scholarly journals Bounds on $2$-torsion in class groups of number fields and integral points on elliptic curves

2020 ◽  
Vol 33 (4) ◽  
pp. 1087-1099 ◽  
Author(s):  
M. Bhargava ◽  
A. Shankar ◽  
T. Taniguchi ◽  
F. Thorne ◽  
J. Tsimerman ◽  
...  
Author(s):  
N. P. SMART ◽  
N. M. STEPHENS

In recent years there has been an interest in using elliptic logarithms to find integral points on elliptic curves defined over the rationals, see [23], [17], [6] and [12]. This has been partly due to work of David [5], who gave an explicit lower bound for linear forms in elliptic logarithms. Previously, integral points on elliptic curves had been found by Siegel's method; that is, a reduction to a set of Thue equations which could be solved, in principle, by the methods in [19]. For examples of this method see [3], [7], [16], [18], [21], [22] and [8]. Other techniques can be used to find all integral points in some special cases, see, for instance, [14].


2015 ◽  
Vol 111 (2) ◽  
pp. 484-518 ◽  
Author(s):  
Xavier Guitart ◽  
Marc Masdeu ◽  
Mehmet Haluk Şengün

2010 ◽  
Vol 13 ◽  
pp. 370-387
Author(s):  
Sonal Jain

AbstractWe determine the smallest possible canonical height$\hat {h}(P)$for a non-torsion pointPof an elliptic curveEover a function field(t) of discriminant degree 12nwith a 2-torsion point forn=1,2,3, and with a 3-torsion point forn=1,2. For eachm=2,3, we parametrize the set of triples (E,P,T) of an elliptic curveE/with a rational pointPandm-torsion pointTthat satisfy certain integrality conditions by an open subset of2. We recover explicit equations for all elliptic surfaces (E,P,T) attaining each minimum by locating them as curves in our projective models. We also prove that forn=1,2 , these heights are minimal for elliptic curves over a function field of any genus. In each case, the optimal (E,P,T) are characterized by their patterns of integral points.


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