Some Recent Research Papers

• Split Orders and Convex Polytopes in Buildings

  As part of his work to develop an explicit trace formula for Hecke operators on congruence subgroups of $SL_2(\Z)$, Hijikata \cite{Hijikata-traces} defines and characterizes the notion of a split order in $M_2(k)$, where $k$ is a local field. In this paper, we generalize the notion of a split order to $M_n(k)$ for $n>2$ and give a natural geometric characterization in terms of the affine building for $SL_n(k)$. In particular, we show that there is a one-to-one correspondence between split orders in $M_n(k)$ and a collection of convex polytopes apartmentsilding such that the split order is the intersection of all the maximal orders representing the vertices in the polytope. This generalizes the geometric interpretation in the $n=2$ case in which split orders correspond to geodesics in the tree for $SL_2(k)$ with the split order given as the intersection of the endpoints of the geodesic.

  pdf version 26 August 2008

• Constructing Simultaneous Hecke Eigenforms

  It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie-Kohnen \cite{Choie-Kohnen} who considered diagonalization of ``bad'' Hecke operators on spaces with square free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character.

  pdf version 1 July 2008

• Inverting the Satake Map for $Sp_n$, and applications to Hecke Operators

  By compatibly grading the $p$-part of the Hecke algebra associated to $Sp_n(\Z)$ and the subring of $\Q[x_0^{\pm1}, \dots,x_n^{\pm1}]$ invariant under the associated Weyl group, we produce a matrix representation of the Satake isomorphism restricted to the corresponding finite dimensional components. In particular, using the elementary divisor theory of integral matrices, we show how to determine the entries of this matrix representation restricted to double cosets of a fixed similitude. The matrix representation is upper-triangular, and can be explicitly inverted.

  To address the specific question of characterizing families of Hecke operators whose generating series have ``Euler'' products, we define $(n+1)$ families of polynomial Hecke operators $t_k^n(p^\ell)$ (in $\Q[x_0^{\pm1}, \dots,x_n^{\pm1}]$) for $Sp_n$ whose generating series $\sum t_k^n(p^\ell) v^\ell$ are rational functions of the form $q_k(v)^{-1}$, where $q_k$ is a polynomial in $\Q[x_0^{\pm1}, \dots,x_n^{\pm1}][v]$ of degree $2^k\binom{n}{k}$ ($2^n$ if $k = 0$). For $k=0$ and $k=1$ the form of the polynomial is essentially that of the local factors in the spinor and standard zeta functions. For $k>1$, these appear to be new expressions. Taking advantage of the generating series and our ability to explicitly invert the Satake isomorphism, we explicitly compute the classical operators with the analogous properties in the case of genus 2. It is of interest to note that these operators lie in the full, but not generally the integral, Hecke algebra.

  pdf version 9 May 2006

• The Arithmetic and Combinatorics of Buildings for $Sp_n$

  In this paper, we investigate both arithmetic and combinatorial aspects of buildings and associated Hecke operators for $Sp_n(K)$ with $K$ a local field. We characterize the action of the affine Weyl group in terms of a symplectic basis for an apartment, characterize the special vertices as those which are self-dual with respect to the induced inner product, and establish a one-to-one correspondence between the special vertices in an apartment and the elements of the quotient $\Z^{n+1}/\Z(2,1,\dots,1)$.

  We then give a natural representation of the local Hecke algebra over $K$ acting on the special vertices of the Bruhat-Tits building for $Sp_n(K)$. Finally, we give an application of the Hecke operators defined on the building by characterizing minimal walks on the building for $Sp_n$.

  original pdf version 19 Dec 2003 (revised PDF) 12 July 2005 (to appear Transactions AMS)

• Hecke Operators, Zeta Functions and the Satake Map

  Hecke operators $t_k^n(p^\ell)$ for $Sp_n$ whose generating series $\sum t_k^n(p^\ell) u^\ell$ are rational functions of the form $q_k(u)^{-1}$, where $q_k$ is a polynomial of degree $2^k\binom{n}{k}$ ($2^n$ if $k = 0$). For $k=0$ and $k=1$ the form of the polynomial is essentially that of the local factors in the spinor and standard zeta functions. For $k>1$, these appear to be new expressions.

  We also offer some insight (disjoint from the representation theory) for why there should be a correspondence between the local Hecke algebra and a ring of polynomials invariant under an associated Weyl group.

  pdf version 19 Dec 2003

• Hecke Operators for GL_n and buildings

  We describe a representation of the local Hecke algebra for $GL_n$ in which the Hecke operators act on the vertices of the Bruhat-Tits building for $SL_n(\Qp)$. We also give a geometric interpretation of this representation, characterizing the action of our operators on a vertex in terms of the endpoints of minimal walks in the building. This generalizes work of Serre who defined Hecke operators acting on the vertices of a tree (the building for $SL_2(\Qp)$).

  pdf version 9 Aug 2002 (Acta Arith 112 (2004))

• Rationality Theorems for Hecke Operators on GL_n

  We define $n$ families of Hecke operators $T_k^n(p^\ell)$ for $GL_n$ whose generating series $\sum T_k^n(p^\ell) u^\ell$ are rational functions of the form $q_k(u)^{-1}$ where $q_k$ is a polynomial of degree $\binom{n}{k}$, and whose form is that of the $k$th exterior product. This work can be viewed as a refinement of work of Andrianov \cite{Andrianov70}, in which he defined Hecke operators the sum of whose generating series was a rational function with nontrivial numerator and whose denominator was essentially $\prod_k q_k$.

  By a careful analysis of the Satake map which defines an isomorphism between a local Hecke algebra and a ring of symmetric polynomials, we define $n$ families of (polynomial) Hecke operators and characterize their generating series as rational functions. We then give an explicit means by which to locally invert the Satake isomorphism, and show how to translate these polynomial operators back to the classical double coset setting. The classical Hecke operators have generating series of exactly the same form as their polynomial counterparts, and hence are of number-theoretic interest. We give explicit examples for $GL_3$ and $GL_4$.

  pdf version 31 July 2002 (JNT 102 278 -- 297)

• Rolles' Theorem for Local Fields

  In this paper we show that no non-archimedean local field has Rolle's property.

  pdf version 11 Apr 2002

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