DOCSLIB.ORG

PART 2: SIEGEL MODULAR FORMS 1. Siegel Modular

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
PART 2: SIEGEL MODULAR FORMS 1. Siegel Modular PART 2: SIEGEL MODULAR FORMS AARON POLLACK 1. Siegel modular forms We now begin to discuss Siegel modular forms. Siegel modular forms are special automorphic forms for the symplectic group Sp2n. Recall that Sp = fg 2 GL : g 0 1n gt = 0 1n g: 2n 2n −1n 0 −1n 0 Equivalently, and more canonically if you prefer, Sp2n is the group preserving a non-degenerate alternating bilinear form on a 2n-dimensional vector space. Equivalently, if k is a field and W is a 2n-dimensional k-vector space, then Sp2n is the subgroup of GL(W ) that preserves a non-degenerate alternating form on W . In more detail, a bilinear form h; i : W × W ! k is a symplectic form if • hx; yi = −hy; xi for all x; y 2 W and • hx; yi = 0 for all y 2 W implies x = 0. Then Sp2n = Sp(W ) is the algebraic group of g 2 GL(W ) such that hgx; gyi = hx; yi for all x; y 2 W . A symplectic basis of W is a basis e1; : : : en; f1; : : : ; fn such that hei; eji = hfi; fji = 0 for all 2n i; j and hei; fji = δij. Every symplectic space W has a symplectic basis. In case W = k with symplectic form hx; yi = xtJ y where J = 0 1n , the usual coordinate basis is a symplectic n n −1n 0 basis. We now describe a few subgroups of Sp2n. First, there is the Siegel parabolic subgroup P ⊆ Sp2n. a b It is the subgroup of g = c d in n × n block form with c = 0. We have the following lemma. a 0 t −1 1 x Lemma 1. One has 0 d in Sp2n if and only if d = a . The element ( 0 1 ) is in Sp2n if and only if x = xt. t Proof. This is a direct matrix calculation with the definition g Jng = Jn for g 2 Sp2n. Another subgroup of Sp2n is a unitary group of size n: Example This is example works over any field, but we explain for k = R. Consider Cn, with the hermitian form (x; y) = x∗y where x; y 2 Cn are column vectors and ∗ denotes conjugate transpose. Now, let W = Cn, but considered as a 2n-dimensional real vector space. Put on W the R-bilinear form given as hx; yi = Im((x; y)) = Im(x∗y). This a non-degenerate alternating form. As U(n) is the subgroup of GLn(C) that fixes the Hermitian pairing ( ), we see that U(n) fixes h ; i as well, so U(n) ⊆ Sp2n(R). Let us now define the Siegel upper half-space hn. The Siegel modular forms will be holomorphic functions on hn that satisfy a certain equivariance property. Set t hn = fZ 2 Mn(C): Z = Z; Im(Z) > 0g: So, if Z 2 hn then Z = X + iY with X; Y symmetric and Y positive-definite. −1 a b The group Sp2n(R) acts on hn via gZ = (aZ + b)(cZ + d) for g = c d 2 Sp2n(R). More precisely, one has the following proposition. a b Proposition 2. Suppose z 2 hn and g = c d Sp2n. Then 1 (1) The n × n matrix cz + d is invertible; −1 (2) the matrix (az + b)(cz + d) is in hn; −1 (3) the map Sp2n(R) × hn ! hn given by (g; z) 7! (az + b)(cz + d) defines an action of Sp2n(R) on hn. Proof. For Z; W 2 Mn(C), note the identity ∗ ∗ ∗ (Z; W )Jn(Z; W ) = ZW − WZ : We now prove (1). For this, note that we have ∗ ∗ 2iy = z − z = (z; 1)Jn(z; 1) t ∗ = (z; 1)gJng (z; 1) ∗ = (az + b; cz + d)Jn(az + b; cz + d) = (az + b)(cz + d)∗ − (cz + d)(az + b)∗: To prove that cz + d is invertible, we prove that ξ 2 Cn and ξ(cz + d) = 0 implies ξ = 0. To see this, suppose ξ(cz + d) = 0. Then from the above, we get 2iξyξ∗ = xi ((az + b)(cz + d)∗ − (cz + d)(az + b)∗) ξ∗ = 0: But since y is positive definite, ξyxi∗ = 0 implies ξ = 0, as desired. For part (2), we first prove that gZ := (az + b)(cz + d)−1 is symmetric and then that Im(gz) is positive definite. For the symmetry, first note the identity t t t (Z; W )Jn(Z; W ) = ZW − WZ : Thus t t 0 = z − z = (z; 1)Jn(z; 1) t t = (z; 1)gJng (z; 1) t = (az + b; cz + d)Jn(az + b; cz + d) t t = cz + d(gz; 1)Jn(gz; 1) (cz + d) = (cz + d)(gz − (gz)t)(cz + d)t: From