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On Hermitian theta series and modular forms [Elektronische Ressource] / vorgelegt von Michael Hentschel

77 pages
On Hermitian theta seriesand modular formsMichael Hentschel ⋆ On Hermitian theta series and modular forms„On Hermitian theta seriesand modular forms“Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen Uni-versity zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigteDissertationvorgelegt vonDiplom-MathematikerMichael Hentschelaus Rhede, Kreis BorkenBerichter: Universitätsprofessor Dr. rer. nat. Aloys KriegUniversitätsprofessorin Dr. rer. nat. Gabriele NebeTag der mündlichen Prüfung: 17. Juli 2009Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar.A few words on the leitmotif of this thesis:The main examples for Hermitian modular forms come from the classical case, from HermitianEisenstein series due to Hel Braun [Br3] or from liftings, see Gritsenko, Ikeda, Krieg, and lotsof others. Cohen and Resnikoff [CoRe] introduced the method of constructing modular formsvia theta-series to the theory of Hermitian modular forms and gave a construction for latticeswhich yield Hermitian modular forms for an arbitrary imaginary quadratic field. In [DeKr]then one finds an elementary method for the construction of those lattices. One is especiallyinterested in the number of distinct isometry classes (in the genus) of lattices which then yielddifferent theta-series, a problem which was already sketched in [CoRe], page 336, ”...its [thegenus] class number remains unknown.”.
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On Hermitian theta and modular forms
series
M
ichael
Hentschel
On
Herm
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theta
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and
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form
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„On Hermitian theta series and modular forms“
Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen Uni-versity zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation
vorgelegt von
Diplom-Mathematiker Michael Hentschel
aus Rhede, Kreis Borken
Berichter: Universitätsprofessor Dr. rer. nat. Aloys Krieg Universitätsprofessorin Dr. rer. nat. Gabriele Nebe
Tag der mündlichen Prüfung: 17. Juli 2009
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar.
A few words on the leitmotif of this thesis: The main examples for Hermitian modular forms come from the classical case, from Hermitian Eisenstein series due to Hel Braun [Br3] or from liftings, see Gritsenko, Ikeda, Krieg, and lots of others. Cohen and Resnikoff [CoRe] introduced the method of constructing modular forms via theta-series to the theory of Hermitian modular forms and gave a construction for lattices which yield Hermitian modular forms for an arbitrary imaginary quadratic eld. In [DeKr] then one nds an elementary method for the construction of those lattices. One is especially interested in the number of distinct isometry classes (in the genus) of lattices which then yield different theta-series, a problem which was already sketched in [CoRe], page 336, ”...its [the genus] class number remains unknown.”. So far, just the situation with respect to the Gaussian number eld, see [KiMu] or [Sc1], was known. This thesis generalizes these results to arbitrary imaginary quadratic elds of class number one. In special, well-arranged cases we investigate the isometry classes of the lattices of interest. Together with Grabriele Nebe we have developed a mass formula which can be applied to imaginary quadratic elds of class number 1 easily and can be adopted to other class numbers. Then we compute the ltration of cusp forms analogous to [HeKr] where we considered the ltration of cusp forms arising from lattices over the Gaussian number eld, but this is limited to very easy cases as bounds for dimension estimations get out of reach very soon. We give some information on the situation with respect to higher class numbers as 1. The natural continuation of this thesis then is to investigate the situation with respect to higher class numbers. But things seem to get very ugly very fast. On the other hand one can step deeper into the theory of modular forms itself and try to make some advance with respect to the ltration.
I wish to thank my advisor, Prof. Dr. Aloys Krieg, for the opportunity to write my Phd. thesis under his supervision and his imperturbable support. Furthermore I am deeply grateful to Prof. Dr. Gabriele Nebe for her help and ideas she was willing to share with me.
