results for au:Stevens_S in:math

- Let $G$ be a symplectic group over a nonarchimedean local field of characteristic zero and odd residual characteristic. Given an irreducible cuspidal representation of G, we determine its Langlands parameter (equivalently, its Jordan blocks in the language of Moeglin) in terms of the local data from which the representation is explicitly constructed, up to a possible unramified twist in each block of the parameter. We deduce a Ramification Theorem for $G$, giving a bijection between the set of endo-parameters for $G$ and the set of restrictions to wild inertia of discrete Langlands parameters for $G$, compatible with the local Langlands correspondence. The main tool consists in analysing the intertwining Hecke algebra of a good cover, in the sense of Bushnell--Kutzko, for parabolic induction from a cuspidal representation of $G\times\mathrm{GL}_n$, seen as a maximal Levi subgroup of a bigger symplectic group, in order to determine its (ir)reducibility; a criterion of Moeglin then relates this to Langlands parameters.
- By computing reducibility points of parabolically induced representations, we construct, to within at most two unramified quadratic characters, the Langlands parameter of an arbitrary depth zero irreducible cuspidal representation $\pi$ of a classical group (which may be not-quasi-split) over a nonarchimedean local field of odd residual characteristic. From this, we can explicitly describe all the irreducible cuspidal representations in the union of one, two, or four L-packets, containing $\pi$. These results generalize the work of DeBacker-Reeder (in the case of classical groups) from regular to arbitrary tame Langlands parameters.
- Nov 16 2016 math.RT arXiv:1611.04796v1Let $\mathfrak{o}$ be the ring of integers in a non-Archimedean local field with finite residue field, $\mathfrak{p}$ its maximal ideal, and $r\geq2$ an integer. An irreducible representation of the finite group $G_{r}=\mathrm{GL}_{N}(\mathfrak{o}/\mathfrak{p}^{r})$ is called regular if its restriction to the principal congruence kernel $K^{r-1}=1+\mathfrak{p}^{r-1}\mathrm{M}_{N}(\mathfrak{o}/\mathfrak{p}^{r})$ consists of representations whose stabilisers modulo $K^{1}$ are centralisers of regular elements in $\mathrm{M}_{N}(\mathfrak{o}/\mathfrak{p})$. The regular representations form the largest class of representations of $G_{r}$ which is currently amenable to explicit construction. Their study, motivated by constructions of supercuspidal representations, goes back to Shintani, but the general case remained open for a long time. In this paper we give an explicit construction of all the regular representations of $G_{r}$.
- Nov 15 2016 math.RT arXiv:1611.04317v1We show how the modular representation theory of inner forms of general linear groups over a non-Archimedean local field can be brought to bear on the complex theory in a remarkable way. Let F be a non-Archimedean locally compact field of residue characteristic p, and let G be an inner form of the general linear group GL(n,F). We consider the problem of describing explicitly the local Jacquet--Langlands correspondence between the complex discrete series representations of G and GL(n,F), in terms of type theory. We show that the congruence properties of the local Jacquet--Langlands correspondence exhibited by A. Mínguez and the first named author give information about the explicit description of this correspondence. We prove that the problem of the invariance of the endo-class by the Jacquet--Langlands correspondence can be reduced to the case where the representations $\pi$ and its Jacquet--Langlands transfer JL($\pi$) are both cuspidal with torsion number 1. We also give an explicit description of the Jacquet--Langlands correspondence for all essentially tame discrete series representations of G, up to an unramified twist, in terms of admissible pairs, generalizing previous results by Bushnell and Henniart. In positive depth, our results are the first beyond the case where $\pi$ and JL($\pi$) are both cuspidal.
- Nov 09 2016 math.RT arXiv:1611.02667v1For a unitary, symplectic, or special orthogonal group over a non-archimedean local field of odd residual characteristic, we prove that two intertwining cuspidal types are conjugate in the group. This completes work of the third author who showed that every irreducible cuspidal representation of such a classical group is compactly induced from a cuspidal type, now giving a classification of irreducible cuspidal representations of classical groups in terms of cuspidal types. Our approach is to completely understand the intertwining of the so-called self dual semisimple characters, which form \emphthe fundamental step in the construction. To this aim, we generalise Bushnell--Henniart's theory of endo-class for simple characters of general linear groups to a theory for self dual semisimple characters of classical groups, and introduce (self dual) endo-parameters which parametrise intertwining classes of (self dual) semisimple characters.