this, one concludes that gz is symmetric. By the same argument, one also proves Im(gz) = (cz + d)−1Im(z)((cz + d)∗)−1, from which one concludes that Im(gz) is positive definite. −1 We leave the proof that z 7! (az + b)(cz + d) is an action to the reader. We also have Lemma 3. The Siegel parabolic subgroup P ⊆ Sp2n, and thus Sp2n, acts transitively on hn, and the stabilizer of i 2 hn is the unitary group U(n). Proof. For the first part, suppose that x + iy 2 hn. Because y is positive definite, there exists t 1 x m m 2 GLn(R) such that mm = y. Then note that ( 1 )( tm−1 )·i = x+iy, so P acts transitively. AB For the second part, if A + iB 2 U(n), with A; B 2 Mn(R), one can check that −BA 2 Sp2n directly from the definition. This is how U(n) is embedded in Sp2n via matrices. One computes immediately that such elements fix i, and moreover that they are the full stabilizer. As a corollary of the proof of the lemma, one has that Sp2n(R) = P (R)U(n). We now come to the definition of Siegel modular forms. 2 Definition 4. Suppose w ≥ 1 is an integer. A holomorphic function f : hn ! C is said to be a Siegel modular form of weight w if f(γz) = j(γ; z)wf(z) for all γ in some congruence subgroup Γ of Sp2n(Z). If n = 1 one also imposes a slow growth condition on f(z), which turns out to be automatic for n > 1. Siegel modular forms have a very nice Fourier expansion, as follows: Denote by Sn(Z) the n × n _ _ symmetric matrices, and Sn (Z) the dual space under trace pairing, so that Sn (Z) consists of the half-integral symmetric matrices. Then 1 V Lemma 5. Suppose f is a Siegel modular form of level Γ, and that Γ contains the elements 0 1 P 2πi(T;Z) for every V 2 S (Z). Then f(Z) = _ a (T )e for some complex numbers a (T ). n T 2Sn(Z) f f Proof. We have f(Z) = f(Z + V ) for every V 2 Sn(Z). This plus holomorphy immediately implies the Fourier expansion. We will shortly prove that af (T ) 6= 0 implies T ≥ 0, i.e. that Siegel modular forms can only have nonzero Fourier coefficients corresponding to T that are positive semidefinite. We begin with a lemma, which is useful to know regardless. For simplicity, from now on we assume (for simplicity) that if f is a Siegel modular form of level Γ, where Γ contains P (Q) \ Sp2n(Z), with P the Siegel parabolic. t Lemma 6. Suppose f is a Siegel modular form of weight w and m 2 GLn(Z). Then af ( mT m) = w det(m) af (T ). Proof. One has X 2πi(T;Z) f(Z) = af (T )e _ T 2Sn(Z) X t 2πi(mtT m;Z) = af (m T m)e _ T 2Sn(Z) and w t X w 2πi(T;mZmt) det(m) f(mZm ) = det(m) af (T )e _ T 2Sn(Z) X w 2πi(mtT m;Z) = det(m) af (T )e : _ T 2Sn(Z) By the equivariance of f, one has det(m)wf(mZmt) = f(Z). Comparing the two displayed equa- tions now gives the lemma. We now prove one of the first very interesting properties of Siegel modular forms. Denote by Sn the n × n symmetric matrices. Proposition 7. Suppose f is a Siegel modular form of genus n > 1. Then af (T ) 6= 0 implies that T is positive semi-definite. Proof. We have Z −2π tr(T;Y0) −2πi tr(T;X0) af (T )e = e f(Y0 + X) dX Sn(Z)nSn(R) 2π tr(T ) from which one obtains the bound jaf (T )j ≤ Cf e for a positive constant Cf that is indepen- dent of T . Thus for m 2 GLn(Z), t 2π tr(mtT m) jaf (T )j = jaf (m T m)j ≤ e : 3 But now if T is not positive semi-definite, then tr(mtT m) can be made arbitrarily negative as m varies over GLn(Z). Thus af (T ) = 0 for such T . We now give some examples of Siegel modular forms. Example 1 Siegel Eisenstein series. For ` sufficiently large (in terms of n), set E`(Z) = P j(γ; Z)−`. When ` is so large that the series converges absolutely, it is clear γ2Pn(Z)n Sp2n(Z) ` that E`(δZ) = j(δ; Z) E`(Z). Example 2 Pullbacks. Denote by W2m a symplectic space of dimension 2m. Then W2n1 ⊕W2n2 = W2(n1+n2) defines a symplectic space of dimension 2n1 +2n2 by taking a direct sum of the symplectic pairings.
