Diamantis and Rolen Res Math Sci (2018) 5:9
Period polynomials, derivatives of
L-functions, and zeros of polynomials
and Larry Rolen
University of Nottingham,
Full list of author information is
available at the end of the article
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.
Keywords: Periods of modular forms, Derivatives of L-functions, Eichler cohomology,
Mathematics Subject Classiﬁcation: 11F11, 11F67
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. As such, 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.
Let f be an element of the space S
of weight k cusp forms for SL
(Z). The period
polynomial of f is the polynomial in X given by
f (τ )(τ − X)
A relation with the L-function of f is provided by the identity (cf eg. )
k − 2
(n + 1)X
(s) is the “completed” L-function of f .
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 Theo-
rem. The algebraic properties of r
(cocycle relations) combined with the arithmetic nature
of f (as a Hecke eigenform) lead to a certain one-dimensionality statement for r
with (1.1) translates to the following proportionality relation.
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