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The Modular Function
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abelian varieties adic algebraic closure assume Chapter characteristic coefficients commutative complex multiplication complex numbers congruence conjugate constant contains Corollary degree denote Deuring differential form discrete valuation divisor element elliptic curve elliptic curve defined elliptic function End(A endomorphism equation exists expansion factor finite number fixed follows Frobenius automorphism function field Galois extension Galois group given Hasse Hence homomorphism idele infinity integral closure invariant isogeny isomorphism kernel Kronecker lattice Lemma matrix maximal ideal modular function modular function field morphism non-degenerate reduction normalized number field open subgroup parametrization pole polynomial positive integer power series prime number prime to f Proof proper o-ideal proves our theorem purely inseparable rational ray class field reduction mod residue class field ring roots of unity Shimura splits completely subfield suffices to prove supersingular Suppose Tate Tp(A transcendental trivial unramified values Vp(A Weber function Weierstrass whence zero zeta function