# Read e-book online p-Adic Automorphic Forms on Shimura Varieties PDF

By Haruzo Hida

This booklet covers the subsequent 3 issues in a way available to graduate scholars who've an knowing of algebraic quantity conception and scheme theoretic algebraic geometry:

1. An hassle-free development of Shimura types as moduli of abelian schemes.

2. p-adic deformation idea of automorphic kinds on Shimura varieties.

3. an easy evidence of irreducibility of the generalized Igusa tower over the Shimura variety.

The publication starts off with a close examine of elliptic and Hilbert modular types and reaches to the leading edge of study of Shimura kinds linked to normal classical teams. the strategy of making p-adic analytic households and the facts of irreducibility used to be lately came across through the writer. the world coated during this e-book is now a focus of analysis around the world with many far-reaching functions that experience ended in strategies of longstanding difficulties and conjectures. in particular, using p-adic elliptic and Hilbert modular varieties have confirmed crucial in fresh breakthroughs in quantity idea (for instance, the facts of Fermat's final Theorem and the Shimura-Taniyama conjecture through A. Wiles and others).

Haruzo Hida is Professor of arithmetic at college of California, l. a.. His past books comprise Modular types and Galois Cohomology (Cambridge college Press 2000) and Geometric Modular varieties and Elliptic Curves (World medical Publishing corporation 2000).

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**Example text**

Thus t'p/tp E V; and if we write the power series expansion of f with respect to tp as f(tp), then we get f(t'p) by substituting tp(t'p) for tp (check that this substitution gives a well-defined power series in t'p because tp(P) = 0). The divisor div(f) of f -=J 0 is defined by a formal sum Lp vp(f)P (in some classical books in number theory, they use multiplicative notation, such as div(f) = TIp pVP(J), because it corresponds to the prime decomposition of the ideal (f), but we use additive symbols following the geometric tradition).

X. The prime p is totally ramified in G, and the prime factor p of p is principal and is generated by (pn - 1. In particular, Ip = Dp = Gal(Q[(pn ]/Q) for the inertia subgroup Ip C Dp. 3 Adelic Reformulation Recall that p and q are distinct primes, and put N commutative diagram = pmqn. Then we have a where Ip denotes the inertia group at p. Since p is unramified in Q[(qn]' we find Ip n Iq = {I}. ,y by XN. 8 The cyclotomic character XN induces an isomorphism of groups. The decomposition group of a prime q f N is cyclic and generated by the Frobenius element ¢q with XN (¢q) = q E (Z/ NZ) x.

R "'" K(V). R trivial over K X with residue field K}. R' . R = for the valuation ring Vp corresponding to P E V'. Vrr(P) Proof. R for a general algebraic function field over a field K. We start with K(p l ) = K(x) with one transcendental element x. The main tool is the fact that K[x] is a principal ideal domain. Pick a nonzero prime ideal P of K[x]. Pick f E P, and factorize f into a product f = c TIt t(xy(t) of irreducible monic polynomials t(x) with a constant C E KX. Since P is a prime, one of the irreducible factors t(x) has to be in P.