Nonvanishing of Hecke L-functions and the Bloch–Kato conjecture

Abstract

In this paper we study the central values of L-functions associated to a large class of algebraic Hecke characters of imaginary quadratic fields. When these central values are nonzero, the Bloch-Kato conjecture predicts an exact formula for the algebraic parts of the central values in terms of periods and arithmetic data, most notably the Selmer groups corresponding to the Hecke characters. We investigate the nonvanishing of these central values, and prove the p-part of the Bloch-Kato conjecture in these cases for primes p which split in K .

Diagonal hypersurfaces and the Bloch-Kato conjecture, I

“…In that work, the underlying variety was the product of a number of elliptic curves of the form Y 2 = X 3 − AX; but the resulting motivic L-series are the same as those in the present paper. Subsequent related work on such products can be found in [6] and in other papers cited there. I am indebted to the referee for this reference.)…”

Nonvanishing of Hecke L-Functions for CM Fields and Ranks of Abelian Varieties

“…[Roh1,2], [MoR], [RoVY], [Y2], [MilY], [M1,2], [KMY]). In [KMY,Th. 1.1], the authors and Kim established an asymptotic formula for the average of the central values L(χ μ , k + 1) using an explicit formula for the central value due to the second author [Y2], a spectral regularization, and the equidistribution of Heegner points on modular curves (see also [T]).…”

“…1.1], the authors and Kim established an asymptotic formula for the average of the central values L(χ μ , k + 1) using an explicit formula for the central value due to the second author [Y2], a spectral regularization, and the equidistribution of Heegner points on modular curves (see also [T]). By combining the asymptotic formula with a subconvexity bound for L(χ μ , k + 1) due to Duke, Friedlander, and Iwaniec, the authors and Kim obtained a quantitative nonvanishing theorem for the central values (see [KMY,Th. 1.4]).…”

“…This allows us to apply the equidistribution theorem of Michel and Venkatesh directly to F μ,k . In section 8 we use representation theory to give a criterion for the cuspidality of θ μ,k (see also [KMY,Prop. 4.3]).…”

“…4.3]). In order to remove the cuspidality assumption, we would need to proceed as in the proof of [KMY,Th. 1.1], where we used a delicate spectral regularization to establish a version of equidistribution of Heegner points on modular curves for test functions which grow moderately in the cusps (see also [FM1,2]).…”