• Login
• Admin

Pimm, Joshua (2025) Real-analytic modular forms for Gamma_0(N) and their L-series. PhD thesis, University of Nottingham.

Preview

PDF (Thesis with corrections) (Thesis - as examined) - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Available under Licence Creative Commons Attribution. Download (567kB) | Preview

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:	Thesis (University of Nottingham only) (PhD)
Supervisors:	Diamantis, Nikolaos Stromberg, Fredrik
Keywords:	Modular forms, L-series, number theory
Subjects:	Q Science > QA Mathematics
Faculties/Schools:	UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID:	82333
Depositing User:	Pimm, Joshua
Date Deposited:	12 Dec 2025 04:40
Last Modified:	12 Dec 2025 04:40
URI:	https://eprints.nottingham.ac.uk/id/eprint/82333

Actions (Archive Staff Only)

Edit View