Period-like polynomials for L-series associated with half-integral weight cusp forms

Area of Science:

• Number Theory
• Automorphic Forms
• Complex Analysis

Background:

• Cusp forms are fundamental objects in number theory, particularly in the study of modular forms.
• L-series encode arithmetic information about number theoretic objects.
• Integral and half-integral weight cusp forms have distinct properties and applications.

Purpose of the Study:

• To generalize the concept of period polynomials to half-integral weight cusp forms.
• To establish a method for lifting half-integral weight cusp forms to integral weight cusp forms.
• To ensure the compatibility of L-series between the lifted and original forms.

Main Methods:

• Construction of polynomials derived from the L-series of half-integral weight cusp forms.
• Development of a novel lifting procedure from half-integral to integral weight cusp forms.
• Verification of L-series compatibility using established properties of automorphic forms.

Main Results:

• Successfully constructed polynomials analogous to classical period polynomials for half-integral weight cusp forms.
• Defined a lift that maps half-integral weight cusp forms to integral weight cusp forms.
• Demonstrated that the L-series of the lifted integral weight form are compatible with the L-series of the original half-integral weight form.

Conclusions:

• The study extends the theory of period polynomials to a broader class of cusp forms.
• The developed lift provides a valuable tool for relating half-integral and integral weight automorphic forms.
• This work deepens the understanding of the interplay between L-series and the structure of cusp forms.