 
Summary: ON INVARIANT SUBSPACES AND EIGENFUNCTIONS
FOR REGULAR HECKE OPERATORS ON
SPACES OF MULTIPLE THETA CONSTANTS
Anatoli Andrianov and Fedor Andrianov
Abstract. Invariant subspaces and eigenfunctions for regular Hecke operators actinng
on spaces spanned by products of even number of Igusa theta constants with rational
characteristics are constructed. For some of the eigenfunctions of genuses g = 1 and
2, the corresponding zeta functions of Hecke and Andrianov are explicitely calculated.
Introduction
The Igusa theta constant of genus g N with characteristic m C2g is the
function on the upper halfplane of genus g,
Hg
= Z = X + iY Cg
g
t
Z = Z, Y > 0 ,
defined by the series
m(Z) =
nZg
exp i (n + m )Z t
