International Congress of Mathematicians

Eisenstein Series and Arithmetic Geometry 181structure fj, and an action of Oc ® Ok, where Oc is a maximal order in the Cliffordalgebra C(V) of V and Ok is the ring of integers in the quadratic field k = Q(Vd) ford = discr(V), the discriminant field of V, [14]. Again, a space V(A, i) = V(A, X, i, fj)of special endomorphisms is defined; it is a Z-module of finite rank equipped witha positive definite quadratic form Q. For T £ Sym r (Z), we let Z(T) be the locusof (A,\,i,fj,x)'s where x = [xi,... ,x r ], xi £ V(A,i) is a collection of r specialendomorphisms with matrix of inner products Q(x) = |((XJ,£J)) = ^ •One would like to define a family of generating functions according to the followingconjectural chart. Again there is a metrized Hodge line bundle û £ CH(A4).r = l, Z(t)q = HZ - curved(r) = w+? + ]T t#0 Jj(£, v)