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2-Selmer groups and Heegner points o...

Li, Chao.

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2-Selmer groups and Heegner points on elliptic curves.

Record Type:	Electronic resources : Monograph/item
Title/Author:	2-Selmer groups and Heegner points on elliptic curves./
Author:	Li, Chao.
Description:	104 p.
Notes:	Source: Dissertation Abstracts International, Volume: 77-04(E), Section: B.
Contained By:	Dissertation Abstracts International77-04B(E).
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3738882
ISBN:	9781339293875

2-Selmer groups and Heegner points on elliptic curves.
Li, Chao.

2-Selmer groups and Heegner points on elliptic curves. - 104 p.

Source: Dissertation Abstracts International, Volume: 77-04(E), Section: B.

Thesis (Ph.D.)--Harvard University, 2015.

This thesis studies several aspects of the arithmetic of elliptic curves. In particular, we explore the prediction of the Birch and Swinnerton-Dyer conjecture when the 2-Selmer group has rank one.

ISBN: 9781339293875Subjects--Topical Terms:

515831
Mathematics.

2-Selmer groups and Heegner points on elliptic curves.
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100 1 $a Li, Chao. $3 1263714
245 1 0 $a 2-Selmer groups and Heegner points on elliptic curves.
300 $a 104 p.
500 $a Source: Dissertation Abstracts International, Volume: 77-04(E), Section: B.
500 $a Adviser: Benedict Gross.
502 $a Thesis (Ph.D.)--Harvard University, 2015.
520 $a This thesis studies several aspects of the arithmetic of elliptic curves. In particular, we explore the prediction of the Birch and Swinnerton-Dyer conjecture when the 2-Selmer group has rank one.
520 $a For certain elliptic curves E/Q : y2 = F(x) with additive reduction at 2, we determine their 2-Selmer ranks in terms of the 2-rank of the class group of the cubic field L = Q[x]/F(x). We then interpret this result as a mod 2 congruence between the Hasse-Weil L-function of E and a degree two Artin L-function associated to the cubic field L.
520 $a When the class number of L is odd, the Birch and Swinnerton-Dyer conjecture predicts that E should have rank one over Q(i). To construct such a point on E, we study Heegner points on Shimura curves with non-maximal level at a prime p ramified in the quaternion algebra (in the special case when p = 2). These curves have a p-adic uniformization by a tame etale covering of Drinfeld's p-adic half-plane. We use the covering to describe the geometry of their reduction mod p and compute the Neron model of their Jacobians.
520 $a For certain elliptic curves E/Q with good or multiplicative reduction at 2, we study their 2-Selmer groups over imaginary quadratic fields using the method of level raising of modular forms mod p = 2. We prove a parity result (predicted by the Birch and Swinnerton-Dyer conjecture) for 2-Selmer ranks. We also show that there is an obstruction for lowering the 2-Selmer ranks, revealing a different phenomenon compared to odd p.
590 $a School code: 0084.
650 4 $a Mathematics. $3 515831
690 $a 0405
710 2 $a Harvard University. $b Mathematics. $3 2049814
773 0 $t Dissertation Abstracts International $g 77-04B(E).
790 $a 0084
791 $a Ph.D.
792 $a 2015
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3738882