Slopes and signatures of links

We define the slope of a colored link in an integral homology sphere, associated to admissible characters on the link group. Away from a certain singular locus, the slope is a rational function which can be regarded as a multivariate generalization of the Kojima--Yamasaki $\eta$-function. It is the ratio of two Conway potentials, provided that the latter makes sense; otherwise, it is a new invariant. The slope is responsible for an extra correction term in the signature formula for the splice of two links, in the previously open exceptional case where both characters are admissible. Using a similar construction for a special class of tangles, we formulate generalized skein relations for the signature

An Explicit CM Type Norm Formula and Effective Nonvanishing of Class Group L-functions for CM Fields

We show that the central value of class group L-functions of CM fields can be expressed in terms of derivatives of real-analytic Hilbert Eisenstein series at CM points. Then, following an idea of Iwaniec and Kowalski we obtain a conditional explicit lower bound of class numbers of CM fields under a weaker assumption. Some results in the proof lead to an effective nonvanishing result for class group L-functions of general CM fields, generalizing the only known ineffective results.Comment: Some typos are corrected. To appear in the Pacific Journal of Mat

Nonvanishing of quadratic Dirichlet L-functions at s=1/2

We show that for a positive proportion of fundamental discriminants d, L(1/2,chi_d) != 0. Here chi_d is the primitive quadratic Dirichlet character of conductor d.Comment: 42 pages, published versio

Simultaneous nonvanishing of Dirichlet LL-functions and twists of Hecke-Maass L-functions

We prove that given a Hecke-Maass form $f$ for $\text{SL}(2, \mathbb{Z})$ and a sufficiently large prime $q$, there exists a primitive Dirichlet character $\chi$ of conductor $q$ such that the $L$-values $L(\tfrac{1}{2}, f \otimes \chi)$ and $L(\tfrac{1}{2}, \chi)$ do not vanish. We expect the same method to work for any large integer $q$