Download PDF by Fred Diamond, Jerry Shurman: A First Course in Modular Forms (Graduate Texts in

Download PDF by Fred Diamond, Jerry Shurman: A First Course in Modular Forms (Graduate Texts in

By Fred Diamond, Jerry Shurman

This publication introduces the speculation 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 thought; Hecke eigenforms and their mathematics homes; the Jacobians of modular curves and the Abelian types linked to Hecke eigenforms. because it offers those principles, the booklet states the Modularity Theorem in a number of kinds, bearing on them to one another and relating their purposes to quantity idea. The authors imagine no history in algebraic quantity thought and algebraic geometry. routines are incorporated.

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If N is the order of K as a subgroup then K ⊂ E[N ] ∼ = Z/N Z × Z/N Z, and so by the theory of finite Abelian groups K ∼ = Z/nZ × Z/nn Z for some positive integers n and n . The multiply-by-n isogeny [n] of C/Λ takes K to a cyclic subgroup nK isomorphic to Z/n Z, and then the quotient isogeny π from C/Λ to C/nK has kernel nK. Follow this by the map C/nK −→ C/Λ given by z + nK → (m/n)z + (m/n)nK, now viewing nK as a lattice in C. This map makes sense and is an isomorphism since (m/n)nK = mK = Λ .

5. 3 the desired lattice is Λ = mΛµ3 for a suitably chosen m. 3 similarly. 3 that two complex elliptic curves C/Λ and C/Λ are holomorphically group-isomorphic if and only if mΛ = Λ for some m ∈ C. Viewing two such curves as equivalent gives a quotient set of equivalence classes of complex elliptic curves. Similarly, view two points τ and τ of the upper half plane as equivalent if and only if γ(τ ) = τ for some γ ∈ SL2 (Z), and consider the resulting quotient set as well. This section shows that there is a bijection from the first quotient set to the second.

E[5]: the 5-torsion points of a torus Cyclic quotient maps are also isogenies but not isomorphisms. Let C/Λ be a complex torus, let N be a positive integer, and let C be a cyclic subgroup of E[N ] isomorphic to Z/N Z. The elements of C are cosets {c + Λ} and so as a set C forms a superlattice of Λ. Slightly abusing notation we use the same symbol for the subgroup and the superlattice. Then the cyclic quotient map π : C/Λ −→ C/C, z+Λ→z+C 28 1 Modular Forms, Elliptic Curves, and Modular Curves is an isogeny with kernel C.

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A First Course in Modular Forms (Graduate Texts in Mathematics) by Fred Diamond, Jerry Shurman

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