Load more

Recommended publications
  • An Approach to BPS Black Hole Microstate Counting in an N= 2 STU
    An approach to BPS black hole microstate counting in an N = 2 STU model G. L. Cardoso, S. Nampuri and D. Polini Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected], [email protected], [email protected] ABSTRACT We consider four-dimensional dyonic single-center BPS black holes in the N = 2 STU model of Sen and Vafa. By working in a region of moduli space where the real part of two of the three complex scalars S; T; U are taken to be large, we evaluate the quantum entropy function for these BPS black holes. In this regime, the subleading corrections point to a microstate counting formula partly based on a Siegel modular form of weight two. This is supplemented by another modular object that takes into account the dependence on Y 0, a complex scalar field belonging to one of the four off-shell vector multiplets of the underlying supergravity theory. We also observe interesting connections to the rational Calogero model and to formal deformation of a Poisson algebra, and suggest a string web picture of our counting proposal. arXiv:1903.07586v2 [hep-th] 20 Aug 2020 August 24, 2020 Contents 1 Introduction1 2 Quantum entropy function for BPS black holes in N = 2 supergravity theories 6 2.1 Generic structure6 2.2 Saddle-point approximation9 3 Quantum entropy function for the N = 2 STU model of Sen and Vafa 10 3.1 The N = 2 STU model with χ = 0 11 3.2 Properties
    [Show full text]
  • Quadratic Forms and Automorphic Forms
    Quadratic Forms and Automorphic Forms Jonathan Hanke May 16, 2011 2 Contents 1 Background on Quadratic Forms 11 1.1 Notation and Conventions . 11 1.2 Definitions of Quadratic Forms . 11 1.3 Equivalence of Quadratic Forms . 13 1.4 Direct Sums and Scaling . 13 1.5 The Geometry of Quadratic Spaces . 14 1.6 Quadratic Forms over Local Fields . 16 1.7 The Geometry of Quadratic Lattices – Dual Lattices . 18 1.8 Quadratic Forms over Local (p-adic) Rings of Integers . 19 1.9 Local-Global Results for Quadratic forms . 20 1.10 The Neighbor Method . 22 1.10.1 Constructing p-neighbors . 22 2 Theta functions 25 2.1 Definitions and convergence . 25 2.2 Symmetries of the theta function . 26 2.3 Modular Forms . 28 2.4 Asymptotic Statements about rQ(m) ...................... 31 2.5 The circle method and Siegel’s Formula . 32 2.6 Mass Formulas . 34 2.7 An Example: The sum of 4 squares . 35 2.7.1 Canonical measures for local densities . 36 2.7.2 Computing β1(m) ............................ 36 2.7.3 Understanding βp(m) by counting . 37 2.7.4 Computing βp(m) for all primes p ................... 38 2.7.5 Computing rQ(m) for certain m ..................... 39 3 Quaternions and Clifford Algebras 41 3.1 Definitions . 41 3.2 The Clifford Algebra . 45 3 4 CONTENTS 3.3 Connecting algebra and geometry in the orthogonal group . 47 3.4 The Spin Group . 49 3.5 Spinor Equivalence . 52 4 The Theta Lifting 55 4.1 Classical to Adelic modular forms for GL2 ..................
