## Elementary Number Theory (2nd Edition) (Dover Books on by Underwood Dudley

By Underwood Dudley

Minimum necessities make this article perfect for a primary direction in quantity conception. Written in a full of life, attractive sort via the writer of well known arithmetic books, it positive aspects approximately 1,000 ingenious routines and difficulties. recommendations to some of the difficulties are integrated, and a teacher's advisor is accessible. 1978 version.

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7 There is a bijection (in fact a homeomorphism of Riemann surfaces) φ : C/Λ → EΛ (C) given by z → (℘(z), ℘ (z))(z ∈ Λ), z → ∞(z ∈ Λ). Proof: Ellipticity of ℘ and ℘ implies that φ is well-defined and (∗) shows that the image is in EΛ (C). To show surjectivity, given (x, y) ∈ EΛ (C) − {∞}, we consider ℘(z) − x, a nonconstant elliptic function with a pole (at 0) and so a zero, say at z = a. By (∗), ℘ (a)2 = y 2 . By oddness of ℘ and evenness of ℘, we see that φ(a) or φ(−a) is (x, y). To show injectivity, if φ(z1 ) = φ(z2 ) with 2z1 ∈ Λ, then consider ℘(z) − ℘(z1 ), which has a pole of order 2 and zeros at z1 , −z1 , z2 , so z2 ≡ ±z1 (mod Λ).

23 See [19], p. 336 on. ΦN (x) has coefficients in Z[j] and is irreducible over C(j) (and so is the minimal polynomial of jN over C(j)). The function field K(X0 (N )) = C(j, jN ). This enables us to define X0 (N ), a priori a curve over C, over Q. This means that it can be given by equations over Q. 4 Modular forms lxiii words, we have a model for X0 (N ) over Q with good reduction at primes not dividing N ). e. satisfy f (z) = f (σz) for all σ ∈ Γ0 (N ). This clearly holds for j since j is a modular function of weight 0 on all of SL2 (Z).

Proof: (a) Let σ ∈ G. Then σ acts on A, and sends m to m. Hence it acts on A/m = k. This defines a map φ : G → Gal(k/Fp ) by sending σ to the map x + m → σ(x) + m. We now examine the kernel of this map. ker φ = {σ ∈ G|σ(x) − x ∈ m for all x ∈ A} = {σ ∈ G|w(σ(x) − x) ≥ 1 for all x ∈ A} = G0 . This shows that φ induces an injective homomorphism from G/G0 to Gal(k/Fp ). As for surjectivity, choose a ∈ A such that the image a ¯ of a in k has k = Fp (¯ a). Let (x − σ(a)). p(x) = σ∈G Then p(x) is a monic polynomial with coefficients in A, and 3.