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Ideal class group annihilators. (English) Zbl 1180.11018
Stickelberger-type theorems exist in many different situations, for example for class groups of Galois extensions of global number or function fields, or for Picard or Chow groups of certain arithmetic schemes. Their general shape is as follows: Some abelian group (e.g. the class group of the field \({\mathbb Q}( \mu_n)\) of \(n\)-th roots of unity), which by construction comes with an action of some group \(G\) (here: \(G = \mathrm{Gal}(\mathbb Q(\mu_n)| \mathbb Q) = \mathbb Z/(n)^*)\) is annihilated by a certain ideal \(St\) of the group ring \(\mathbb Z[G]\), whose definition uses “geometric” information about the group action in question (here: the Galois action on Gauss sums and their prime factorization).
In the paper it is shown (Theorem 1) that a certain correspondence, derived from the Hecke correspondences, on the moduli scheme \(\mathcal{E}_n^{I\infty}\) of “elliptic sheaves of rank \(n\) with structure of level \(I\infty\)” is trivial, which in the context means, rationally equivalent to zero. Definitions and details are too technical to be reproduced here. The proof relies on a detailed study of isogenies of elliptic sheaves.
MSC:
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11G20 Curves over finite and local fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F52 Modular forms associated to Drinfel’d modules
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