    [Show full text]
  • RESEARCH SUMMARY 1. Introduction My Current Research Concerns the Theory and Computation of Siegel Modular Forms. Complex Analyt
    RESEARCH SUMMARY CRIS POOR 1. Introduction My current research concerns the theory and computation of Siegel modular forms. Complex analytic techniques are the main tool al- though, besides being a part of several complex variables and tran- scendental algebraic geometry, the field of automorphic functions in- creasingly falls under the domain of number theory. I also publish results in the geometry of numbers, although I do not stray far from applications to Siegel modular forms. In the past I have published in the fields of mixed Hodge theory, Riemann surfaces, theta functions and particle coagulation. I may return to these fields and write up unpublished results but currently I am completely occupied with questions about Siegel modular forms. Let Γ be a group commensurable with Spg(Z). Siegel modular forms of weight k for Γ are holomorphic functions on the Siegel upper half space that are bounded at the cusps and invariant under an action of Γ that depends upon the weight k. They give mappings of Ag(Γ), the moduli space of principally polarized abelian varieties with level structure Γ, into projective spaces. The moduli space Ag has many important subvarieties such as modular curves, the hyperelliptic locus and the Jacobian locus. One important topic is the behavior of Siegel modular forms on these subvarieties; for example, the Schottky prob- lem asks for the ideal of Siegel forms vanishing on the Jacobian locus. Articles 1, 2, 3, 5 and 15 concern these subvarieties. The L-functions of Siegel modular Hecke eigenforms are a second important topic. As part of the Langlands program, we should look for generalizations of A.
    [Show full text]
  • Weight One Jacobi Forms and Umbral Moonshine 2
    Weight One Jacobi Forms and Umbral Moonshine Miranda C. N. Cheng∗ John F. R. Duncan† Jeffrey A. Harvey‡ Abstract We analyze holomorphic Jacobi forms of weight one with level. One such form plays an important role in umbral moonshine, leading to simplifications of the statements of the umbral moonshine conjectures. We prove that non-zero holomorphic Jacobi forms of weight one do not exist for many combinations of index and level, and use this to establish a characterization of the McKay–Thompson series of umbral moonshine in terms of Rademacher sums. arXiv:1703.03968v1 [math.NT] 11 Mar 2017 MSC2010: 11F22, 11F37, 11F46, 11F50, 20C34 ∗Institute of Physics and Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, the Netherlands. On leave from CNRS, France. E-mail: [email protected] †Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, U.S.A. E-mail: [email protected] ‡Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, IL 60637, U.S.A. E-mail: [email protected] 1 Weight One Jacobi Forms and Umbral Moonshine 2 Contents 1 Introduction 3 2 Umbral Moonshine 7 2.1 UmbralMoonshineModules. .. .. .. .. .. .. .. .. .. .. .. .. 7 2.2 UmbralMockModularForms.. .. .. .. .. .. .. .. .. .. .. .. 10 2.3 ParamodularForms .................................. 11 3 Weight One Jacobi Forms 13 3.1 WeilRepresentations................................ .. 15 3.2 Exponents........................................ 16 3.3 PrimePowerParts ................................... 18 3.4 ProofoftheMainResult ............................... 19 A Levels 27 B Characters at ℓ =9 31 C Coefficients at ℓ =9 32 D Decompositions at ℓ =9 36 Weight One Jacobi Forms and Umbral Moonshine 3 1 Introduction Umbral moonshine [1,2] attaches distinguished vector-valued mock modular forms to automor- phisms of Niemeier lattices.
