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Modular points, modular curves, modular surfaces and modular forms

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Zagier,  Don
Max Planck Institute for Mathematics, Max Planck Society;

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Zagier, D. (1985). Modular points, modular curves, modular surfaces and modular forms. In Arbeitstagung 1984: Proceedings of the meeting held by the Max-Planck-Institut für Mathematik, Bonn, June 15-22, 1984 (pp. 225-248). Berlin: Springer.


Cite as: http://hdl.handle.net/21.11116/0000-0004-396B-5
Abstract
[For the entire collection see Zbl 0547.00007.] \\par This is the written version of a talk at the Arbeitstagung at Bonn. It is centered around one example: the modular curve X\\sb 0(37). The elliptic curve E:\\quad y(y-1)=(x+1)x(x-1) is a factor of the Jacobian J\\sb 0(37). The article treats special values of L-series attached to E and its twists, Heegner points on E, the Gross-Zagier theorem and illustrates the interplay between classical algebraic geometry over \\bbfC and Arakelov geometry over \\bbfZ. It also gives an extension of the Gross-Zagier result: \\sum P\\sb dq\\sp d\\quad is a modular form of weight 3/2 and level 37. Here P\\sb d is the Heegner point on X\\sb 0(37) associated to d. This has now been proved for arbitrary N (rather than for N=37) by Gross/Kohnen/Zagier. The proof for the special case treated here uses an ad hoc method. This article is written to wet one's appetite and no doubt it will.