrary poles.
Let $L/K$ be a finite Galois extension of local fields. The Hasse-Arf theorem says that if Gal$(L/K)$ is abelian then the upper ramification breaks of $L/K$ must be integers. We prove the following converse to the Hasse-Arf theorem: Let $G$ be a nonabelian group which is isomorphic to the Galois group of some totally ramified extension $E/F$ of local fields with residue characteristic $p>2$. Then there is a totally ramified extension of local fields $L/K$ with residue characteristic $p$ such that Gal$(L/K)\cong G$ and $L/K$ has at least one nonintegral upper ramification break.
We prove a companion forms theorem for ordinary n-dimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author, and a technique for deducing companion forms theorems due to the first author. We deduce results about the possible Serre weights of mod l Galois representations corresponding to automorphic representations on unitary groups. We then use functoriality to prove similar results for automorphic representations of GSp4 over totally real fields.
We give a proof of a soft version of the $p$-converse to a theorem of Gross--Zagier and Kolyvagin for non-CM elliptic curves with good ordinary reduction at $p >3$ under the irreducibility assumption on the residual representation. In particular, no condition on the conductor is imposed. Combining with the known results, we obtain that the Mordell-Weil rank is one and the Tate-Shafarevich group is finite if and only if the analytic rank is one for every elliptic curve over the rationals.
We prove a general result relating the shape of the Euler product of an $L$-function to the analytic properties of certain linear twists of the $L$-function itself. Then, by a sharp form of the transformation formula for linear twists, we check the required analytic properties in the case of $L$-functions of degree 2 and conductor 1 in the Selberg class. Finally we prove a converse theorem, showing that $ζ(s)^2$ is the only member of the Selberg class satisfying the above conditions and, moreover, having a pole at $s=1$.
We show that every Fricke invariant meromorphic modular form for $Γ_0(N)$ whose divisor on $X_0(N)$ is defined over $\mathbb{Q}$ and supported on Heegner divisors and the cusps is a generalized Borcherds product associated to a harmonic Maass form of weight $1/2$. Further, we derive a criterion for the finiteness of the multiplier systems of generalized Borcherds products in terms of the vanishing of the central derivatives of $L$-function of certain weight $2$ newforms. We also prove similar results for twisted Borcherds products.
Associated to a newform $f(z)$ is a Dirichlet series $L_f(s)$ with functional equation and Euler product. Hecke showed that if the Dirichlet series $F(s)$ has a functional equation of a particular form, then $F(s)=L_f(s)$ for some holomorphic newform $f(z)$ on $Γ(1)$. Weil extended this result to $Γ_0(N)$ under an assumption on the twists of $F(s)$ by Dirichlet characters. Conrey and Farmer extended Hecke's result for certain small $N$, assuming that the local factors in the Euler product of $F(s)$ were of a special form. We make the same assumption on the Euler product and describe an approac…
We prove a descent criterion for certain families of smooth representations of GL_n(F) (F a p-adic field) in terms of the gamma factors of pairs constructed in previous work of the second author. We then use this descent criterion, together with a theory of gamma factors for families of representations of the Weil group W_F (developed previously by both authors), to prove a series of conjectures, due to the first author, that give a complete description of the integral Bernstein center in terms of Galois theory and the local Langlands correspondence. An immediate consequence is the conjectural…
We prove a converse theorem for split even special orthogonal groups over finite fields. This is the only case left on converse theorems of split classical groups and the difficulty is the existence of the outer automorphism. In this paper, we develop new ideas and overcome this difficulty.
In this paper, we completely prove a standard conjecture on the local converse theorem for generic representations of GLn(F), where F is a non-archimedean local field.
We prove a local converse theorem for $GL_n$ over the archimedean local fields which characterizes an infinitesimal equivalence class of irreducible admissible representations of $GL_n(\mathbb{R})$ or $GL_n(\mathbb{C})$ in terms of twisted local gamma factors.
Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove that if $E$ has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two odd primes and if the rank of $E(\mathbb{Q})$ is one and the Tate-Shafarevich group of $E$ has finite order, then $\mathrm{ord}_{s=1}L(E,s)=1$. We also prove the corresponding result for the abelian variety associated with a weight two newform $f$ of trivial character. These, and other related results, are consequences of our main theorem, which establishes criteria for $f$ and $H^1_f(\mathbb{Q},V)$, wh…
London Institute for Mathematical Sciences（伦敦数学科学研究所）；Merton College, Oxford（牛津大学默顿学院）；University of Connecticut（康涅狄格大学）；Korea Institute for Advanced Study（韩国高等研究院）；University of Westminster（威斯敏斯特大学）；Northeastern University（东北大学）
In this paper, we prove the local converse theorem for split even special orthogonal groups over a non-Archimedean local field of characteristic zero. This is the only case left on local converse theorems of split classical groups and the difficulty is the existence of the outer automorphism. We apply new ideas of considering the summation of partial Bessel functions and overcome this difficulty. As a direct application, we obtain a weak rigidity theorem for irreducible generic cuspidal representations of split even special orthogonal groups.
