Number Theory

Some applications of modular forms by Peter Sarnak

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By Peter Sarnak

The idea of modular varieties and particularly the so-called 'Ramanujan Conjectures' have lately been utilized to solve difficulties in combinatorics, machine technology, research and quantity conception. This tract, according to the Wittemore Lectures given at Yale college, is worried with describing a few of these purposes. so as to retain the presentation kind of self-contained, Professor Sarnak starts off via constructing the required heritage fabric in modular varieties. He then considers the answer of 3 difficulties: the Ruziewicz challenge referring to finitely additive rotationally invariant measures at the sphere; the specific development of hugely attached yet sparse graphs: 'expander graphs' and 'Ramanujan graphs'; and the Linnik challenge in regards to the distribution of integers that symbolize a given huge integer as a sum of 3 squares. those functions are conducted intimately. The ebook hence may be obtainable to a large viewers of graduate scholars and researchers in arithmetic and desktop technological know-how.

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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.

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