CHAPTER I AUTOMORPHIC FORMS AN D FAMILIES OF ABELIAN VARIETIES 1. Algebraic preliminaries 1.1. The algebraic or Lie groups we treat in this book are symplectic and unitary, and the hermitian symmetric domains associated with them belong to Types A and C. Our methods are in fact applicable to groups and domains of other types, but it is naturally cumbersome to treat all cases. Therefore, in order to keep the book a reasonable length, we confine ourselves to those two types, though at some points we shall indicate that other cases can be handled in a similar way by citing relevant papers. We take a basic field F of characteristic different from 2 and a couple (K, p) consisting of an F-algebra K of rank 2 and an F-linear automorphism p of K belonging to the following three types: (I) K = F and p = idF (II) K is a quadratic extension of F and p is the generator of Gal(K/F) (III) i^ = F x F a n d (x, y)p = (y, x) for (x, y) G F x F. In our later discussion, objects of type (III) will appear as the localizations of the global objects of type (II). Given left if-modules V and W, we denote by Hom(W, V K) the set of all K- linear maps of W into V. We then put End(V, K) = Hom(V, V] K), GL(V, K) = End(y, K)x, and SX(V, K) = { a G GL(V, K) | det(a) = 1 } . We drop the letter K if that is clear from the context. We let Hom(W, V) act on W on the right namely we denote by wa the image of w G W under a G Hom(W, V). Let V be a left K-module isomorphic to K^. Given e = ±1, by an e-hermitian form on V we understand an F-linear map if : V x V K such that (1.1) tp(x, y)p = s(p{y, x), (1.2) p(ax, by) = cup(x, y)bp for every a, b G K. Assuming p to be nondegenerate, we define groups GU(ip) U(ip), and SU(tp) by (1.3) GC%) = { a G GL(V, K) \ p(xa, ya) = v(a)ip(x, y) with v(a) G Fx }, (1.4) t/(^) = { a G Gtffc) | i/(a) = 1 }, SC%) = t % ) n SL(F, # ) . We call ip isotropic if ^(x, x) = 0 for some x G V, 7^ 0 we call p anisotropic if y?(x, x) = 0 only for x = 0. Given (V, p) and another structure (V7, (//) of the same type, we denote by (V, p)® (V, (£') the structure (IV, -0) given by W = V e V " and ij)(x + x\ y + y') = 7
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