2010/05/05 by Cornélius Greither, Cornelius Greither, Greither, Cornelius +2 · 4 citations
Arts and Humanities · Mathematics · Social Sciences · #Abelian group #Algebra over a field #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Cover (algebra) #Discrete mathematics #FOS: Mathematics #Field (mathematics) #Function field #Galois group #Galois module #Group (periodic table) #Historical Studies and Socio-cultural Analysis #Mathematics #Number Theory (math.NT) #Picard group #Pure mathematics #Realization (probability) #Vietnamese History and Culture Studies #math.AG #math.NT
paper · pdf · doi:10.48550/arxiv.1005.0661
published in arXiv (Cornell University) (Cornell University) · 32 pages
arxiv created 2010/05/05 · openalex publication_date 2010/05/05 · arxiv updated 2010/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We show that the l-adic realizations of certain Picard 1-motives associated to a G-Galois cover of smooth, projective curves defined over an algebraically closed field are G-cohomologically trivial, for all primes l. In the process, we generalize a well-known theorem of Nakajima on the Galois module structure of certain spaces of Kahler differentials associated to the top curve of the cover, assuming that the field of definition is of characteristic p. If the cover and the Picard 1-motive are defined over a finite field and if G is abelian, we compute the first Fitting ideal of each l-adic realization over a certain profinite l-adic group algebra in terms of an equivariant Artin L-function. This is a refinement of earlier work of Deligne and Tate on Picard 1-motives associated to global function fields. As a consequence, we prove refined versions of the Brumer-Stark and the (étale) Coates-Sinnott Conjectures for global function fields. In upcoming work, we use these results to prove several other conjectures on special values of global L-functions in characteristic p. Also, we prove analogous results in characteristic 0, where the l-adic realizations of Picard 1-motives will be replaced by a new class of Iwasawa modules.