In [Iha86b], Ihara constructs a universal cocycle Gal ( Q/Q ) −→ Zp[[t0, t1, t∞]]/ ((t0 + 1)(t1 + 1)(t∞ + 1)− 1) arising from the action of Gal ( Q/Q ) on certain quotients of the Jacobians of the Fermat curves

In this paper, we show how to determine several properties of a finite graph G from its Ihara zeta function ZG(u). If G is connected and has minimal degree at least 2, we show how to calculate the number of vertices of G. To do so we use a result of Bass, and in the case that G is nonbipartite, we give an elementary proof of Bass’ result. We further show how to determine whether G is regular, a...

To a polynomial f over a p{adic eld K and a character of the group of units of the valuation ring of K one associates Igusa's local zeta function Z(s; f;), which is a meromorphic function on C. Several theorems and conjectures relate the poles of Z(s; f;) to the monodromy of f; the so{called holomorphy conjecture states roughly that if the order of does not divide the order of any eigenvalue of...

In this manuscript we show that function fields K|k with td(K|k) > dim(k)+1 can be recovered from their maximal pro-` abelian-by-central Galois theory, where dim(k) is the Kronecker dimension. This extends the main result of Pop [P5] beyond the case where k is an algebraic closure of a finite field. We also show how this implies the pro-` form of Ihara’s question / Oda-Matsumoto conjecture I/OM...

For K a field of characteristic zero, let M(K) denote the set of Drinfel’d associators defined over K [Dr]. The variety M is of great interest because of its connection to the deformations of universal enveloping algebras, fundamental group of the Teichmuller tower, and to Gal(Q/Q). There is a natural map M → A and for λ ∈ K, and Mλ is a torsor under GRT1 (= M0). If K << X,Y >> denotes the form...

In [16, 17], Pajitnov considers the closed orbit structure of generic gradient flows of circle-valued Morse functions. It turns out that the torsion of a chain homotopy equivalence between the Novikov complex and the completed simplicial chain complex of the universal cover detects the eta function of the flow. This eta function counts the closed orbits and reduces to the logarithm of the zeta ...

Call diffeomorphisms satisfying (2) Artin-Mazur (A-M) diffeomorphisms. In [I.1] we give an elementary proof of an extension of Artin–Mazur’s result, namely, we prove that A-M diffeomorphisms with only hyperbolic periodic points are dense in Diffr(M). In [I.2] we construct Baire generic diffeomorphisms inside of an open set (a Newhouse domain) in Diffr(M) of with superexponential (even an arbitr...

Abstract. Let K be a p−adic field, and ZΦ(s, f), s ∈ C, with Re(s) > 0, the Igusa local zeta function associated to f(x) = (f1(x), .., fl(x)) ∈ [K (x1, .., xn)] , and Φ a Schwartz-Bruhat function. The aim of this paper is to describe explicitly the poles of the meromorphic continuation of ZΦ(s, f). Using resolution of singularities is possible to express ZΦ(s, f) as a finite sum of p−adic monom...

We consider the problem of fast zeta and Möbius transforms in finite posets, particularly in lattices. It has previously been shown that for a certain family of lattices, zeta and Möbius transforms can be computed in O(e) elementary arithmetic operations, where e denotes the size of the covering relation. We show that this family is exactly that of geometric lattices. We also extend the algorit...

The T cell receptor complex (TCR) zeta chain is constitutively tyrosine phosphorylated specifically at two of the six zeta immunoreceptor tyrosine-based activation motif (ITAM) tyrosine residues in resting peripheral T cells. Further phosphorylation of zeta is induced by both agonist and antagonist ligands of the TCR, with agonists inducing complete phosphorylation of the zeta ITAM tyrosines. A...