Abstract: The canonical height of a point on an elliptic curve measures the arithmetic complexity of the point, with height zero characterizing points of finite order. An elliptic curve analogue of Lehmer's conjecture provides a lower bound for the height of non-torsion points defined over extension fields. In this talk I will give a survey of what is currently known about Lehmer's conjecture, discuss a method of Hindry-Silverman (1990) that uses local heights and Fourier averaging, and as time permits describe recent work of Nicole Looper and myself that generalizes the Fourier averaging method to abelian surfaces.
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