2003/05/31 by Douglas Ulmer
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Conjecture #Elliptic curve #Field (mathematics) #Finite field #Function field #Galois module #Interpretation (philosophy) #Monodromy #Order (exchange) #math.AG #math.NT #msc:11G05 #msc:11G40 #msc:14G10
paper · pdf · doi:10.1007/s00222-004-0386-z
46 pages. New version corrects minor errors. To appear in Inventiones Math
arxiv created 2004/05/13 · openalex publication_date 2004/09/02 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We consider L-functions attached to representations of the Galois group of the function field of a curve over a finite field. Under mild tameness hypotheses, we prove non-vanishing results for twists of these L-functions by characters of order prime to the characteristic of the ground field and by certain representations with solvable image. We also allow local restrictions on the twisting representation at finitely many places. Our methods are geometric, and include the Riemann-Roch theorem, the cohomological interpretation of L-functions, and some monodromy calculations of Katz. As an application, we prove a result which allows one to deduce the conjecture of Birch and Swinnerton-Dyer for non-isotrivial elliptic curves over function fields whose L-function vanishes to order at most 1 from a suitable Gross-Zagier formula.