Here [[??].sub.i] are the base change maps; [??] is induced from the natural [alpha]-action on the bundle [C.sup.2] and gives an isogeny of degree det([alpha]) at each smooth fibre.
On the other hand, under [Laplace] [equivalent] [??]]K [K.sub.[??]], the pullback of weight 1 modular forms agrees with the det [([alpha]).sup.-1] multiple of the pullback of 1-forms on the fibres by the isogeny [alpha] : [S.sub.[[omega]]] [right arrow] [S.sub.[[omega][alpha]]], because [alpha] multiplies the symplectic form by det([alpha]).
Our characterization of the Igusa curves will depend on the isogeny invariance of the p-adic Galois character associated to an elliptic curve in characteristic p.
Specifically, [w.sub.N] maps the triple [mathematical Expressions Omitted] onto [Mathematical Expressions Omitted], where [Mathematical Expressions Omitted] is the isogeny which is dual to [phi].
Since [rho] : [D.sub.[xi] right arrow] [D'.sub.[xi]] is an isogeny
, if I'((D) is defined as in the remark after Theorem 3, then the absolute convergence of I'([phi]) for every [phi] in S([X.sub.A]) there is invariant under strong equivalence relation and castling transforms.
In this paper we shall study from the "differential algebraic viewpoint" the moduli spaces [A.sub.g,n] of principally polarized abelian F-varieties with level n structure; we will be especially concerned with understanding from this viewpoint the isogeny equivalence relation on [A.sub.g,n].
So we are provided with an equivalence relation on F called isogeny.
The Isogeny Theorem due to Faltings (, [section]5 Korollar 2) implies that
Then the Isogeny Theorem implies that [V.sub.l](E) and[V.sub.l](E') are isomorphic as [G.sub.F]'-modules for any prime '.
Let [Y.sup.0] [subset] Y be a neighbourhood of [t.sub.0] on which the isogeny [B.sub.t0] [right arrow] [B.sub.t0]/[H.sub.t0] can be extended to an isogeny:
[MATHEMATICAL EXPRESSION OMITTED] and by [[psi].sub.y] [[psi].sub.y]; = [rho] [omicron] [[phi].sub.y] (3.2) the composition with an isogeny [rho] in such a way that [A.sub.y] := [rho]([A'.sub.y]) is principally polarized (cf.
Ono [12,17] whose study of the behavior of [tau] under an isogeny
explains the presence of Pic(H), and reduces the semisimple case to the simply connected case.