By Fred Diamond, Jerry Shurman

This e-book introduces the idea of modular varieties, from which all rational elliptic curves come up, with a watch towards the Modularity Theorem. dialogue covers elliptic curves as advanced tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves; Hecke operators and Atkin-Lehner idea; Hecke eigenforms and their mathematics houses; the Jacobians of modular curves and the Abelian kinds linked to Hecke eigenforms. because it provides those rules, the publication states the Modularity Theorem in numerous kinds, referring to them to one another and pertaining to their functions to quantity thought. The authors think no historical past in algebraic quantity idea and algebraic geometry. routines are integrated.

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For one more example, some complex tori have endomorphisms other than the multiply-by-N maps [N ], in which case they have complex multiplication. √ Let τ = d for some squarefree d ∈ Z− such that d ≡ 2, 3 (mod 4), or √ let τ = (−1 + d)/2 for squarefree d ∈ Z− , d ≡ 1 (mod 4). Then the set O = τ Z ⊕ Z is a ring. ) Let Λ be any ideal of O and let m be any element of O. Then mΛ ⊂ Λ, so multiplying by m gives an endomorphism of C/Λ. In particular, the ring of endomorphisms of C/Λi is isomorphic to Λi = iZ ⊕ Z rather than to Z, and similarly for the ring of endomorphisms of C/Λµ3 where µ3 = e2πi/3 .

3(b–d) for more properties of the Weil pairing, in particular that the Weil pairing is preserved under isomorphisms of complex tori. 5. 1. 1. 2. 3. 3. (a) Show that the Weil pairing is independent of which basis {ω1 , ω2 } is used, provided ω1 /ω2 ∈ H. (b) Show that the Weil pairing is bilinear, alternating, and nondegenerate. ) (c) Show that the Weil pairing is compatible with N . This means that for positive integers N and d, the diagram E[dN ] × E[dN ] edN (·,·) / µdN d(·,·) E[N ] × E[N ] eN (·,·) ·d / µN commutes, where the vertical maps are suitable multiplications by d.

The relation between ℘ and ℘ from (b) shows that the corresponding values xi = ℘(zi ) for i = 1, 2, 3 are roots of the cubic polynomial pΛ (x) = 4x3 − g2 (Λ)x − g3 (Λ), so it factors as claimed. 1(b)), this makes the three xi distinct. That is, the cubic polynomial pΛ has distinct roots. Part (b) of the proposition shows that the map z → (℘Λ (z), ℘Λ (z)) takes nonlattice points of C to points (x, y) ∈ C2 satisfying the nonsingular cubic equation of part (c), y 2 = 4x3 −g2 (Λ)x−g3 (Λ). The map bijects since generally a value x ∈ C is taken by ℘Λ twice on C/Λ, that is, x = ℘Λ (±z +Λ), and then the two y-values satisfying the cubic equation are ℘ (±z + Λ) = ±℘ (z + Λ).