Chow groups of ind-schemes and extensions of Saito's filtration

Let $K$ be a field of characteristic zero and let $Sm/K$ be the category of smooth and separated schemes over $K$. For an ind-scheme $\mathcal X$ (and more generally for any presheaf of sets on $Sm/K$), we define its Chow groups $\{CH^p(\mathcal X)\}_{p\in \mathbb Z}$. We also introduce Chow groups $\{\mathcal{CH}^p(\mathcal G)\}_{p\in \mathbb Z}$ for a presheaf with transfers $\mathcal G$ on $Sm/K$. Then, we show that we have natural isomorphisms of Chow groups $$CH^p(\mathcal X)\cong \mathcal{CH}^p(Cor(\mathcal X))\qquad\forall\textrm{ }p \in \mathbb Z$$ where $Cor(\mathcal X)$ is the presheaf with transfers that associates to any $Y\in Sm/K$ the collection of finite correspondences from $Y$ to $\mathcal X$. Additionally, when $K=\mathbb C$, we show that Saito's filtration on the Chow groups of a smooth projective scheme can be extended to the Chow groups $CH^p(\mathcal X)$ and more generally, to the Chow groups of an arbitrary presheaf of sets on $Sm/\mathbb C$. Similarly, there exists an extension of Saito's filtration to the Chow groups of a presheaf with transfers on $Sm/\mathbb C$. Finally, when the ind-scheme $\mathcal X$ is ind-proper, we show that the isomorphism $CH^p(\mathcal X)\cong \mathcal{CH}^p(Cor(\mathcal X))$ is actually a filtered isomorphism.Comment: Exposition improve

L-functions for holomorphic forms on GSp(4) x GL(2) and their special values

We provide an explicit integral representation for L-functions of pairs (F,g) where F is a holomorphic genus 2 Siegel newform and g a holomorphic elliptic newform, both of squarefree levels and of equal weights. When F,g have level one, this was earlier known by the work of Furusawa. The extension is not straightforward. Our methods involve precise double-coset and volume computations as well as an explicit formula for the Bessel model for GSp(4) in the Steinberg case; the latter is possibly of independent interest. We apply our integral representation to prove an algebraicity result for a critical special value of L(s, F \times g). This is in the spirit of known results on critical values of triple product L-functions, also of degree 8, though there are significant differences.Comment: 48 pages, typos corrected, some changes in Sections 6 and 7, other minor change

Absolute convergence of the twisted Arthur-Selberg trace formula

We show that the distributions occurring in the geometric and spectral side of the twisted Arthur-Selberg trace formula extend to non-compactly supported test functions. The geometric assertion is modulo a hypothesis on root systems proven when the group is split. The result extends the work of Finis-Lapid (and M\"uller, spectral side) to the twisted setting. We use the absolute convergence to give a geometric interpretation of sums of residues of certain Rankin-Selberg L-functions.Comment: Accepted to be published in Mathematische Zeitschrift. Removed proof of RCL for base change; Section 8 now requires Assumption 8.1. Also, minor correction

Heat conduction in a one-dimensional gas of elastically colliding particles of unequal masses

We study the nonequlibrium state of heat conduction in a one-dimensional system of hard point particles of unequal masses interacting through elastic collisions. A BBGKY-type formulation is presented and some exact results are obtained from it. Extensive numerical simulations for the two-mass problem indicate that even for arbitrarily small mass differences, a nontrivial steady state is obtained. This state exhibits local thermal equilibrium and has a temperature profile in accordance with the predictions of kinetic theory. The temperature jumps typically seen in such studies are shown to be finite-size effects. The thermal conductivity appears to have a very slow divergence with system size, different from that seen in most other systems.Comment: 5 pages, 4 figures, Accepted for publication in Phys. Rev. Let

Non-extensive Statistical Mechanics and Black Hole Entropy From Quantum Geometry

Using non-extensive statistical mechanics, the Bekenstein-Hawking area law is obtained from microstates of black holes in loop quantum gravity, for arbitrary real positive values of the Barbero-Immirzi parameter$(\gamma)$. The arbitrariness of $\gamma$ is encoded in the strength of the "bias" created in the horizon microstates through the coupling with the quantum geometric fields exterior to the horizon. An experimental determination of $\gamma$ will fix this coupling, leaving out the macroscopic area of the black hole to be the only free quantity of the theory.Comment: 6 pages, published versio

Classifying subcategories and the spectrum of a locally noetherian category

Let $\mathcal A$ be a locally noetherian Grothendieck category. In this paper, we study subcategories of $\mathcal A$ using subsets of the spectrum $\mathfrak Spec(\mathcal A)$. Along the way, we also develop results in local algebra with respect to the category $\mathcal A$ that we believe to be of independent interest.Comment: 40 pages, some new results adde