Julia Gordon

The fundamental lemma of Jacquet and Rallis

We prove both the group version and the Lie algebra version of the Fundamental Lemma appearing in a relative trace formula of Jacquet-Rallis in the function field case when the characteristic is greater than the rank of the relevant groups.We prove both the group version and the Lie algebra version of the Fundamental Lemma appearing in a relative trace formula of Jacquet-Rallis in the function field case when the characteristic is greater than the rank of the relevant groups.

Integrability of oscillatory functions on local fields: transfer principles

For oscillatory functions on local fields coming from motivic exponential functions, we show that integrability over Q n p implies integrability over F p ((t)) n for large p , and vice versa. More generally, the integrability only depends on the isomorphism class of the residue field of the local field, once the characteristic of the residue field is large enough. This principle yields general local integrability results for Harish-Chandra characters in positive characteristic as we show in other work. Transfer principles for related conditions such as boundedness and local integrability are also obtained. The proofs rely on a thorough study of loci of integrability, to which we give a geometric meaning by relating them to zero loci of functions of a specific kind.

VIRTUAL TRANSFER FACTORS

The Langlands-Shelstad transfer factor is a function defined on some reductive groups over a p-adic field. Near the origin of the group, it may be viewed as a function on the Lie algebra. For classical groups, its values have the form q c sign, where sign 2 { 1,0,1}, q is the cardinality of the residue field, and c is a rational number. The sign function partitions the Lie algebra into three subsets. This article shows that this partition into three subsets is independent of the p-adic field in the following sense. We define three universal objects (virtual sets in the sense of Quine) such that for any p-adic field F of sufficiently large residue characteristic, the F-points of these three virtual sets form the partition. The theory of arithmetic motivic integration associates a virtual Chow motive with each of the three virtual sets. The construction in this article achieves the first step in a long program to determine the (still conjectural) virtual Chow motives that control the behavior of orbital integrals.

On the computability of some positive-depth supercuspidal characters near the identity

This paper is concerned with the values of Harish-Chandra characters of a class of positive-depth, toral, very supercuspidal representations of

Endoscopic transfer of orbital integrals in large residual characteristic

p

Motivic nature of character values of depth-zero representations

-adic symplectic and special orthogonal groups, near the identity element. We declare two representations equivalent if their characters coincide on a specific neighbourhood of the identity (which is larger than the neighbourhood on which Harish-Chandra local character expansion holds). We construct a parameter space

Motivic Haar Measure on Reductive Groups

B

Motivic Proof of a Character Formula for SL(2)

(that depends on the group and a real number

Elliptic curves, random matrices and orbital integrals

r>0

Transfer principles for bounds of motivic exponential functions

) for the set of equivalence classes of the representations of minimal depth