Shaun Stevens

Semisimple characters for p-adic classical groups

Let G be a unitary, symplectic, or orthogonal group over a non-Archimedean local field of residual characteristic different from 2, considered as the fixed-point subgroup in a general linear group of an involution. Following previous work of Bushnell and Kutzko, and of the author, we generalize the notion of a semisimple character for and for G. In particular, following the formalism of Bushnell and Henniart, we show that these semisimple characters have certain functorial properties. Finally, we show that any positive level supercuspidal representation of G contains a semisimple character.

Primes Generated by Recurrence Sequences

We 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.

Double coset decompositions and intertwining

Abstract: Let F be a non-archimedean local field of residual characteristic different from 2. This paper is a first step in the description of the smooth representation theory of a unitary group G⊂ GL(N,F) via types. We intersect certain double cosets in GL(N,F) with G and hence obtain the intertwining of certain characters ψβ− of open compact subgroups of G, for β∈ Lie G. In the case when β is elliptic regular “modulo ?F”, we will then obtain supercuspidal representations of G.

Towards the Jacquet conjecture on the Local Converse Problem for p-adic GLn

The Local Converse Problem is to determine how the family of twisted local gamma factors characterizes the isomorphism class of an irreducible admissible generic representation of GL(n,F), with F a non-archimedean local field, where the twists run through all irreducible supercuspidal representations of GL(r,F) and r runs through positive integers. The Jacquet conjecture asserts that it is enough to take r = 1,2,...,[n/2]. 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.

Semisimple strata for p-adic classical groups☆

Let F0 be a non-archimedean local field, of residual characteristic different from 2, and let G be a unitary, symplectic or orthogonal group defined over F0. In this paper, we prove some fundamental results towards the classification of the representations of G via the types of Bushnell and Kutzko. In particular, we show that any positive level supercuspidal representation of G contains a semisimple skew stratum, that is, a special character of a certain compact open subgroup of G. The intertwining of such a stratum has previously been calculated.

Dirichlet series for finite combinatorial rank dynamics

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.

Primitive Divisors on Twists of Fermat's Cubic

We 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.

The uniform primality conjecture for elliptic curves

An 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 Langs 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.

Genericity of supercuspidal representations of p -adic Sp 4

We describe the supercuspidal representations of Sp4(F), for F a non-archimedean local field of residual characteristic different from 2, and determine which are generic.

The regular representations of GLN over finite local principal ideal rings

Let