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Explicit Construction of Spaces of Hilbert Modular Cusp Forms Using Quaternionic Theta Series

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Explicit Construction of Spaces of Hilbert Modular Cusp Forms Using Quaternionic Theta Series Explicit construction of spaces of Hilbert modular cusp forms using quaternionic theta series Dissertation zur Erlangung des Grades des Doktors der Naturwissenschaften vorgelegt der Mathematisch-Naturwissenschaftlichen Fakult¨at I der Universit¨at des Saarlandes von Ute Gebhardt Saarbr¨ucken 2009 Tag des Kolloquiums: 12. Juni 2009 Dekan: Prof. Dr. Joachim Weickert Vorsitzender der Kommission: Prof. Dr. J¨org Eschmeier Berichterstattende: Prof. Dr. Rainer Schulze-Pillot-Ziemen Prof. Dr. Siegfried B¨ocherer (Mannheim) akademischer Mitarbeiter: Dr. Dominik Faas Abstract After preliminary work [19], [21] in the 1950s, a paper by Martin Eichler appeared in [50] in 1972 in which the basis problem for modular forms was solved. Eichler describes explicitly how theta series attached to quaternion ideals can be used to construct a basis for a given space Sk Γ0(N), 1) of cusp forms of even weight k > 2 and trivial character for the group Γ0(N). In the same year an article by Hideo Shimizu ([65]) was published that can be viewed as a generalization of Eichler’s work to the case of an arbitrary totally real number field F . Like Eichler, Shimizu is able to find a set of generators—although not necessarily a basis—for spaces of cusp forms. His result, however, is far less explicit than Eichler’s. A reason is that in his proof Shimizu makes intensive use of the representation theory of the group GL2(AF ) over the adele ring AF , so that this group and its representations still play a crucial role in the description of the generating system that he constructs, and a connection to the classical theta series can hardly be seen. Therefore, it is the aim of this thesis to prove an explicit version of Shimizu’s Theorem. This means that we will show how Shimizu’s “adelic theta series” can be transformed into classical theta series not unlike those in Eichler’s work, which have the advantage of being explicit enough to be evaluated on a computer. To this end, we begin by explaining the correspondence between adelic and classical modular forms, which is well-known in the case F = Q, but only seldom treated in the case of general number fields. Afterwards we provide the basics of the theory of admissible representations of GL2(AF ) that are necessary for the understanding of Shimizu’s result. With this background material we are then able to prove our Main Theorem 5.2.1, which is an explicit version of Shimizu’s Theorem. Finally, the results are used to construct theta series on a computer with the help of the computer algebra system Magma. We are thus able to find explicit sets of generators—in some cases even bases—for a selection of spaces of cusp forms. Zusammenfassung Nach Vorarbeiten [19], [21] aus den 1950er Jahren erschien 1972 in [50] ein Artikel Martin Eichlers ¨uber das Basisproblem f¨ur Modulformen, worin er explizit beschreibt, wie man aus Thetareihen zu gewissen Quaternionenidealen eine Basis eines vorgegebenen Raumes Sk Γ0(N), 1) von Spitzenformen geraden Gewichts k > 2 und mit trivialem Charakter zur Gruppe Γ0(N) konstruiert. Im selben Jahr ver¨offentlichte Hideo Shimizu einen Artikel ([65]), den man als Verallge- meinerung von Eichlers Arbeit auf den Fall eines beliebigen total reellen Zahlk¨orpers F auffassen kann. Auch ihm gelingt es, ein Erzeugendensystem — wenngleich nicht notwendi- gerweise eine Basis — f¨ur R¨aumevon Spitzenformen anzugeben. Allerdings ist sein Resultat weit weniger explizit als Eichlers. Dies liegt in erster Linie daran, dass er in seinem Beweis intensiven Gebrauch von der Darstellungstheorie der Gruppe GL2(AF ) ¨uber dem Adelring AF macht, so dass auch in der Beschreibung des von ihm angegebenen Erzeugendensys- tems der Gruppe GL2(AF ) und Darstellungen derselben eine tragende Rolle zukommt und zun¨achst keinerlei Ahnlichkeit¨ mit klassischen Thetareihen erkennbar scheint. Ziel der vorliegenden Arbeit ist es nun, eine explizite Version von Shimizus Theorem zu beweisen. Das heißt, es soll gezeigt werden, wie sich Shimizus ,,adelische Thetareihen“ in klassische Thetareihen ¨ubersetzen lassen, die — wie in Eichlers urspr¨unglicher Arbeit — so konkret angegeben werden k¨onnen, dass dadurch eine Berechnung mit dem Computer m¨oglich wird. Zu diesem Zweck wird zun¨achst der Zusammenhang zwischen adelischen und klassischen Modulformen hergeleitet, der im Fall F = Q hinl¨anglich bekannt ist, im Fall allgemeinerer Zahlk¨orper in der Literatur aber kaum behandelt wird. Ferner werden diejenigen Aus- sagen ¨uber die Theorie der zul¨assigen Darstellungen von GL2(AF ) bereitgestellt, die f¨ur das Verst¨andnis von Shimizus Resultat n¨otigsind. Mit diesen Hilfsmitteln kann dann der Hauptsatz 5.2.1, d. h. eine explizite Version von Shimizus Theorem, bewiesen werden. Schließlich wird dieses Resultat benutzt, um mit Hilfe des Computeralgebrasystems Magma Thetareihen zu konstruieren und somit ganz konkret Erzeugendensysteme — und teilweise sogar Basen — einiger ausgew¨ahlter R¨aume von Spitzenformen zu berechnen. Contents Introduction 5 1 Hilbert modular forms in the classical setting 9 1.1 Introduction to Hilbert modular forms . 