Real-analytic modular forms for Gamma_0(N) and their L-series
Publication Date
December 12, 2025
Creator
Abstract
A class of real-analytic modular forms for SL2(Z) was recently introduced by Francis Brown. The purpose of this thesis is to generalise these objects to higher level congruence subgroups, and construct L-series for them which satisfy a functional equation and a converse theorem. In particular, we investigate higher level modular iterated integrals and real-analytic Eisenstein series.
We recall the basic theory of Dirichlet characters and of holomorphic modular forms in Chapter 2, and the classical theory of L-series due to Hecke and Weil. We give important applications of this theory due to e.g. Shimura, Kohnen. We define Brown’s real-analytic modular forms, our key differential operators and modular iterated integrals. The inspiration for our L-series construction is presented in the context of harmonic Maass forms.
In Chapter 3, we define the objects of Brown in higher level, and construct higher level Eisenstein series. Then, in Chapter 4, we define an L-series for a certain class of real-analytic modular form. We show that it has a functional equation and converse theorem. We also use a construction with an auxiliary variable to define L-series in the more general setting, and prove a functional equation and converse theorem in this case, too. Finally, in Chapter 5, we use this functional equation to prove a summation formula for cusp forms.
Item Type
ethesis
Thesis Type
PhD
Supervisors
Subjects (LC)
Associated Schools / Departments
School of Mathematical Sciences (UK)
eprints ID
82333
UoN Repository URI
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Joshua Pimm - Thesis with corrections.pdf
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Examined. Thesis with corrections
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