Period functions associated to real-analytic modular forms
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RESEARCH
Period functions associated to real-analytic modular forms Nikolaos Diamantis
and Joshua Drewitt
* Correspondence:
[email protected] University of Nottingham, Nottingham, UK
Abstract We define L-functions for the class of real-analytic modular forms recently introduced by F. Brown. We establish their main properties and construct the analogue of period polynomial in cases of special interest, including those of modular iterated integrals of length one.
1 Introduction A well-known difference between the behaviour of holomorphic cusp forms and that of Maass cusp forms is that the former are “motivic” whereas the latter are, in general, not expected to be. This difference is fundamental for applications to arithmetic, and it is reflected in the possibility of proving results such as Manin’s Periods Theorem. We state it here in a slightly weakened form to make the comparison with one of our theorems easier: Proposition 1.1 [15] Let f be a cusp form of weight k for Γ :=SL2 (Z) which is a normalised eigenfunction of the Hecke operators. Let L∗f denote the “completed” L-function of f . Then there are ω+ (f ), ω− (f ) ∈ C such that L∗f (j) ∈ ω+ (f )Kf , for odd j ∈ {2, . . . k − 2} and L∗f (j) ∈ ω− (f )Kf , for even j ∈ {2, . . . k − 2}. where Kf is the field generated by the Fourier coefficients of f . A key tool used to prove Proposition 1.1 was the period polynomial. This encodes the “critical values” of L-functions of cusp forms in an algebraically convenient way (Eichler cohomology). Period polynomials have been studied extensively, from various standpoints since at least the 70s and have been used to prove many important results besides Manin’s Periods Theorem. By contrast, the analogue of period polynomials for Maass cusp forms proved to be harder to construct. It was introduced and studied by Lewis and Zagier in the late 90s [13,14]. This period function found important uses in various contexts, though not in arithmetic applications.
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N. Diamantis and J. Drewitt Res Math Sci (2020)7:21
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Against this contrasting picture, Brown [3–5] identified a “hybrid” space, which, though infinite dimensional and c
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