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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.

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