## 5-Cycle Systems with Holes by Bryant D.E., Hoffman D.G., Rodger C.A.

By Bryant D.E., Hoffman D.G., Rodger C.A.

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Number Theory 115 (2005), 100–122. [Zh2] Q. Zhang, Integral mean values of Maass L–functions, IMRN, Art. ID 41417, 19 pp. (2006).

Let Φν be any Schwartz function on kν2 invariant under Kν (under the obvious right action of GL2 ), and put ην (g) = χν (det g) | det g|sν · kν× χ2ν (t) |t|2s ν · Φν (t · e2 · g) dt where e2 = e2, ν is the second basis element in kν2 . This ην has the same left Pν –equivariance as ην , namely a ∗ ην · g = |a/d|sν · χν (a/d) · ην (g) 0 d For Φν invariant under the standard maximal compact Kν of GL2 (kν ), the function ην is right Kν –invariant. By the Iwasawa decomposition, up to constant multiples, there is only one such function, so ην (g) = ην (1) · ην (g) (since ην (1) = 1) and1 ην (1) = kν× χ2 (t) |t|2s · Φ(t · e2 · 1) dt = ζν (2s, χ2 , Φ(0, ∗)) (a Tate-Iwasawa zeta integral) 1 From now on, to avoid clutter, suppress the subscript ν where there is no risk of confusion.

Diaconu, P. Garrett and D. Goldfeld, The function Z3 (w) has a natural boundary, 6 pages, preprint, 2006. [DGG2] A. Diaconu, P. Garrett and D. Goldfeld, Integral Moments for GLr , in preparation. [DGH] A. Diaconu, D. Goldfeld and J. Hoffstein, Multiple Dirichlet series and moments of zeta and L– functions, Comp. Math 139 (2003), 297–360. 44 ADRIAN DIACONU PAUL GARRETT [Do] H. Donnelly, On the cuspidal spectrum for finite volume symmetric spaces, J. Diff. Geom. 17 (1982), 239–253. [GJ] S. Gelbart and H.