Epsilon factors as algebraic characters on the smooth dual of GLn\mathrm{GL}_n

Let $K$ be a non-archimedean local field and let $G = \mathrm{GL}_n(K)$. We have shown in previous work that the smooth dual $\mathbf{Irr}(G)$ admits a complex structure: in this article we show how the epsilon factors interface with this complex structure. The epsilon factors, up to a constant term, factor as invariant characters through the corresponding complex tori. For the arithmetically unramified smooth dual of $\mathrm{GL}_n$, we provide explicit formulas for the invariant characters.Comment: 12 pages. Minor improvements, new titl

Base change and K-theory for GL(n,R)

We investigate base change $C/R$ at the level of $K$-theory for the general linear group $GL(n,R)$. In the course of this study, we compute in detail the $C*$-algebra $K$-theory of this disconnected group. We investigate the interaction of base change with the Baum-Connes correspondence for $GL(n,R)$ and $GL(n,C)$. This article is the archimedean companion of our previous article in the Journal of Noncommutative Geometry.Comment: 17 pages, introduction and section 5 completely rewritte

L-packets and depth for SL_2(K) with K a local function field of characteristic 2

Let G = SL_2(K) with K a local function field of characteristic 2. We review Artin-Schreier theory for the field K, and show that this leads to a parametrization of certain L-packets in the smooth dual of G. We relate this to a recent geometric conjecture. The L-packets in the principal series are parametrized by quadratic extensions, and the supercuspidal L-packets of cardinality 4 are parametrized by biquadratic extensions. Each supercuspidal packet of cardinality 4 is accompanied by a singleton packet for SL_1(D). We compute the depths of the irreducible constituents of all these L-packets for SL_2(K) and its inner form SL_1(D).Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:1302.603

K-theory and the connection index

Let G denote a split simply connected almost simple p-adic group. The classical example is the special linear group SL(n). We study the K-theory of the unramified unitary principal series of G and prove that the rank of K_0 is the connection index f(G). We relate this result to a recent refinement of the Baum-Connes conjecture, and show explicitly how generators of K_0 contribute to the K-theory of the Iwahori C*-algebra I(G).Comment: 11 page

Base change and K-theory for GL(n)

Let F be a nonarchimedean local field and let G = GL(n) = GL(n,F). Let E/F be a finite Galois extension. We investigate base change E/F at two levels: at the level of algebraic varieties, and at the level of K-theory. We put special emphasis on the representations with Iwahori fixed vectors, and the tempered spectrum of GL(1) and GL(2). In this context, the prominent arithmetic invariant is the residue degree f(E/F).Comment: 20 pages. Completely rewritten, much more concis

A new bound for the smallest xx with π(x)>li(x)\pi(x) > li(x)

We reduce the leading term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays and Hudson. Entering $2,000,000$ Riemann zeros, we prove that there exists $x$ in the interval $[exp(727.951858), exp(727.952178)]$ for which \pi(x)-\li(x) > 3.2 \times 10^{151}. There are at least $10^{154}$ successive integers $x$ in this interval for which \pi(x)>\li(x). This interval is strictly a sub-interval of the interval in Bays and Hudson, and is narrower by a factor of about 12.Comment: Final version, to be published in the International Journal of Number Theory [copyright World Scientific Publishing Company][www.worldscinet.com/ijnt