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Heegner points and the class number of imaginary quadratic fields Verones, Deanna Lynn
Abstract
Gauss' class number problem is that of finding an upper bound for |D| with given class number h(D) where D is a negative fundamental discriminant. A theorem of Goldfeld reduces the class number problem to finding an elliptic curve defined over Q with rank r > 3 which satisfies the Birch and Swinnerton-Dyer conjecture. A theorem of Gross and Zagier gives a method of predicting when a Heegner point yields rational point of infinite order on an elliptic curve. In some cases their theorem allows us to say for certain whether the derivative of the L-series of an elliptic curve vanishes. Applying their theorem to a particular elliptic curve with rank r = 3, Gross and Zagier were able to show that their curve satisfied the Birch and Swinnerton-Dyer conjecture, thus solving the class number problem. This thesis examines closely the theory of Heegner points including computational results varifying the Gross-Zagier theorem.
Item Metadata
Title |
Heegner points and the class number of imaginary quadratic fields
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
1999
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Description |
Gauss' class number problem is that of finding an upper bound for |D| with given
class number h(D) where D is a negative fundamental discriminant. A theorem of
Goldfeld reduces the class number problem to finding an elliptic curve defined over Q
with rank r > 3 which satisfies the Birch and Swinnerton-Dyer conjecture. A theorem
of Gross and Zagier gives a method of predicting when a Heegner point yields rational
point of infinite order on an elliptic curve. In some cases their theorem allows us to
say for certain whether the derivative of the L-series of an elliptic curve vanishes.
Applying their theorem to a particular elliptic curve with rank r = 3, Gross and
Zagier were able to show that their curve satisfied the Birch and Swinnerton-Dyer
conjecture, thus solving the class number problem. This thesis examines closely the
theory of Heegner points including computational results varifying the Gross-Zagier
theorem.
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Extent |
1913046 bytes
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Genre | |
Type | |
File Format |
application/pdf
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Language |
eng
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Date Available |
2009-06-29
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Provider |
Vancouver : University of British Columbia Library
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Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
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DOI |
10.14288/1.0080027
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
1999-11
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Campus | |
Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.