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Notation: Despite some misgivings, we introduce the following notation. For a global field K, we let ΣK denote the set of all places of K, i.e., equivalence classes of nontrivial norms on K. (Recall that at this point we have associated a canonical normalized Artin absolute value to each place in K.) Further we write Σ K for the subset of non-Archimeden places of K, and let Σ K denote the subse...

We show that, for all characteristic p global fields k and natural numbers n coprime to the order of the non–p–part of the Picard group Pic0(k) of k, there exists an abelian extension L/k whose local degree at every prime of k is equal to n. This answers in the affirmative in this context a question recently posed by Kisilevsky and Sonn. As a consequence, we show that, for all n and k as above,...

We compute the anomalous divergence of currents associated with global transformations in the antifield formalism, by introducing compensating fields that gauge these transformations. We consider the explicit case of the global axial current in QCD but the method applies to any global transformation of the fields. PACS: 03.70.+k, 11.10.Ef, 11.15.-q [email protected]; [email protected]; henneaux@...

We prove a strong form of the Brumer–Stark Conjecture and, as a consequence, a strong form of Rubin’s integral refinement of the abelian Stark Conjecture, for a large class of abelian extensions of an arbitrary characteristic p global field k. This class includes all the abelian extensions K/k contained in the compositum kp∞ := kp · k∞ of the maximal pro-p abelian extension kp/k and the maximal...

Let K/k be a finite extension of a global field. Such an extension can be generated over k by a single element. The aim of this article is to prove the existence of a ”small“ generator in the function field case. This answers the function field version of a question of Ruppert on small generators of number fields.

Fix a number field k with its adele ring A. Let G = O(n+3) be an orthogonal group of k-rank 1 and H = O(n + 2) a k-anisotropic subgroup. We have previously described how to factor the global period