- Sep 21 2016 math.CO arXiv:1609.06284v4We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field $\mathbb{F}$, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that $m$ points and $n$ lines in $\mathbb{F}^2$, with $m^{7/8}<n<m^{8/7}$, determine at most $O(m^{11/15}n^{11/15})$ incidences (where, if $\mathbb{F}$ has positive characteristic $p$, we assume $m^{-2}n^{13}\ll p^{15}$). This improves on the previous best known bound, due to Jones. To obtain our bound, we first prove an optimal point-line incidence bound on Cartesian products, using a reduction to a point-plane incidence bound of Rudnev. We then cover most of the point set with Cartesian products, and we bound the incidences on each product separately, using the bound just mentioned. We give several applications, to sum-product-type problems, an expander problem of Bourgain, the distinct distance problem and Beck's theorem.
- We prove new exponents for the energy version of the Erdős-Szemerédi sum-product conjecture, raised by Balog and Wooley. They match the previously established milestone values for the standard formulation of the question, both for general fields and the special case of real or complex numbers, and appear to be the best ones attainable within the currently available technology. Further results are obtained about multiplicative energies of additive shifts and a strengthened energy version of the "few sums, many products" inequality of Elekes and Ruzsa. The latter inequality enables us to obtain a minor improvement of the state-of the art sum-product exponent over the reals due to Konyagin and the second author, up to $\frac{4}{3}+\frac{1}{1509}$. An application of energy estimates to an instance of arithmetic growth in prime residue fields is presented.
- Sep 09 2015 math.RT arXiv:1509.02212v2For a classical group over a non-archimedean local field of odd residual characteristic p, we construct all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p, as representations induced from a cuspidal type. We also give a fundamental step towards the classification of cuspidal representations, identifying when certain cuspidal types induce to equivalent representations; this result is new even in the case of complex representations. Finally, we prove that the representations induced from more general types are quasi-projective, a crucial tool for extending the results here to arbitrary irreducible representations.
- Jul 01 2015 math.NT arXiv:1506.08897v1In this paper we showed that under two assumptions we are able to define interesting functions that we call generalized local coefficients. We showed that in the quasi-split case generalized local coefficients are up to a positive constant the same as Shahidi's local coefficients. We provide a proof that the non quasi-split group $GL_m(D)$, for a central division algebra $D$ satisfies those assumptions. We also showed that generalized local coefficients satisfy nice properties, like the relation to Plancherel measures and multiplicativity inherited by that of intertwining operators. Generalized local coefficients are only defined for representations that are $(Y,\varphi)$-generic which is a generalization of generic representations in the quasi-split case. Here $Y$ denotes a nilpotent element in the Lie algebra of the group and $\varphi$ is a co-character related to $Y$.
- The Local Converse Problem is to determine how the family of the local gamma factors $\gamma(s,\pi\times\tau,\psi)$ characterizes the isomorphism class of an irreducible admissible generic representation $\pi$ of $\mathrm{GL}_n(F)$, with $F$ a non-archimedean local field, where $\tau$ runs through all irreducible supercuspidal representations of $\mathrm{GL}_r(F)$ and $r$ runs through positive integers. The Jacquet conjecture asserts that it is enough to take $r=1,2,\ldots,\left[\frac{n}{2}\right]$. Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet conjecture, we formulate a general approach to prove the Jacquet conjecture. With this approach, the Jacquet conjecture is proved under an assumption which is then verified in several cases, including the case of level zero representations.
- Apr 01 2015 math.NT arXiv:1503.08882v3Let~$G$ be a unitary group of an~$\epsilon$-hermitian form~$h$ given over a nonarchimedean local field~$F_0$ of odd residue characteristic. We introduce a geometric combinatoric condition under which we prove "Intertwining implies Conjugacy" for semisimple characters of~$G$ and the general linear group of the ambient vector space of~$G$. Further we prove a Skolem-Noether result for the action of~$G$ on its Lie algebra, more precisely two Lie algebra elements of~$G$ which have the same characteristic polynomial over~$F$ must be conjugate under an element of~$G$ if there are corresponding semisimple characters which intertwine over an element of~$G$ Let~$G$ be a unitary group over a nonarchimedean local field of odd residual characteristic. This paper concerns the study of the "wild part" of the irreducible smooth representations of~$G$, encoded in a so-called "semisimple character". We prove two fundamental results concerning them, which are crucial steps towards a classification of the cuspidal representations of~$G$. First we introduce a geometric combinatoric condition under which we prove an "intertwining implies conjugacy" theorem for semisimple characters, both in~$G$ and in the ambient general linear group. Second, we prove a Skolem--Noether theorem for the action of~$G$ on its Lie algebra; more precisely, two semisimple elements of the Lie algebra of~$G$ which have the same characteristic polynomial must be conjugate under an element of~$G$ if there are corresponding semisimple strata which are intertwined by an element of~$G$.
