Neponištavanja Poincaréovih redova na metaplektičkoj grupi i primjene

doctoral thesis

• Title (english): Non-vanishing of Poincaré series on the metaplectic group and applications
• Author: Sonja Žunar
• Mentor: Goran Muić (mentor)
• Committee member: Marcela Hanzer (predsjednik povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Committee member: Ivica Gusić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-05-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

U ovoj se disertaciji proučava neponištavanje Poincaréovih redova na metaplektičkom natkrivaču, \(SL_2(\mathbb{R})^{\sim}\), grupe \(SL_2(\mathbb{R})\) i istražuju se primjene tih redova u teoriji modularnih formi polucijele težine. Pritom se koriste tri osnovna alata: Harish-Chandrin pristup teoriji reprezentacija povezanih poluprostih Liejevih grupa s konačnim centrom, klasična korespondencija između kusp-formi polucijele težine i kuspidalnih automorfnih formi na grupi \(SL_2(\mathbb{R})^{\sim}\) i integralni kriterij neponištavanja za Poincaréove redove na lokalno kompaktnim Hausdorffovim grupama koji je dokazao G. Muić. U disertaciji su pomoću Poincaréovih redova \(K\)-konačnih matričnih koeficijenata integrabilnih reprezentacija grupe \(SL_2(\mathbb{R})^{\sim}\) konstruirani sustavi izvodnica za prostore kusp--formi polucijele težine \(m \geq \frac{5}{2}\). Dokazana je i formula za Peterssonov skalarni produkt spomenutih Poincaréovih redova s proizvoljnom kuspidalnom automorfnom formom na \(SL_2(\mathbb{R})^{\sim}\) iste težine. Iz te formule i njena dokaza proizašao je niz rezultata o kusp-formama polucijele težine. Nadalje, dokazane su blago ojačane varijante Muićeva integralnog kriterija neponištavanja za Poincaréove redove na Liejevim grupama i za Poincaréove redove polucijele težine na gornjoj kompleksnoj poluravnini. Njihovom su primjenom dokazani rezultati o neponištavanju Poincaréovih redova pridruženih \(K\)-konačnim matričnim koeficijentima integrabilnih reprezentacija grupe \(SL_2(\mathbb{R})^{\sim}\) i pripadnih kusp-formi polucijele težine, rezultati o neponištavanju nekih klasičnih Poincaréovih redova polucijele težine i rezultati o neponištavanju \(L\)-funkcija pridruženih kusp-formama polucijele težine \(m \geq \frac{9}{2}\) u točkama pruge \(\mathbb{C}_{\frac{m}{2} < \Re (s) < m-1}\).

Abstract (english)

In this thesis, we study the non-vanishing of Poincaré series on the metaplectic cover, \(SL_2(\mathbb{R})^{\sim}\), of \(SL_2(\mathbb{R})\) and investigate their applications in the theory of modular forms of halfintegral weight. Our methods rely on the following three tools: Harish-Chandra’s approach to representation theory of connected semisimple Lie groups with finite center, the classical correspondence between cusp forms of half-integral weight and cuspidal automorphic forms on \(SL_2(\mathbb{R})^{\sim}\), and the integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups that was proved by G. Muić. In the thesis, we use the Poincaré series of \(K\)-finite matrix coefficients of integrable representations of \(SL_2(\mathbb{R})^{\sim}\) to construct spanning sets for spaces of cusp forms of halfintegral weight \(m \geq \frac{5}{2}\). We prove a formula for the Petersson inner product of these Poincaré series with any cuspidal automorphic form on \(SL_2(\mathbb{R})^{\sim}\) of the same weight. From this formula and its proof, we derive a series of results on cusp forms of half-integral weight. Next, we prove strengthened variants of Muić’s integral non-vanishing criterion for Poincaré series on Lie groups and for Poincaré series of half-integral weight on the upper half-plane. Using these criteria, we prove results on the non-vanishing of Poincaré series of \(K\)-finite matrix coefficients of integrable representations of \(SL_2(\mathbb{R})^{\sim}\) and of corresponding cusp forms, results on the non-vanishing of some classical Poincaré series of half-integral weight, and results on the non-vanishing of \(L\)-functions associated to cusp forms of halfintegral weight \(m \geq \frac{9}{2}\) in points of the strip \(\mathbb{C}_{\frac{m}{2} < \Re (s) < m-1}\).

• Keywords: neponištavanje Poincaréovih redova; metaplektički natkrivač grupe \(SL_2(\mathbb{R})\); modularne forme polucijele težine; \(K\)-konačni matrični koeficijenti; \(L\)-funkcije kusp-formi polucijele težine
• Keywords (english): non-vanishing of Poincaré series; metaplectic cover of \(SL_2(\mathbb{R})\); modular forms of half-integral weight; \(K\)-finite matrix coefficients; \(L\)-functions associated to cusp forms of half-integral weight
• Language: croatian
• URN:NBN: urn:nbn:hr:217:792242
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: xi, 171 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-03-22 12:39:23