9 1.2 Quaternion algebras and their ideals . 12 1.3 An example: Theta series attached to quaternion ideals . 16 2 Adelic Hilbert modular forms 21 2.1 Group representations . 21 2.2 Hilbert modular forms in the adelic setting . 26 2.3 Correspondence between classical and adelic Hilbert modular forms . 36 3 The action of the Hecke algebra on automorphic forms 43 3.1 The local Hecke algebra (non-archimedean case) . 43 3.2 The local Hecke algebra (real case) . 46 3.3 The global Hecke algebra . 50 3.4 Representation of the Hecke algebra on the space of Hilbert automorphic forms 51 4 Harmonic homogeneous polynomials 59 4.1 Gegenbauer polynomials . 59 4.2 The action of the Hamiltonians on homogeneous polynomials . 65 4.3 Harmonic polynomials and automorphic forms . 68 3 4 Contents 5 Theta series as generators of the space of Hilbert modular cusp forms 77 5.1 The Weil representation . 77 5.2 An explicit version of Shimizu’s Theorem . 79 5.3 Proof of Theorem 5.2.1 . 84 6 Explicit computations and results 99 6.1 Sketches of the algorithms . 99 6.1.1 Algorithms concerning the number field . 100 6.1.2 Algorithms concerning the quaternion ideals . 100 6.1.3 Algorithms concerning the harmonic polynomials . 103 √ 6.2 First example over Q( 5) explained in detail . 104 6.3 Further results . 107 √ 6.3.1 Theta series for Q( 5)........................... 107 √ 6.3.2 Theta series for Q( 3)........................... 110 √ 6.3.3 Theta series for Q( 2)........................... 114 6.3.4 Theta series for Q[X]/(X3 − X2 − 2X + 1) . 116 List of Symbols 119 Index 123 Bibliography 125 Introduction Let F be a totally real number field of degree [F : Q] = n > 1 of arbitrary narrow class number h+, and let k ∈ Zn, k ≥ 2 be a (possibly non-parallel) weight vector. Hilbert modular forms in the classical sense are holomorphic functions on the n-fold complex upper half plane with the well-known transformation rule f|kγ = χ(γ)f + for all γ in some congruence subgroup of GL2 (F ) and some character χ. This definition is completely analogous to the elliptic situation, i. e. to the case of modular forms over Q. But when delving deeper into the theory and trying to carry methods and proofs over from Q to the number field F , one will soon face more problems than expected. The theory of Hecke operators, for example, which is known to be a highly useful tool in the treatment of modular forms, does not generalize as easily to the number field situation, and it becomes particularly technical if F is of narrow class number h+ > 1. On the other hand, one may also define Hilbert modular forms in the adelic setting as certain automorphic forms on GL2(F )\GL2(AF ) (the precise definition is given in Definition 2.2.9). Although it looks rather unwieldy, the adelic approach turns out to be more promising for many purposes and theoretical results. For example, the adelic Hecke theory allows a uniform treatment of all number fields, so that a non-trivial narrow class group no longer causes complications. But above all, automorphic forms on GL2(F )\GL2(AF ) give naturally rise to admissible representations of GL2(AF ), which can then be approached by representation theoretic means as explained in Herv´eJacquet’s and Robert Langlands’s famous work [41]. A natural question to ask when examining a given space of modular forms for a certain weight, character and congruence subgroup is: What is the dimension of this space? More- over, which forms constitute a basis? In preliminary works [19], [21] and finally in [22],[23], Martin Eichler solved the so-called “basis problem” for Sk Γ0(N), 1 , the space of elliptic cusp forms for the group Γ0(N) of even weight k and trivial character χ = 1. By proving that Sk Γ0(N), 1 is the direct sum of spaces of newforms and translates thereof, each of which is spanned by theta series attached to a certain quaternion algebra, he gave an explicit description of a basis of the space Sk Γ0(N), 1 . 5 6 Introduction In the same year, a paper by Hideo Shimizu was published in which the author presents an alternative proof of Jacquet-Langlands [41, Thm. 14.4] and—as a byproduct—obtains 0 generators for certain spaces Hk K0(d, n), 1 of adelic Hilbert modular cusp forms of weight k > 2 and level n. His result can be viewed as a generalization of Eichler’s theorem to the 0 number field case in the sense that the space Hk K0(d, n), 1 decomposes into a direct sum of spaces U(m) of newforms and translates thereof, each of which is generated by functions that we will call, for lack of a better word, adelic theta series.
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