TY - JOUR
T1 - Geometric non-vanishing
AU - Ulmer, Douglas
PY - 2005/1
Y1 - 2005/1
N2 - 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 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 of L-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 whose L-function vanishes to order at most 1 from a suitable Gross-Zagier formula.
AB - 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 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 of L-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 whose L-function vanishes to order at most 1 from a suitable Gross-Zagier formula.
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U2 - 10.1007/s00222-004-0386-z
DO - 10.1007/s00222-004-0386-z
M3 - Article
AN - SCOPUS:12144264069
VL - 159
SP - 133
EP - 186
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
SN - 0020-9910
IS - 1
ER -