Explicit Class Field Theory for Global Function Fields

Introduction to Drinfeld Modules

(1) Explicit class field theory for global function fields (just as torsion of Gm gives abelian extensions of Q, and torsion of CM elliptic curves gives abelian extension of imaginary quadratic fields). Here global function field means Fp(T ) or a finite extension. (2) Langlands conjectures for GLn over function fields (Drinfeld modular varieties play the role of Shimura varieties). (3) Modular...

Applications of the Class Field Theory of Global Fields

Class field theory of global fields provides a description of finite abelian extensions of number fields and of function fields of transcendence degree 1 over finite fields. After a brief review of the handling of both function and number fields in Magma, we give a introduction to computational class field theory focusing on applications: We show how to construct tables of small degree extensio...

Adelic Approach to the Zeta Function of Arithmetic Schemes in Dimension Two

This paper suggests a new approach to the study of the fundamental properties of the zeta function of a model of elliptic curve over a global field. This complex valued commutative approach is a two-dimensional extension of the classical adelic analysis of Tate and Iwasawa. We explain how using structures which come naturally from the explicit two-dimensional class field theory and working with...

Explicit class field theory in function fields: Gross-Stark units and Drinfeld modules

Globalization and Employment

In this paper we study the compatibility of Cohen-Lenstra heuristics with Leopoldt’s Spiegelungssatz (the reflection theorem). We generalize Dutarte’s ([1983, in “Théorie des nombres, Besançon, 1983-1984”]) work to every prime number p : he proved the compatibility of the Cohen-Lenstra conjectures with the Spiegelungssatz in the case p = 3. We also show that the Spiegelungssatz is compatible wi...