 
Summary: ACTION OF HECKE OPERATORS ON PRODUCTS OF IGUSA
THETA CONSTANTS WITH RATIONAL CHARACTERISTICS
Anatoli Andrianov (St. Petersburg)
and Fedor Andrianov (Los Angeles)
Abstract. We derive explicit formulas for the images of products of Igusa theta
constants with rational characteristics under the action of regular Hecke operators.
In particular, we prove that when the class number of the sum of 2k squares is one,
images of products of 2k Igusa theta constants with rational characteristics under
the action of regular Hecke operators are, in general, linear combinations of similar
products with explicitly given coefficients.
Introduction
In 1980 H.Yoshida ([Yo(80)], p.243) proposed a twodimensional analog of the
famous ShimuraTaniyama relation between Hasse zeta functions of elliptic curves
over the field of rational numbers and Hecke zeta functions of elliptic modular
forms. In confirmation of the Yoshida conjecture, R.SalvatiManni and J.Top have
considered in [SMT(93)] a number of products of four Igusa theta constants with
rational characteristics which (in part hypothetically) are eigenfunctions of all regu
lar Hecke operators with Andrianov zeta function coinciding (up to a finite number
of Euler factors) with Hasse zeta function of appropriate twodimensional Abelian
variety.
