Julia Yulia Gordon

The main goal of this thesis is to provide results to explicitly compute Tamagawa numbers for maximal tori of similitude groups. These numbers correspond to volumes inherently linked to local-global principles, and the ones studied in this thesis have direct applications for evaluations of a mass formula for isogeny classes of principally polarized abelian varieties over finite fields. The polarization yields a symplectic structure on their ℓ-adic cohomology groups, and the torus studied corresponds to the centralizer of the Frobenius element of the abelian variety in the similitude group associated with the polarization. We present many theoretical results, as well as algorithms that were developed to study the the cohomology of any algebraic torus and helped establish the right conjectures leading to the aforementioned results.We also present work and partial results on descent-type problems for orbital integrals, aiming to compare masses of abelian varieties with distinguished type. The work presented aims to establish the proportion of products of elliptic curves within the isogeny class of such a product. We also provide extensive background material, reviewing the most important concepts and results used in the thesis, as well as references.

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In 1991, James Arthur published a local trace formula ([Art91, Theorem 12.2]), which is an equality of distributions on the Lie algebra of a connected, reductive algebraic group G over a field F of characteristic zero. His approach was later used by Jean-Loup Waldspurger to give a slight reformulation, identifying the value of a particular distribution on a test function with that of its Fourier transform ([Wal95, Théorème V.2]). We show that this identity may be formulated as an identity of motivic distributions on definable manifolds. By so doing, we would make available the use of the transfer principle to establish the trace formula for groups defined over fields of positive characteristic.

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Let k be a non-Archimedean local field, and Cc∞(GLn) the space of locally constant compactly supported complex-valued functions on the general linear group GLn over k. For every irreducible representation (?,V) of GLn, the space Hom(Cc∞(GLn), V ⊗ Ṽ) is one-dimensional. This space is generated by an element denoted by ?, which can be thought of as an integral against matrix coefficients. In this thesis, we are interested in the so-called "extension problem" of ?. More explicitly, for 0≤m≤n, GLn can be embedded into the space R≥m of all n×n matrices over k of rank at least m. If ? lies in the image of the induced map of this embedding, then we say that ? can be extended to rank at least m. For m=0, the extension problem of ? to rank at least 0 has been completely answered in Tate's thesis for n=1, and by Moeglin, Vignéras, Waldspurger, and Minguez for general n. Our goal is to determine the least value m for a given representation such that ? can be extended to rank at least m. A representation is said to appear in rank m if Hom(Cc∞(R≥m), V ⊗ Ṽ) is non-trivial. It is natural to conjecture that ? extends to rank at least m+1 but does not extend to rank at least m where m is the highest rank less than n that ? appears in. In this thesis, this conjecture is proved for spherical representations, by means of extending Satake transform to the space of K-bi-invariant functions on Mn and obtaining a partial description of the image of the rank filtration under this extended Satake transform.Some explicit computations for spherical representations of GL₃ are included as motivating examples of the general case.There are also some suggestive calculations for non-spherical representations of GL₂.

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In this dissertation, we combine the work of A. Aizenbud and D. Gourevitch on Schwartz functions on Nash manifolds, and the work of F. du Cloux on Schwartz inductions, to develop a toolbox of Schwartz analysis. We then use these tools to study the intertwining operators between parabolic inductions, and study the behavior of intertwining distributions on certain open subsets. Finally we use our results to give new proof of results in Bruhat's thesis on irreducibilities of degenerate principal series and minimal principal series.

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