Geometric non-vanishing

Details Geometric non-vanishing Geometric non-vanishing

2003-05-22

arxiv.org/abs/math/0305321v2

Mathematics

Number Theory

46 pages. New version corrects minor errors. To appear in Inventiones Math

Scientific paper

10.1007/s00222-004-0386-z

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.

Affiliated with

Also associated with

No associations

Rating

Geometric non-vanishing does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Geometric non-vanishing, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Geometric non-vanishing will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-133572