    [Show full text]
  • A Finite Field Hypergeometric Function Associated to Eigenvalues of a Siegel Eigenform
    A FINITE FIELD HYPERGEOMETRIC FUNCTION ASSOCIATED TO EIGENVALUES OF A SIEGEL EIGENFORM DERMOT McCARTHY AND MATTHEW A. PAPANIKOLAS Abstract. Although links between values of finite field hypergeometric functions and eigenvalues of elliptic modular forms are well known, we establish in this paper that there are also connections to eigenvalues of Siegel modular forms of higher degree. Specifically, we relate the eigenvalue of the Hecke operator of index p of a Siegel eigenform of degree 2 and level 8 to a special value of a 4F3-hypergeometric function. 1. Introduction and Statement of Results One of the more interesting applications of hypergeometric functions over finite fields is their links to elliptic modular forms and in particular Hecke eigenforms [1, 3, 8, 11, 12, 13, 21, 22, 24, 27, 28]. It is anticipated that these links represent a deeper connection that also encompasses Siegel modular forms of higher degree, and the purpose of this paper is to provide new evidence in this direction. Specifically, we relate the eigenvalue of the Hecke operator of index p of a certain Siegel eigenform of degree 2 and level 8 to a special value of a 4F3-hypergeometric function over Fp. We believe this is the first result connecting hypergeometric functions over finite fields to Siegel modular forms of degree > 1. Hypergeometric functions over finite fields were originally defined by Greene [15], who first established these functions as analogues of classical hypergeometric functions. Functions of this type were also introduced by Katz [17] about the same time. In the present article we use a normalized version of these functions defined by the first author [23], which is more suitable for our purposes.
    [Show full text]
  • Notes on Siegel Modular Forms
    Notes on Siegel modular forms. Kevin Buzzard April 26, 2012 Last modified Oct 2005. 1 Definitions and the basics. this document was written in about 2004 when supervising Joshua Harris for his 4th year undergraduate project at Imperial. For simplicity I will stick to \full level structure", indicating why I've made this choice a bit later on. Let n ≥ 1 be an integer (the case n = 0 is sometimes useful but we shall refrain from doing empty-set theory and just give ad-hoc definitions). If A and B are 2n by 2n matrices then by the notation A[B], I mean (Bt)AB. Let I denote the n by n identity matrix and let J denote the 0 I 2n by 2n matrix −I 0 . For a ring R we define GSp(2n; R) to be the invertible 2n by 2n matrices M such that J[M] = ν(M)J, for ν(M) 2 R×, and Sp(2n; R) to be the kernel of ν. + We define Siegel upper half space Hn to be the set of n by n complex matrices Ω = X + iY such that Ωt = Ω and such that Y is positive definite. This space has dimension n(n + 1)=2 and useful coordinates are the (i; j)th entries of Ω for + AB −1 i ≤ j. Recall that Sp(2n; R) acts on Hn by CD Ω = (AΩ + B)(CΩ + D) . Here of course A; B; C; D denote n by n matrices. The proof that this is well- defined and an action is in my notes on Shimura varieties.
    [Show full text]
  • Modular Forms on G2: Lecture 1
    MODULAR FORMS ON G2: LECTURE 1 AARON POLLACK 1. Classical modular forms The purpose of this course is to (eventually) define and discuss modular forms on the exceptional group G2. I will not assume that you know what G2 is, and certainly I will not assume that you know what a modular form is on G2. Instead, the prerequisite for this course is basic graduate representation theory, complex analysis, algebra etc, and a familiarity with classical modular forms on GL2 or the upper half plane. Today I'll give an overview of what we will discuss over the course of the term, and the key results we will prove, time-permitting. Let me recall the definition of classical modular forms: SL2(R) acts on the upper half-plane h = fz 2 C : Im(z) > 0g −1 a b by fractional linear transformations γz = (az + b)(cz + d) for γ = c d 2 SL2(R). Suppose n ≥ 1, and f : h ! C is a holomorphic function. One says that f is a modular form of weight n if n • f(γz) = (cz + d) f(z) for all γ in a congruence subgroup of SL2(Z) • f(x + iy) is O(yN ) for some N as y ! 1 Recall that such an f has a Fourier expansion: if f is level 1 then f(z) = f(z + n) for all n 2 Z P n and this plus growth condition plus holomorphy implies f(z) = n≥0 af (n)q for some complex numbers af (n). Here q = exp(2πi), as usual.
