What question did this study set out to answer?

This research aims to explore the rational structures of cusp forms on Γ0(N) in relation to their periods.

April 23, 2026

Rational structures on the space of cusp forms on Γ0(N) through the rationality of periods

Key Points

This research aims to explore the rational structures of cusp forms on Γ0(N) in relation to their periods.
Examined cusp forms categorized by binary quadratic forms on Γ0(N).
Computed even and odd period polynomials using locally harmonic Maass forms.
Analyzed the implications for rational structures of cusp forms of a given weight.
Validated that cusp forms have rational even and odd periods.
Established the space of cusp forms on Γ0(N) possesses rational structures.
Provided explicit forms of period polynomials for higher level cusp forms.

Abstract

Kohnen and Zagier studied two types of cusp forms Formula: see text on Formula: see text defined by binary quadratic forms. They showed the cusp forms Formula: see text have rational even periods, while Formula: see text have rational odd periods. In this paper, we examine analogous cusp forms Formula: see text on Formula: see text for higher levels Formula: see text. Using locally harmonic Maass forms on Formula: see text, we explicitly compute the even (resp. odd) period polynomials of Formula: see text (resp. Formula: see text). Consequently, we establish that the space of cusp forms of weight Formula: see text on Formula: see text has rational structures, attributed to the periods of these cusp forms.

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Rational structures on the space of cusp forms on Γ0(N) through the rationality of periods

Key Points

Abstract

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