We prove a new converse theorem for Borcherds' multiplicative theta lift which improves the previously known results. To this end we develop a newform theory for vector valued modular forms for the Weil representation, which might be of independent interest. We also derive lower bounds for the ranks of the Picard groups and the spaces of holomorphic top degree differential forms of modular varieties associated to orthogonal groups.
Let F be a non-archimedean local field of characteristic zero. In this paper we construct examples of supercuspidal representations showing that the bound $[N/2]$ for the local converse theorem of $GL_N(F)$ is sharp, N general, when the residual characteristic of $F$ is bigger than $N$.
We show that the distribution of large values of an additive function on the integers, and the distribution of values of the additive function on the primes are related to each other via a Levy Process. As a consequence we obtain a converse to an old theorem of Halasz. Halasz proved that if f is an strongly additive function with f (p) \in {0, 1}, then f is Poisson distributed on the integers. We prove, conversely, that if f is Poisson distributed on the integers then for most primes p, f(p) = o(1) or f(p) = 1 + o(1).
In this paper we characterize irreducible generic representations of $\SO_{2n+1}(k)$ where $k$ is a $p$-adic field) by means of twisted local gamma factors (the Local Converse Theorem). As applications, we prove that two irreducible generic cuspidal automorphic representations of $\SO_{2n+1}({\Bbb A})$ (where ${\Bbb A}$ is the ring of adeles of a number field) are equivalent if their local components are equivalent at almost all local places (the Rigidity Theorem);and prove the Local Langlands Reciprocity Conjecture for generic supercuspidal representations of $\SO_{2n+1}(k)$.
Mod $\ell$ gamma factors and a converse theorem for finite general linear groups
Boston University（波士顿大学）；University of Toronto（多伦多大学）；Brooklyn College, City University of New York（纽约市立大学布鲁克林学院）；Johns Hopkins University（约翰斯·霍普金斯大学）；University of Maine（缅因大学）；Massachusetts Institute of Technology（麻省理工学院）
This paper verifies $n\times 1$ Local Converse Theorem for twisted gamma factors of irreducible cuspidal representations of ${\rm GL}_n({\mathbb F}_p)$, for $n\leq 5,$ and of irreducible generic representations, for $n<\frac{q-1}{2\sqrt{q}}+1$ in the appendix by Zhiwei Yun, where $p$ is a prime and q is a power of $p$. The counterpart of $n\times 1$ converse theorem for level zero cuspidal representations also follows the established relation between gamma factors of ${\rm GL}_n({\mathcal F})$ and that of ${\rm GL}_n({\mathbb F}_q)$, where ${\mathcal F}$ denotes a $p$-adic field whose residue …
Heilbronn Institute for Mathematical Research, School of Mathematics University of Bristol（布里斯托大学数学学院海布隆数学研究所）；Dept of Mathematics and Statistics Williams College（威廉姆斯学院数学与统计系）
Given two arbitrary almost periodic functions with associated Fourier exponents which are linearly independent over the rational numbers, we prove that the existence of a common open vertical strip $V$, where both functions assume the same set of values on every open vertical substrip included in $V$, is a necessary and sufficient condition for both functions to have the same region of almost periodicity and to be $^*$-equivalent or Bohr-equivalent. This result represents the converse of Bohr's equivalence theorem for this particular case.
Associated to a newform $f(z)$ is a Dirichlet series $L_f(s)$ with functional equation and Euler product. Hecke showed that if the Dirichlet series $F(s)$ has a functional equation of the appropriate form, then $F(s)=L_f(s)$ for some holomorphic newform $f(z)$ on $Γ(1)$. Weil extended this result to $Γ_0(N)$ under an assumption on the twists of $F(s)$ by Dirichlet characters. We show that, at least for small $N$, the assumption on twists can be replaced by an assumption on the local factors of the Euler product of $F(s)$.
Wall's theorem on arithmetic progressions says that if $0.a_1a_2a_3\dots$ is normal, then for any $k,\ell\in \mathbb{N}$, $0.a_ka_{k+\ell}a_{k+2\ell}\dots$ is also normal. We examine a converse statement and show that if $0.a_{n_1}a_{n_2}a_{n_3}\dots$ is normal for periodic increasing sequences $n_1<n_2<n_3<\dots$ of asymptotic density arbitrarily close to $1$, then $0.a_1a_2a_3\dots$ is normal. We show this is close to sharp in the sense that there are numbers $0.a_1a_2a_3\dots$ that are not normal, but for which $0.a_{n_1}a_{n_2}a_{n_3}\dots$ is normal along a large collection of sequences w…
