Representations of Higher Adelic Groups and Arithmetic

Abstract

What do we mean by local ? To get an answer to this question let us start from the following two problems.First problem is from number theory. When does the diophantine equationhave a non-trivial solution in rational numbers ? In order to solve the problem, let us consider the quadratic norm residue symbol (−, −) p where p runs through all primes p and also ∞. This symbol is a bi-multiplicative map (−, −) p : Q * × Q * → {±1} and it is easily computed in terms of the Legendre symbol. Then, a non-trivial solution exists if and only if, for any p, (a, b) p = 1. However, these conditions are not independent:This is essentially the Gauss reciprocity law in the Hilbert form. The "points" p correspond to all possible completions of the field Q of rational numbers, namely to the p-adic fields Q p and the field R of real numbers. One can show that the equation f = 0 has a non-trivial solution in Q p if and only if (a, b) p = 1.The second problem comes from complex analysis. Let X be a compact Riemann surface (= complete smooth algebraic curve defined over C). For a point P ∈ X, denote by K P = C((t P )) the field of Laurent formal power series in a local coordinate t P at the point P . The field K P contains the ring O P = C[[t P ]] of Taylor formal power series. These have an invariant meaning and are called the local field and the local ring at P respectively. Let us now fix finitely many points P 1 , . . . , P n ∈ X and assign to every P in X some elements f P such that f P 1 ∈ K P 1 , . . . , f Pn ∈ K Pn and f P = 0 for all other points. *

Arithmetic surfaces and adelic quotient groups

“…From this diagram and formula (14) it follows that we prove equality (18) if we prove that the group Im α contains a subgroup…”

“…The ultimate goal of the higher adeles program is the generalization of the Tate-Iwasawa method from one-dimensional case to the case of higher dimensions, see [13,14]. The Tate-Iwasawa method allows to obtain a meromorphic continuation to all of C and a functional equation for zeta-and L -functions of number fields and the fields of rational functions of curves defined over finite fields, and this method works simultaneously in the number theory case and in the geometric case, see [17].…”

Harmonic analysis on the rank- value group of a two-dimensional local field

“…The report [11] raised the question (see also work [10]) about the extension of the Tate-Iwasawa method to the case of two-dimensional schemes of finite type over Spec Z using the theory of higher adeles and harmonic analysis constructed in works [6,7].…”

“…With these groups one connects subgroups of the multiplicative group A * of invertible elements. In paper [11] the discrete adele groups, in particular various groups of divisors, were constructed in this situation. The question on construction of harmonic analysis on these groups (such that this analysis is based on already constructed harmonic analysis from papers [6] and [7]) by analogy with the case of curves in [12] was not considered at that time.…”

“…The second part is much more complicated, but, nevertheless, it can be assumed that the main tool for establishing such a connection should be adele Heisenberg groups on an algebraic surface, which were also introduced in [11]. In this paper we consider only the first part and only the local discrete groups associated with fixed two-dimensional local field of an algebraic surface.…”