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- Title
Geometric non-vanishing.
- Authors
Ulmer, Douglas
- Abstract
We considerL-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 theseL-functions by characters of order prime to the characteristic of the ground field and more generally 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 ofL-functions, and 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 whoseL-function vanishes to order at most 1 from a suitable Gross-Zagier formula.
- Subjects
L-functions; RIEMANN-Roch theorems; FINITE fields; ELLIPTIC curves; NUMBER theory; ALGEBRAIC functions
- Publication
Inventiones Mathematicae, 2005, Vol 159, Issue 1, p133
- ISSN
0020-9910
- Publication type
Article
- DOI
10.1007/s00222-004-0386-z