Borcherds Forms and Generalizations of Singular Moduli
dc.contributor.advisor | Kudla, Stephen S | en_US |
dc.contributor.author | Schofer, Jarad | en_US |
dc.contributor.department | Mathematics | en_US |
dc.contributor.publisher | Digital Repository at the University of Maryland | en_US |
dc.contributor.publisher | University of Maryland (College Park, Md.) | en_US |
dc.date.accessioned | 2005-08-03T13:58:53Z | |
dc.date.available | 2005-08-03T13:58:53Z | |
dc.date.issued | 2005-04-18 | en_US |
dc.description.abstract | In the first part of this thesis, we prove an explicit formula for the average of a Borcherds form over CM points associated to a quadratic form of signature (n, 2). One step in the proof extends a theorem of Kudla to the case n = 0. The formula we obtain involves the negative Fourier coefficients of a modular form F, and the second terms in the Laurent expansions (at s = 0) of the Fourier coefficients of an Eisenstein series of weight one. These Laurent expansion terms were calculated by Kudla, Rapoport and Yang in a special case. We extend their results to a more general case. In the second part of this thesis, we consider examples of our main theorem for n = 0 and n = 1 in more detail. When n = 0, we let k be an imaginary quadratic field and we obtain a function on the product of the ideal class group of k with the squares of the ideal class group of k. The example for n = 1 allows us to reproduce the well-known singular moduli result of Gross and Zagier. This result gives an explicit factorization of the function J(D, d), defined as a product of j(z)-j(w) over points z and w of discriminant D and d, respectively, where D and d are negative relatively prime fundamental discriminants. | en_US |
dc.format.extent | 590415 bytes | |
dc.format.mimetype | application/pdf | |
dc.identifier.uri | http://hdl.handle.net/1903/2417 | |
dc.language.iso | en_US | |
dc.subject.pqcontrolled | Mathematics | en_US |
dc.title | Borcherds Forms and Generalizations of Singular Moduli | en_US |
dc.type | Dissertation | en_US |
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