    [Show full text]
  • Siegel Modular Forms of Genus 2 Attached to Elliptic Curves
    Math. Res. Lett. 14 (2007), no. 2, 315–332 c International Press 2007 SIEGEL MODULAR FORMS OF GENUS 2 ATTACHED TO ELLIPTIC CURVES Dinakar Ramakrishnan and Freydoon Shahidi Introduction The object of this article is to construct certain classes of arithmetically significant, holomorphic Siegel cusp forms F of genus 2, which are neither of Saito-Kurokawa type, in which case the degree 4 spinor L-function L(s, F ) is divisible by an abelian L-function, nor of Yoshida type, in which case L(s, F ) is a product of L-series of a pair of elliptic cusp forms. In other words, for each F in the class of holomorphic Siegel cusp forms we construct below, which includes those of scalar weight ≥ 3, the cuspidal automorphic representation π of GSp(4, A) generated by F is neither CAP, short for Cuspidal Associated to a Parabolic, nor endoscopic, i.e., one arising as the transfer of a cusp form on GL(2) × GL(2), and consequently, L(s, F ) is not a product of L-series of smaller degrees. A key example of the first kind of Siegel modular forms we construct is furnished by the following result, which will be used in [34]: Theorem A Let E be an non-CM elliptic curve over Q of conductor N. Then there exists a holomorphic, Hecke eigen-cusp form F of weight 3 acting on the 3-dimensional Siegel upper half space H2, such that L(s, F ) = L(s, sym3(E)). Moreover, F has the correct level, i.e., what is given by the conductor M of the Galois representation on the symmetric cube of the Tate module T`(E) of E.
    [Show full text]
  • On the Mod $ P $ Kernel of the Theta Operator and Eisenstein Series
    On the mod p kernel of the theta operator and Eisenstein series Shoyu Nagaoka and Sho Takemori Abstract Siegel modular forms in the space of the mod p kernel of the theta op- erator are constructed by the Eisenstein series in some odd-degree cases. Additionally, a similar result in the case of Hermitian modular forms is given. 1 Introduction The theta operator is a kind of differential operator operating on modular forms. Let F be a Siegel modular form with the generalized q-expansion F = a(T )qT , qT := exp(2πitr(TZ))). The theta operator Θ is defined as P Θ : F = a(T )qT Θ(F ) := a(T ) det(T )qT , 7−→ · which is a generalizationX of the classical Ramanujan’sX θ-operator. It is known that the notion of singular modular form F is characterized by Θ(F ) = 0. For a prime number p, the mod p kernel of the theta operator is defined as the set of modular form F such that Θ(F ) 0 (mod p). Namely, the element in the kernel of the theta operator can be interpreted≡ as a mod p analogue of the singular modular form. In the case of Siegel modular forms of even degree, several examples are known (cf. Remark 2.3). In [16], the first author constructed such a form by using Siegel Eisenstein series in the case of even degree. However little is known about the existence of such a modular form in the case of odd degree. arXiv:1708.00564v1 [math.NT] 2 Aug 2017 In this paper, we shall show that some odd-degree Siegel Eisenstein series give examples of modular forms in the mod p kernel of the theta operator (see Theorem 2.4).
    [Show full text]
  • Siegel Modular Forms: Classical Approach and Representation Theory
    Siegel modular forms: Classical approach and representation theory Ameya Pitale Contents 0. Introduction2 1. Lecture 1: Introduction to Siegel modular forms5 1.1. Motivation5 1.2. The symplectic group6 1.3. Siegel upper half space7 1.4. Siegel modular forms8 2. Lecture 2: Examples 11 2.1. Examples 11 2.2. Congruence subgroups 13 3. Lecture 3: Hecke Theory and L-functions 15 3.1. The Hecke algebra 15 3.2. Action of the Hecke algebra on Siegel modular forms 16 3.3. Relation between Fourier coefficients and Hecke eigenvalues for genus 2 18 3.4. L-functions 19 4. Lecture 4: Non-vanishing of Fourier coefficients and applications 21 4.1. Generalized Ramanujan conjecture 21 4.2. Non-vanishing of Fourier coefficients 22 4.3. Application of the non-vanishing result 24 4.4. Böcherer’s conjecture 25 5. Lecture 5: Applications of properties of L-functions 27 5.1. Determining cusp forms by size of Fourier coefficients 27 5.2. Sign changes of Hecke eigenvalues 28 5.3. Sign changes of Fourier coefficients 30 6. Lecture 6: Cuspidal automorphic representations corresponding to Siegel modular forms 32 6.1. Classical to adelic 32 6.2. Hecke equivariance 35 6.3. Satake isomorphism 36 6.4. Spherical representations 37 6.5. The representation associated to a Siegel cusp form 38 7. Lecture 7: Local representation theory of GSp4(Qp) 40 7.1. Local non-archimedean representations for GSp4 40 7.2. Generic representations 43 7.3. Iwahori spherical representations 45 7.4. Paramodular theory 47 8. Lecture 8: Bessel models and applications 53 8.1.
