This isogeny invariance property of the character [[chi].sub.j] is the key to Theorems 1.9 and 1.11.

Thus hypothesis 2 may be interpreted as a requirement that the extension K/[K.sub.0] have an isogeny invariance property similar to that of the Igusa extension [K.sub.[infinity]/[K.sub.0].

In particular, for any [phi] [element of] Aut ([G.sub.0]) there is a unique continuous k-linear automorphism [[sigma].sub.[phi]] of R such that [phi] lifts to an isogeny

This formula expresses the isogeny invariance of the character XG.

Let x [element of] F be a point on the line and denote by I(x) [subset] F the isogeny class of x (set of all y [element of] F isogenous to x).

Note that both "strata" are saturated with respect to isogeny. It is trivial to check that if x [element of] C we must have I(x)* = C.

There exists a [Delta]-regular function [Chi] : F\C [approaches] F of order 3 which is constant on each isogeny class and has the following properties:

(5) Any other [Delta]-regular function [Eta] : F\C [approaches] F which is constant on all isogeny classes may be written in the form [Eta](x) = A([Chi](x)), x [element of] F\C for a suitable [Delta]-polynomial A [element of] F{y}.

Then the

Isogeny Theorem (see [section]1) implies that [V.sub.l](E) [equivalent] [V.sub.l](E') as [G.sub.F']-modules for each l [member of] S'.

Moreover Nori [N] (see also [B] for a different approach) proves that, for A generic, [Gr.sup.2](A) is infinitely generated by showing that the cycles [[C.sub.[alpha]]] - [C??] are algebraically independent, where [C.sub.[alpha]] is the image under an isogeny [r.sub.[alpha]] : j([C'.sub.[alpha]]) [right arrow] A of some Abel-Jacobi embedded curve [C'.sub.[alpha]] in J([C'.sub.[alpha]]).

(4.1) Associated to it there is a fiber map: [R.sub.[alpha]] : A [right arrow] [rho]??A which restricted to the fibers is an isogeny.

[MATHEMATICAL EXPRESSION OMITTED] For a fixed PPA threefold A = [A.sub.p] corresponding to the point p [element of] [H.sub.3] the cycle [[C.sub.[alpha]] - [C??], considered by Nori, defined by the isogeny [r.sub.[alpha]] : J([C.sub.[alpha]]) [right arrow] A, corresponds, for t [element of] [M.sub.3], [[pi].sub.[alpha]](t) = p and C := [C.sub.t], to the cycle