Derivatives of L-functions

TL;DR

This paper explores the derivatives of various L-functions, including the Riemann zeta and Artin L-functions, and surveys their connections to the geometry and arithmetic of Shimura varieties.

Contribution

It provides a comprehensive survey of the relationships between derivatives of L-functions and the geometry and arithmetic of Shimura varieties.

Findings

Identifies key relations between L-function derivatives and Shimura varieties

Summarizes known results on derivatives of specific L-functions

Highlights open problems in the field

Abstract

In this paper, we investigate the derivatives of L-functions, in particular, the Riemann zeta function, the Hasse-Weil L-function, the Rankin L-function and the Artin L-function, and survey the relations between the derivatives of L-functions and the geometry and arithmetic of the associated Shimura varieties.