Quadratic polynomials, period polynomials, and Hecke operators
Volume 158 / 2013
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 $F_k(D;x).$