    [Show full text]
  • Explicit Forms for Koecher-Maass Series and Their Applications
    Explicit forms for Koecher-Maass series and their applications Yoshinori Mizuno Abstract The author gives a survey of one aspect about Koecher-Maass series mainly from his results on explicit forms for Koecher-Maass series and their applications. 1 Introduction In this survey, one aspect about Koecher-Maass series will be presented. In particular the author would like to explain his works on explicit forms for Koecher-Maass series and their applications. As for more general and wide topics, there exist a valuable proceedings \On Koecher-Maass series" [25] and beautiful papers \A survey on the new proof of Saito-Kurokawa lifting after Duke and Imamoglu" [23] by T. Ibukiyama, \Bemerkungen ¨uber die Dirichletreichen von Koecher und Maass" [7] by S. B¨ocherer. We recommend these articles to anyone interested in Koecher-Maass series. Initially this series was introduced by Maass [39] and Koecher [33] for holomorphic Siegel modular forms. Let X F (Z) = A(T;F ) exp(2πitr(TZ)) T ≥O be a holomorphic Siegel modular form of degree n, where the summation extends over all half- integral positive semi-definite symmetric matrices T of degree n. Then the Koecher-Maass series ηF (s) associated with F (Z) is defined by X A(T;F ) η (s) = ; F (T )(det T )s fT >Og=SLn(Z) where the summation extends over all half-integral positive definite symmetric matrices T of t degree n modulo the action T ! T [U] = UTU of the group SLn(Z) and (T ) = ]fU 2 SLn(Z); T [U] = T g is the order of the unit group of T .
    [Show full text]
  • Arxiv:Math/0605346V2 [Math.AG] 21 May 2007 Ru SL(2 Group Fhceoeaos Rmtefuircoefficients
    SIEGEL MODULAR FORMS GERARD VAN DER GEER Abstract. These are the lecture notes of the lectures on Siegel modular forms at the Nordfjordeid Summer School on Modular Forms and their Applications. We give a survey of Siegel modular forms and explain the joint work with Carel Faber on vector-valued Siegel modular forms of genus 2 and present evidence for a conjecture of Harder on congruences between Siegel modular forms of genus 1 and 2. 1. Introduction Siegel modular forms generalize the usual modular forms on SL(2, Z) in that the group SL(2, Z) is replaced by the automorphism group Sp(2g, Z) of a unimodular symplectic form on Z2g and the upper half plane is replaced by the Siegel upper half plane g. The integer g 1 is called the degree or genus. Siegel pioneeredH the generalization≥ of the theory of elliptic modular forms to the modular forms in more variables now named after him. He was motivated by his work on the Minkowski-Hasse principle for quadratic forms over the rationals, cf., [95]. He investigated the geometry of the Siegel upper half plane, determined a fundamental domain and its volume and proved a central result equating an Eisenstein series with a weighted sum of theta functions. No doubt, Siegel modular forms are of fundamental importance in number theory and algebraic geometry, but unfortunately, their reputation does not match their importance. And although vector-valued rather than scalar-valued Siegel modular forms are the natural generalization of elliptic modular forms, their reputation amounts to even less. A tradition of ill-chosen notations may have contributed to this, but the lack of attractive examples that can be handled decently seems to be the main responsible.
    [Show full text]