The isomorphism conjecture for constant depth reductions

For any class C closed under TC0 reductions, and for any measure u of uniformity containing Dlogtime, it is shown that all sets complete for C under u-uniform AC0 reductions are isomorphic under u-uniform AC0-computable isomorphisms

On the strictness of the quantifier structure hierarchy in first-order logic

We study a natural hierarchy in first-order logic, namely the quantifier structure hierarchy, which gives a systematic classification of first-order formulas based on structural quantifier resource. We define a variant of Ehrenfeucht-Fraisse games that characterizes quantifier classes and use it to prove that this hierarchy is strict over finite structures, using strategy compositions. Moreover, we prove that this hierarchy is strict even over ordered finite structures, which is interesting in the context of descriptive complexity.Comment: 38 pages, 8 figure

Matching of orbital integrals (transfer) and Roche Hecke algebra isomorphisms

Let $F$ be a non-Archimedan local field, $G$ a connected reductive group defined and split over $F$, and $T$ a maximal $F$-split torus in $G$. Let $\chi_0$ be a depth zero character of the maximal compact subgroup $\mathcal{T}$ of $T(F)$. It gives by inflation a character $\rho$ of an Iwahori subgroup $\mathcal{I}$ of $G(F)$ containing $\mathcal{T}$. From Roche, $\chi_0$ defines a split endoscopic group $G'$ of $G$, and there is an injective morphism of ${\Bbb C}$-algebras $\mathcal{H}(G(F),\rho) \rightarrow \mathcal{H}(G'(F),1_{\mathcal{I}'})$ where $\mathcal{H}(G(F),\rho)$ is the Hecke algebra of compactly supported $\rho^{-1}$-spherical functions on $G(F)$ and $\mathcal{I}'$ is an Iwahori subgroup of $G'(F)$. This morphism restricts to an injective morphism $\zeta: \mathcal{Z}(G(F),\rho)\rightarrow \mathcal{Z}(G'(F),1_{\mathcal{I}'})$ between the centers of the Hecke algebras. We prove here that a certain linear combination of morphisms analogous to $\zeta$ realizes the transfer (matching of strongly $G$-regular semisimple orbital integrals). If ${\rm char}(F)=p>0$, our result is unconditional only if $p$ is large enough.Comment: 82 page