Between Dec 19, 2024 and Jan 2, 2025, datasets can be submitted to DRUM but will not be processed until after the break. Staff will not be available to answer email during this period, and will not be able to provide DOIs until after Jan 2. If you are in need of a DOI during this period, consider Dryad or OpenICPSR. Submission responses to the UDC may also be delayed during this time.

Browsing by Subject "Automorphic forms"

Now showing 1 - 3 of 3
  • Thumbnail Image
    Item
    Automorphic forms on certain affine symmetric spaces.
    (2011-05) Zhang, Lei
    In this thesis, we consider automorphic periods associated to certain affine symmetric spaces such as the symmetric pairs. In this thesis, we consider automorphic periods associated to certain affine symmetric spaces such as the symmetric pairs (Sp4n; ResK=kSp2n) and (GSp4n; ResK=kGSp2n); where k is a number field and K is an Etale algebra over k of dimension 2. We consider the period integral of a cusp forms of Sp4n(Ak) against with an Eisenstein series of the symmetric subgroup ResK=kSp2n. We expect to establish an identity between this period integrals and the special value of the spin L-function of the symplectic group. In the local theory, using Aizenbud and Gourevitch's generalized Harish-Chandra method and traditional methods, i.e. the Gelfand-Kahzdan theorem, we can prove that these symmetric pairs are Gelfand pairs when Kv is a quadratic extension field over kv for any n, or Kv is isomorphic to kv x kv for n <_ 2. Since (U(J2n; kv(p #28; )); Sp2n(kv)) is a descendant of (Sp4n(kv); Sp2n(kv) #2; Sp2n(kv)), we prove that it is a Gelfand pair for both archimedean and non-archimedean fields. According to the Yu' construction in [76] of irreducible tame supercuspidal representations, we give a parametrization of the distinguished tame supercuspidal representation of symplectic groups in this thesis. Applying the dimension formula of the space HomH(#25;; 1) given by Hakim and Murnaghan [28], we prove that if (G;H) is the symmetric pair (U(J2n;Kv); Sp2n(kv)) there is no H-distinguished tame supercuspidal representation, where Kv is a quadratic extension over kv. In addition, for the symmetric pair (Sp4n(kv); Sp2n(Kv)), we give the sufficient and necessary conditions of generic cuspidal data such that the corresponding tame supercuspidal representations are H-distinguished. Note that our case is the first case worked out with none of G and H being the general linear groups. Furthermore, motivated by a sub-question, we also give an example for the distinguished representations of finite groups of Lie Type in a low rank case. In particular, we show that #18;10 is the unique SL2(Fq2) distinguished cuspidal representation of Sp4(Fq). Note: See PDF abstract for the correct interpretation of the mathematical symbols.
  • Thumbnail Image
    Item
    Automorphic Hamiltonians, Epstein zeta functions, and Kronecker limit formulas
    (2020-09) Sands, Adrienne
    We construct an automorphic Hamiltonian which has purely discrete spectrum on $L^2\left(SL_r(\Z)\backslash SL_r(\R)/SO(r,\R)\right)$, identify its ground state, and show how it can characterize a nuclear Fr\'echet automorphic Schwartz space.
  • Thumbnail Image
    Item
    Fourier coefficients of automorphic forms and arthur classification
    (2013-05) Liu, Baiying
    Fourier coefficients play important roles in the study of both classical modular forms and automorphic forms. For example, it is a well-known theorem of Shalika and Piatetski-Shapiro that cuspidal automorphic forms of GLn(A) are globally generic, that is, have non-degenerate Whittaker-Fourier coefficients, which is proved by taking Fourier expansion. For general connected reductive groups, there is a framework of attaching Fourier coefficients to nilpotent orbits. For general linear groups and classical groups, nilpotent orbits are parametrized by partitions. Given any automorphic representation π of general linear groups or classical groups, characterizing the set n^m(π) of maximal partitions with corresponding nilpotent orbits providing non-vanishing Fourier coefficients is an interesting question, and has applications in automorphic descent and construction of endoscopic lifting.