Abstract
Period polynomials have long been fruitful tools for the study of values of L-functions in the context of major outstanding conjectures. In this paper, we survey some facets of this study from the perspective of Eichler cohomology. We discuss ways to incorporate non-cuspidal modular forms and values of derivatives of L-functions into the same framework. We further review investigations of the location of zeros of the period polynomial as well as of its analogue for L-derivatives.
Highlights
The period polynomial provides a way of encoding critical values of L-functions associated with modular cusp forms that has proven very successful in the uncovering of important arithmetic properties of L-values
The example we will more closely be reviewing here is an analogue of the period polynomial associated with derivatives of L-functions
A way to define the period polynomial associated with f is as a polynomial in z of degree ≤ k − 2 given by rf (z) :=
Summary
The period polynomial provides a way of encoding critical values of L-functions associated with modular cusp forms that has proven very successful in the uncovering of important arithmetic properties of L-values. Its structure and properties as an object in its own right have attracted a lot of interest from various perspectives, one of the most important ones being that of Zagier, as will become apparent below. To give an idea of the uses of the period polynomial and its structure, we start by outlining its definition. An example of the manner by which the structure of the period polynomial leads to important arithmetic information about values of L-functions is Manin’s Periods Theorem. The algebraic properties of rf (cocycle relations) combined with the arithmetic nature of f (as a Hecke eigenform) lead to a certain one-dimensionality statement for rf , which with (1.1) translates to the following proportionality relation
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