2015/03/31 by David Mehrle, Steven J. Miller, Tomer Reiter +2
Mathematics · #math.NT #msc:11G05 #msc:11G20 #msc:11G40 #msc:14G10
published as Minnesota Journal Of Undergraduate Mathematics, 2(1), 2017 · Version 1.0, 4 pages, sequel to arXiv:math/0406579
arxiv created 2017/06/03 · arxiv updated 2017/11/10
We generalize a construction of families of moderate rank elliptic curves over ℚ to number fields K/ℚ. The construction, originally due to Steven J. Miller, Álvaro Lozano-Robledo and Scott Arms, invokes a theorem of Rosen and Silverman to show that computing the rank of these curves can be done by controlling the average of the traces of Frobenius, the construction for number fields proceeds in essentially the same way. One novelty of this method is that we can construct families of moderate rank without having to explicitly determine points and calculating determinants of height matrices.