By Yuval Z Flicker

Drinfeld Moduli Schemes and Automorphic types: the speculation of Elliptic Modules with functions is predicated at the author’s unique paintings setting up the correspondence among ell-adic rank r Galois representations and automorphic representations of GL(r) over a functionality box, within the neighborhood case, and, within the worldwide case, below a restrict at a unmarried position. It develops Drinfeld’s concept of elliptic modules, their moduli schemes and overlaying schemes, the easy hint formulation, the mounted element formulation, in addition to the congruence family members and a "simple" speak theorem, now not but released anyplace. This model, in accordance with a up to date direction taught by means of the writer on the Ohio nation college, is up to date with references to analyze that has prolonged and built the unique paintings. using the speculation of elliptic modules within the current paintings makes it available to graduate scholars, and it'll function a beneficial source to facilitate an front to this interesting sector of mathematics.

Table of Contents

Cover

Drinfeld Moduli Schemes and Automorphic varieties - the speculation of Elliptic Modules with Applications

ISBN 9781461458876 ISBN 9781461458883

Contents

1. Introduction

Part 1. Elliptic Moduli

2. Elliptic Modules: Analytic Definition

three. Elliptic Modules: Algebraic Definition

four. Elliptic Modules: Geometric Definition

five. masking Schemes

Part 2. Hecke Correspondences

6. Deligne's Conjecture and Congruence Relations

Part three. hint Formulae

7. Isogeny Classes

eight. Counting Points

nine. round Functions

Part four. greater Reciprocity Laws

10. Purity Theorem

eleven. lifestyles Theorem

12. Representations of a Weil Group

thirteen. basic communicate Theorem

References

Index

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**Extra info for Drinfeld Moduli Schemes and Automorphic Forms: The Theory of Elliptic Modules with Applications**

**Sample text**

Hence ϕ is a standard elliptic module. If P is an endomorphism of ϕ and Spec B is connected, then it follows from (ii) that there is a nonnegative integer s such that P is strictly of degree s. If P is invertible then s = 0 and P is linear. The proposition follows. Let I = 0 be an ideal in A. Let E = Eϕ be an elliptic functor of rank r over S. 3. Let EI = Ann I|E be the subfunctor of E annihilated by I. 5. The annihilator EI of I in E is ﬁnite and ﬂat as an aﬃne group scheme over S. Proof. Up to isomorphism we may assume that E = Eϕ is a standard elliptic functor.

1. Let K be a ﬁeld of characteristic p > 0. Let n be a positive integer with (p, n) = 1. Let X = Y be the aﬃne group scheme Gm,K = Spec K[x, x−1 ]. Then the morphism Y → X deﬁned by x → xn is ´etale, since (n, p) = 1, and we obtain a locally constant sheaf YX on X´et . Z. 1007/978-1-4614-5888-3 6, © Yuval Z. Flicker 2013 45 46 YUVAL Z. FLICKER group of nth roots of unity in the multiplicative group K × of K. We claim that the restriction of YX to the ´etale covering {Z → X} = {Y → X} of X is the constant sheaf deﬁned by N .

8, p. 358) it follows that one has the canonical isomorphisms (Ri b! L)s = Hci (Xs , L|Xs )and(Ri b! L)v = Hci (Xv , L|Xv ). 2. 7. Let f, h : X → X be ﬁnite morphisms, and b : X → S a morphism where S is irreducible. Let L be a smooth sheaf on X, and ﬁx a morphism α : h∗ L → f ! L.