- Sep 18 2014 math.RT arXiv:1409.4790v1Based on previous results of Jiang, Nien and the third author, we prove that any two minimax unitarizable supercuspidals of GL_N that have the same depth and central character admit a special pair of Whittaker functions. This result gives a new reduction towards a final proof of Jacquet's conjecture on the local converse problem for GL_N. As a corollary of our result, we prove Jacquet's conjecture for GL_N, when N is prime.
- Jul 30 2014 math.DS arXiv:1407.7787v1The relationship between two dynamical systems, one of which is obtained from the other by forming the quotient by an action of an involution commuting with the dynamics, is studied. The constraints and the possible extent of freedom in the relationship between the growth of closed orbits in pairs of systems related in this way is explored.
- Feb 24 2014 math.RT arXiv:1402.5349v1Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite dimensional central division F-algebra and let R be an algebraically closed field of characteristic different from p. To any irreducible smooth representation of G=GL(m,D) with coefficients in R, we can attach a uniquely determined inertial class of supercuspidal pairs of G. This provides us with a partition of the set of all isomorphism classes of irreducible representations of G. We write R(G) for the category of all smooth representations of G with coefficients in R. To any inertial class O of supercuspidal pairs of G, we can attach the subcategory R(O) made of smooth representations all of whose irreducible subquotients are in the subset determined by this inertial class. We prove that R(G) decomposes into the product of the R(O), where O ranges over all possible inertial class of supercuspidal pairs of G, and that each summand R(O) is indecomposable.
- We construct, for any symplectic, unitary or special orthogonal group over a locally compact nonarchimedean local field of odd residual characteristic, a type for each Bernstein component of the category of smooth representations, using Bushnell--Kutzko's theory of covers. Moreover, for a component corresponding to a cuspidal representation of a maximal Levi subgroup, we prove that the Hecke algebra is either abelian, or a generic Hecke algebra on an infinite dihedral group, with parameters which are, at least in principle, computable via results of Lusztig.
- We exhibit continua on two different growth scales in the dynamical Mertens' theorem for ergodic automorphisms of one-dimensional solenoids.
- We give a complete description of the category of smooth complex representations of the multiplicative group of a central simple algebra over a locally compact nonarchimedean local field. More precisely, for each inertial class in the Bernstein spectrum, we construct a type and compute its Hecke algebra. The Hecke algebras that arise are all naturally isomorphic to products of affine Hecke algebras of type A. We also prove that, for cuspidal classes, the simple type is unique up to conjugacy.
- Let F be a locally compact nonarchimedean local field. In this article, we extend to any inner form of GL(n) over F the notion of endo-class introduced by Bushnell and Henniart for GL(n,F). We investigate the intertwining relations of simple characters of these groups, in particular their preservation properties under transfer. This allows us to associate to any discrete series representation of an inner form of GL(n,F) an endo-class over F. We conjecture that this endo-class is invariant under the local Jacquet-Langlands correspondence.
- Mar 11 2010 math.NT arXiv:1003.2131v2On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. We prove that the number of prime terms in the sequence is uniformly bounded. When the rational point is the image of another rational point under a certain 3-isogeny, all terms beyond the first fail to be primes.
- Jan 15 2008 math.DS arXiv:0801.2082v4A dynamical Mertens' theorem for ergodic toral automorphisms with error term O(N^-1) is found, and the influence of resonances among the eigenvalues of unit modulus is examined. Examples are found with many more, and with many fewer, periodic orbits than expected.
- Dec 18 2007 math.NT arXiv:0712.2696v1An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang's conjecture, and over the rational function field, unconditionally. In the latter case, a uniform bound is obtained on the index of a prime term. Sharpened versions of these techniques are shown to lead to explicit results where all the irreducible terms can be computed.
- We describe the supercuspidal representations of Sp(4,F), for F a non-archimedean local field of residual characteristic different from 2, and determine which are generic.
- We introduce a class of group endomorphisms -- those of finite combinatorial rank -- exhibiting slow orbit growth. An associated Dirichlet series is used to obtain an exact orbit counting formula, and in the connected case this series is shown to have a closed rational form. Analytic properties of the Dirichlet series are related to orbit-growth asymptotics: depending on the location of the abscissa of convergence and the degree of the pole there, various orbit-growth asymptotics are found, all of which are polynomially bounded.