Contents 1 The basic theory 3 1.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Hermitian modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Lattices and Hermitian modular forms . . . . . . . . . . . . . . . . . . . . . . . 9 2 The cased=311 2.1 Formulating a strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 A mass formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 The computation of the< 17 . . . . . . . . . . . . . . . . . . . . . . . . .-lattices . 2.3.1<-lattices of rankn=4 andn=8 . 17 . . . . . . . . . . . . . . . . . . . . 2.3.2<-lattices of rankn= . . . . . . . . . . . . . . . . . . . . . . . . . 1812 . 2.4 Another approach to the<-lattices of rankn=4,n=8 andn= 2012 . . . . . . . . 2.5 Application to Hermitian modular forms . . . . . . . . . . . . . . . . . . . . . . 23 2.6 A rst look onto situation of rankn= 27 . . . . . . . . . . . . . . . . . . . . .16 . 3 The neighbourhood method 28 3.1 A closer look onto<-lattices of rank 16 overOQ(3) 30. . . . . . . . . . . . . . . 4Latticeswithrespecttootherimaginaryquadraticelds33 4.1 The case of prime discriminants . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 The cased=2 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  36 4.3 Application to Hermitian modular forms of low weight . . . . . . . . . . . . . . 39 4.3.1 Weightk=4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3.2 Weightk=. . . . . . 408 . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Non-free lattices 42 A Appendix 46 A.1 Numbers of isometry classes of< . . . . . . . . . . . . . . . . . . . .-lattices . 46 . . . . . . . 47 A.2 The rank 4<-lattice overOQ(2). . . . . . . . . . . . . . . . . . A.3 The rank 4<-lattice overOQ(3). . . . . . . . . . . . . . . . . . . . . . . . . 47 A.4 The rank 4<-lattice overOQ(7). . . . . . . . . . . . . . . . . . . . . . . . . 47 A.5 The two rank 4<-lattices overOQ(19) 47. . . . . . . . . . . . . . . . . . . . . . A.6 The four rank 4<-lattices overOQ(43) 48. . . . . . . . . . . . . . . . . . . . . . ank 4<-lattices ov A.7 The six r erOQ(67) 48. . . . . . . . . . . . . . . . . . . . . . A.8 The six rank 8<-lattices overOQ(2). . . . . . . . . . . . . . . . . . . . . . . 49 A.9 The rank 8<-lattice overOQ(3) 51. . . . . . . . . . . . . . . . . . . . . . . . . A.10 The three rank 8<-lattices overOQ(7). . . . . . . . . . . . . . . . . . . . . . 51 A.11 The seven rank 8<-lattices overOQ(11) 52. . . . . . . . . . . . . . . . . . . . . A.12 The ve rank 12<-lattices overOQ(3) 54. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.13 The non-isometric rank 16<-lattices overOQ(3)56 A.14 The non-isometric rank 12<-lattices overOQ(7) 58. . . . . . . . . . . . . . . . 1
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On Hermitian theta-series and modular forms
A.15 The Magma Code for neighbour stepping
References
Curriculum Vitae
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The basic theory
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1 The basic theory 1.1 Lattices LetdN Thenbe a squarefree integer.K:=Q(d)is an imaginary quadratic number eld. ObviouslyKis an algebraicQ-eld extension of degree 2. In cased1,2(4)we have disc(K) = 4dand in cased3(4)we have disc(K) =d consider the ring of integers. WeOKof the Q-eld extensionK, we haveOK=h1,wiZ, where w=( di d1,2(4) 1+i2dd3(4)OKis a Dedekind domain. Denition 1.1A lattice of ranknoverOQ(d)is a subset9Q(d)nwhich has the struc-ture of anOK-submodule with respect to the vector space and fulllsQ(d)9=Q(d)n. In full generality one replacesQ(d)nby an abitrary vector space andOKby a Dedekind domain within the underlying eld. When the rank and the underlying eld is not specied we just speak of a lattice. Remark 1.2a) LetKan algebraic number eld, which is aQ-extension of nite index, then the ideal class groupclKis dened as the quotientJKPK, the group of fractional ideals modulo the group of fractional principal ideals. The class numberhKis dened as the orderclK. For d∈ {1,2,3,7,11,19,43,67,163} the imaginary quadratic number eldK=Q(d)fulllshK=1. In case ofhK=1 a lattice9of ranknoverOQ(d)is a necessarily freeOK-module, see [OMe], and so there exists a basis(b1,    ,bn)ofQ(d)nsuch that9=hb1,    ,bniOK.9is a freeOK-module of ranknin the vector spaceQ(d)n. b) Let9be a free lattice within an arbitrary vector spaceV. IfVis equipped with a regular symmetric formh:V×VK, where symmetry may be denied as convenient and regu-larity means that the form is linear in the rst variable andKthe underlying eld, then one can introduce the Gram matrix of a lattice with a bilinear form(9,h)as (Gram(9,h))i,j=h(bi,bj)In the special case of imaginary quadratic elds, the vectorspaceQ(d)n, any square-freed, is canonically equipped with the standard Hermitian form h,i:Q(d)n×Q(d)nC, which is of course linear in the rst variable. So forhK=1 we can always consider Gram matrices of lattices. When we do not specify, we always will use the standard Hermitian form.
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