tailieunhanh - Đề tài " Elliptic units for real quadratic fields "

Elliptic units, which are obtained by evaluating modular units at quadratic imaginary arguments of the Poincar´e upper half-plane, provide us with a rich source of arithmetic questions and insights. They allow the analytic construction of abelian extensions of imaginary quadratic fields, encode special values of zeta functions through the Kronecker limit formula, and are a prototype for Stark’s conjectural construction of units in abelian extensions of number fields. Elliptic units have also played a key role in the study of elliptic curves with complex multiplication through the work of Coates and Wiles | Annals of Mathematics Elliptic units for real quadratic fields By Henri Darmon and Samit Dasgupta Annals of Mathematics 163 2006 301 346 Elliptic units for real quadratic fields By Henri Darmon and Samit Dasgupta Contents 1. A review of the classical setting 2. Elliptic units for real quadratic fields . p-adic measures . Double integrals . Splitting a two-cocycle . The main conjecture . Modular symbols and Dedekind sums . Measures and the Bruhat-Tits tree . Indefinite integrals . The action of complex conjugation and of Up 3. Special values of zeta functions . The zeta function . Values at negative integers . The p-adic valuation . The Brumer-Stark conjecture . Connection with the Gross-Stark conjecture 4. A Kronecker limit formula . Measures associated to Eisenstein series . Construction of the p-adic L-function . An explicit splitting of a two-cocycle . Generalized Dedekind sums . Measures on Zp X Zp . A partial modular symbol of measures on Zp X Zp . From Zp X Zp to X . The measures p and T-invariance Introduction Elliptic units which are obtained by evaluating modular units at quadratic imaginary arguments of the Poincare upper half-plane provide us with a rich source of arithmetic questions and insights. They allow the analytic construction of abelian extensions of imaginary quadratic fields encode special values 302 HENRI DARMON AND SAMIT DASGUPTA of zeta functions through the Kronecker limit formula and are a prototype for Stark s conjectural construction of units in abelian extensions of number fields. Elliptic units have also played a key role in the study of elliptic curves with complex multiplication through the work of Coates and Wiles. This article is motivated by the desire to transpose the theory of elliptic units to the context of real quadratic fields. The classical construction of elliptic units does not give units in abelian extensions of such Naively one could try to .