Some Recent Research Papers

Local Selectivity of Orders in Central Simple Algebras

Let $B$ be a central simple algebra of degree $n$ over a number field $K$, and $L\subset B$ a strictly maximal subfield. We say that the ring of integers $\mathcal O_L$ is selective if there exists an isomorphism class of maximal orders in $B$ no element of which contains $\mathcal O_L$. Many authors have worked to characterize the degree to which selectivity occurs, first in quaternion algebras, and more recently in higher-rank algebras. In the present work, we consider a local variant of the selectivity problem and applications. We first prove a theorem characterizing which maximal orders in a local central simple algebra contain the global ring of integers $\mathcal O_L$ by leveraging the theory of affine buildings for $SL_r(D)$ where $D$ is a local central division algebra. Then as an application, we use the local result and a local-global principle to show how to compute a set of representatives of the isomorphism classes of maximal orders in $B$, and distinguish those which are guaranteed to contain $\mathcal O_L$. Having such a set of representatives allows both algebraic and geometric applications. As an algebraic application, we recover a global selectivity result mentioned above, and give examples which clarify the interesting role of partial ramification in the algebra.

ArXiv version 2017
International Journal of Number Theory. 13, No. 4, pp. 853--884 (2017)

Selective Orders in Central Simple Algebras

Let $B$ be a central simple algebra of degree $n$ over a number field $K$, and let $L/K$ be a field extension of degree $n$ which embeds in $B$. The question of which of the isomorphism classes of maximal orders in $B$ admit an embedding of an $\mathcal{O}_K$-suborder $\Omega$ of $\mathcal{O}_L$ is the question of selectivity of the order $\Omega$. In this paper we are concerned with algebras of degree $n \ge 3$, and continue the work of several authors to characterize the degree to which and conditions under which selectivity will occur. We take a local approach via the theory of affine buildings which provides explicit information about the structure of local embeddings, and leverage that information to produce a global characterization. We clarify that selectivity can never occur in an algebra which is a division algebra at a finite prime as well as construct representatives of the isomorphism classes of maximal orders which do admit embeddings of $\Omega$. An example of selectivity in an algebra with partial ramification is provided.

ArXiv version 12 April 2012

Embedding Orders into Central Simple Algebras

The question of embedding fields into central simple algebras $B$ over a number field $K$ was the realm of class field theory. The subject of embedding orders contained in the ring of integers of maximal subfields $L$ of such an algebra into orders in that algebra is more nuanced. The first such result along those lines is an elegant result of Chevalley \cite{Chevalley-book} which says that with $B = M_n(K)$ the ratio of the number of isomorphism classes of maximal orders in $B$ into which the ring of integers of $L$ can be embedded (to the total number of classes) is $[L \cap \widetilde K : K]^{-1}$ where $\widetilde K$ is the Hilbert class field of $K$. Chinburg and Friedman (\cite{Chinburg-Friedman}) consider arbitrary quadratic orders in quaternion algebras satisfying the Eichler condition, and Arenas-Carmona \cite{Arenas-Carmona} considers embeddings of the ring of integers into maximal orders in a broad class of higher rank central simple algebras. In this paper, we consider central simple algebras of dimension $p^2$, $p$ an odd prime, and we show that arbitrary commutative orders in a degree $p$ extension of $K$, embed into none, all or exactly one out of $p$ isomorphism classes of maximal orders. Those commutative orders which are selective in this sense are explicitly characterized; class fields play a pivotal role. A crucial ingredient of Chinberg and Friedman's argument was the structure of the tree of maximal orders for $SL_2$ over a local field. In this work, we generalize Chinburg and Friedman's results replacing the tree by the Bruhat-Tits building for $SL_p$.

ArXiv version 18 June 2010
Journal de théorie des nombres de Bordeaux, 24 no. 2 (2012), 405 -- 424.

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 $SL_2({\mathbb{Z}})$, 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 Journal of Number Theory Volume 130, Issue 1, January 2010, Pages 101-115

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 $Sp_n({\mathbb{Z}})$ and the subring of ${\mathbb{Q}}[x_0^{\pm1}, \dots,x_n^{\pm1}]$ 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 ${\mathbb{Q}}[x_0^{\pm1}, \dots,x_n^{\pm1}]$) 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 ${\mathbb{Q}}[x_0^{\pm1}, \dots,x_n^{\pm1}][v]$ 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 ${\mathbb{Z}}^{n+1}/{\mathbb{Z}}(2,1,\dots,1)$.
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 $Sp_n$.

(revised PDF) 12 July 2005
Transactions AMS 359, (2007), 3409-3423.

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 $SL_n({\mathbb{Q}}p)$. 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 $SL_2({\mathbb{Q}}p)$).

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