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  Modular points, modular curves, modular surfaces and modular forms

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.

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Item Permalink: https://hdl.handle.net/21.11116/0000-0004-396B-5 Version Permalink: https://hdl.handle.net/21.11116/0000-0004-396C-4
Genre: Book Chapter

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Zagier, Don1, Author           
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1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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 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.

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 Dates: 1985
 Publication Status: Issued
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 Rev. Type: Internal
 Identifiers: eDoc: 744958
Other: 111
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Title: Arbeitstagung 1984 : Proceedings of the meeting held by the Max-Planck-Institut für Mathematik, Bonn, June 15-22, 1984
Source Genre: Book
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Publ. Info: Berlin : Springer
Pages: 481 S. Volume / Issue: - Sequence Number: - Start / End Page: 225 - 248 Identifier: -

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Title: Lecture Notes in Mathematics
Source Genre: Series
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Pages: - Volume / Issue: 1111 Sequence Number: - Start / End Page: - Identifier: -