- Mar 20 2007 math.NT arXiv:math/0703553v1We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u^3+v^3=m, with m cube-free, all the terms beyond the first have a primitive divisor.
- Oct 27 2006 math.RT arXiv:math/0610785v1We study components of the Bernstein category for a p-adic classical group (with p odd) with inertial support a self-dual positive level supercuspidal representation of a Siegel Levi subgroup. More precisely, we use the method of covers to construct a Bushnell-Kutzko type for such a component. A detailed knowledge of the Hecke algebra of the type should have number-theoretic implications.
- Jul 26 2006 math.RT arXiv:math/0607622v2Let G be a unitary, symplectic or special orthogonal group over a locally compact non-archimedean local field of odd residual characteristic. We construct many new supercuspidal representations of G, and Bushnell-Kutzko types for these representations. Moreover, we prove that every irreducible supercuspidal representation of G arises from our constructions.
- Let F be a non Archimedean locally compact field and let D be a central F-division algebra. We prove that any positive level supercuspidal irreducible representation of the group GL(m,D) is compactly induced from a representation of a compact mod center open subgroup of GL(m,D). More precisely, we prove that such representations contain a maximal simple type.
- Jun 01 2006 math.NT arXiv:math/0606003v2We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P)=A_P/B_P^2 denote the $x$-coordinate of the rational point P then we consider when B_P can be a prime power. Using Faltings' Theorem we show that for a fixed power greater than 1, there are only finitely many rational points. Where descent via an isogeny is possible we show, with no restrictions on the power, that there are only finitely many rational points, these points are bounded in number in an explicit fashion, and that they are effectively computable.
- Let $G$ be the group of rational points of a general linear group over a non-archimedean local field $F$. We show that certain representations of open, compact-mod-centre subgroups of $G$, (the maximal simple types of Bushnell and Kutzko) can be realized as concrete spaces. In the level zero case our result is essentially due to Gelfand. This allows us, for a supercuspidal representation $\pi$ of $G$, to compute a distinguished matrix coefficient of $\pi$. By integrating, we obtain an explicit Whittaker function for $\pi$. We use this to compute the epsilon factor of pairs, for supercuspidal representations $\pi_1$, $\pi_2$ of $G$, when $\pi_1$ and the contragredient of $\pi_2$ differ only at the `tame level' (more precisely, $\pi_1$ and $\check{\pi}_2$ contain the same simple character). We do this by computing both sides of the functional equation defining the epsilon factor, using the definition of Jacquet, Piatetskii-Shapiro, Shalika. We also investigate the behaviour of the epsilon factor under twisting of $\pi_1$ by tamely ramified quasi-characters. Our results generalise the special case $\pi_1=\check{\pi}_2$ totally wildly ramified, due to Bushnell and Henniart.
- There are well-known analogs of the prime number theorem and Mertens' theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest non-hyperbolic algebraic systems. The asymptotic behaviour of the orbit-counting function is governed by a rotation on an associated compact group, and in simple examples we exhibit uncountably many different asymptotic growth rates for the orbit-counting function. Mertens' Theorem also holds in this setting, with an explicit rational leading coefficient obtained from arithmetic properties of the non-hyperbolic eigendirections.
- Dec 06 2004 math.NT arXiv:math/0412079v2We consider primitive divisors of terms of integer sequences defined by quadratic polynomials. Apart from some small counterexamples, when a term has a primitive divisor, that primitive divisor is unique. It seems likely that the number of terms with a primitive divisor has a natural density. We discuss two heuristic arguments to suggest a value for that density, one using recent advances made about the distribution of roots of polynomial congruences.
- Let F_o be a non-archimedean locally compact field of residual characteristic not 2. Let G be a classical group over F_o (with no quaternionic algebra involved) which is not of type A_n for n>1. Let b be an element of the Lie algebra g of G that we assume semisimple for simplicity. Let H be the centralizer of b in G and h its Lie algebra. Let I and I_b denote the (enlarged) Bruhat-Tits buildings of G and H respectively. We prove that there is a natural set of maps j_b : I_b --> I which enjoy the following properties: they are affine, H-equivariant, map any apartment of I_b into an apartment of I and are compatible with the Lie algebra filtrations of g and h. In a particular case, where this set is reduced to one element, we prove that j_b is characterized by the last property in the list. We also prove a similar characterization result for the general linear group.