Remarks on the geometry of the extended siegeljacobi. In this paper we study the automorphisms of siegel upper half plane of complex dimension 3. On siegel paramodular forms of degree 2 with small levels 2. That is, h comprises those elements j 2cv \sp v such that j is a euclidean structure on v.
The action on h 2 of the real symplectic group gsp2,r of degree two with similitudes is given by the map 0. Request pdf revisiting the siegel upper half plane ii in this paper we study the automorphisms of siegel upper half plane of complex dimension 3. Then a local degeneration is given by a holomorphic map f. Differential equations and the bergmansilov boundary on. Sp rt r be the full siegel modular group of genus n, and let t b\ 6. Isometries and geodesics of the upper halfplane 8 6. It is proved that the extended siegeljacobi upper halfplane. The siegel threefold is a quasiprojective variety of dimension 3 obtained from the siegel upper halfplane of degree two which by definition is the set of twobytwo symmetric matrices over c whose imaginary part is positive definite, i. Modular form in mathematics, a modular form is a complex analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. Therefore the space sn is a generalization of the complex upper half plane.
Gama pinto 2, 1649003 lisboa, portugal april 30, 2003 abstract. Geometry of shimura varieties in positive characteristics. The symplectic geometry of a new kind of siegel upper half space of. We then compactify the siegel upper half plane, so that we can extend the action continuously to a compact domain. We show that the the set of accumulation points of the orbit. Sh n sp n, r k n is the compactification as a bounded domain. A special homeomorphism acknowledgments 14 references 14 1. Geometric retracts of siegels upper half space ubc. In mathematics, the siegel upper halfspace of degree g or genus g is the set of g. Action of complex symplectic matrices on the siegel upper half space. Freitas, on the action of the symplectic group on the siegel upper half plane, phd thesis, university of illinois at chicago, 1999. Revisiting the siegel upper half plane ii request pdf. We introduce a method in differential geometry to study the derivative operators of siegel modular forms.
On certain vector valued siegel modular forms of degree two. Geometric retracts of siegels upper half space ubc library open. The siegel upperhalf plane and the symplectic group. The siegel upper half space department of mathematics. Invariant metrics and laplacians on siegeljacobi space. Differential equations and the bergmansilov boundary on t. We shall show that h is in fact one connected component of cv\sp v. In the first part of the paper we show that the busemann 1compactification of the siegel upper half plane of rank n. On siegel paramodular forms of degree 2 with small levels. Siegel modular form academic dictionaries and encyclopedias.
Let f be a siegel modular form of weight k such that the invariant obtained in corollary 3. In the second part of the paper we study certain properties of discrete. The siegel halfplane h again x a symplectic structure on v, with induced symplectic form the siegel halfplane determined by is h h. Freitas center for linear and combinatorial structures university of lisbon av. The symplectic geometry on h by the action of the group of linear fractional trans formations, or. Your browser doesnt seem to have a pdf viewer, please download the. Mathematicians sometimes identify the cartesian plane with the complex plane, and then the upper half plane corresponds to the set of complex numbers with positive imaginary part. On the way, we will also obtain a bound for the first nonvanishing taylor coefficient function of a generalized jacobi form. Cn,n z tz, imz0 be the siegel upper half plane of degree n and let spn,r m. Revisiting the siegel upper half plane i sciencedirect. Differential equations and the bergmansilov boundary on the siegel upper half plane. The siegel upper half plane of degree two is denoted by h2.
In mathematics, the siegel upper half space of degree g or genus g also called the siegel upper half plane is the set of g. Richter introduction the theory ofnonanalytic modular forms has a. By determining the coefficients of the invariant levicivita connection on a siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a nonholomorphic derivative operator of the siegel modular forms. It was introduced by siegel in the case g 1, the siegel upper half space is the wellknown upper half plane see also. Here zis a n ncomplex matrix whose real and imaginary parts are xand y respectively. Mathematicians sometimes identify the cartesian plane with the complex plane, and then the upper halfplane corresponds to the set of complex numbers with positive imaginary part.
Characterization of the isometries of the upper halfplane 12 7. Explicit bound for the first sign change of the fourier. R2n,2n tmj nm jn this work was supported by inha university research grant. Contents notation and conventions cornell university. M nc, z tz, imz 0 be the siegel upper halfspace of degree n, and. Introduction to the tangent space in the euclidean plane 1 2. Preliminaries let i n be the n identity matrix and set the symplectic group spn, r is defined as the group of 2n m with real entries for which mjjm where. Berndt received february 7, 1995 let p s nam be the minimal parabolic subgroup of su. Revisiting the siegel upper half plane i shmuel friedland department of mathematics, statistics and computer science, university of illinois at chicago chicago, illinois 606077045, usa pedro j. Revisiting the siegel upper half plane ii sciencedirect. Ants xiii proceedings of the thirteenth algorithmic number. Cl ery abstract we prove that there exist exactly eight siegel modular forms with respect to the congruence subgroups of hecke type of the paramodular groups of genus two vanishing precisely along the diagonal of the siegel upper halfplane. Recovering the familiar formula in the 1dimensional case, where sl 2z acts on the complex upper halfplane, requires compensating for the notational inconsistency that in one variable, vectors z 1 z 2 have their bottom coordinates normalized. There we deal with possible compacti cations of the space.
Siegel domain, a generalization of the siegel upper. Let, and h denote the unit complex disk, punctured disk, and complex upper half plane, respectively, and let t. In the case g 1, the siegel upper halfspace is the wellknown upper halfplane. Let vk,r be a representation space of the holomorphic representation detkqsymst of gl2, c. The siegel modular forms of genus 2 with the simplest divisor. In the second part of the paper we study certain properties of discrete groups. Our result extends to give the existence of a holomorphic siegel cusp form f corresponding to the symmetric cube of any newform f of even weight k.
On the action of the symplectic group on the siegel upper. Geometric retracts of siegels upper half space by justin martel b. In this paper we explicitly construct vector valued siegel modular forms of degree two and the automorphic factor detk for even k where st denotes the. Request pdf revisiting the siegel upper half plane i in the first part of the paper we show that the busemann 1compactification of the siegel upper half plane of rank is the compactification. The siegel modular forms of genus 2 with the simplest divisor v. On the siegeljacobi ball stefan berceanu department of theoretical physics, horia hulubei national institute for physics and nuclear engineering.
Siegel modular varieties i they are quotients of the siegel upper half space by congruence subgroups of sp 2g z. The siegel upper half space and its bergman kernel. Shn spn, rkn is the compactification as a bounded domain. This generalizes a basic result of in the case of classical jacobi forms theorem. We first define two notions which are the main objects studied in this section. It is the subgroup of g ab cd in n nblock form with c 0. Mixed hodge structures and compacti cations of siegels space. Vector valued siegels modular forms of degree two and the. Differential equations and the bergmansilov boundary on the siegel upper half plane kenneth d. Siegel halfspace of degree n is the set h n where 1. For d 1, one easily sees that 1 is the upper half plane h. Revisiting the siegel upper half plane i request pdf.
Revisiting the siegel upper half plane i, linear algebra. Revisiting the siegel upper half plane i revisiting the siegel upper half plane i friedland, shmuel. First, there is the siegel parabolic subgroup p sp 2n. Remarks on the geometry of the extended siegeljacobi upper halfplane. On the action of the symplectic group on the siegel upper half plane. The special linear group sl2,ractsonh transitively by.
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