Title Neponištavanja Poincaréovih redova na metaplektičkoj grupi i primjene 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 Defense date and country 2018-05-28, Croatia Scientific / art field, NATURAL SCIENCES Universal decimal classificationUDC ) 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 Type of resource Text Extent xi, 171 str. File origin Born digital Access conditions Open access Terms of use Created on 2019-03-22 12:39:23
