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Quadratic polynomials, period polynomials, and Hecke operators

Volume 158 / 2013

Marie Jameson, Wissam Raji Acta Arithmetica 158 (2013), 287-297 MSC: Primary 11F67; Secondary 11F33. DOI: 10.4064/aa158-3-7

Abstract

For any non-square $1< D\equiv 0,1$ (mod $4$), Zagier defined $$ F_{k}(D;x) :=\sum_{\substack {a,b,c \in \mathbb {Z},\, a< 0\\ b^2-4ac=D }} \max(0,(ax^2+bx+c)^{k-1}). $$ Here we use the theory of periods to give identities and congruences which relate various values of $F_k(D;x).$

Authors

  • Marie JamesonDepartment of Mathematics
    and Computer Science
    Emory University
    Atlanta, GA 30322, U.S.A.
    e-mail
  • Wissam RajiDepartment of Mathematics
    American University of Beirut
    Beirut, Lebanon
    and fellow at
    Center for Advanced Mathematical Sciences
    Beirut, Lebanon
    e-mail

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