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For all totally positive algebraic numbers α except a finite number of explicit exceptions, the following inequality holds: 1 d (α 1 + · · · + α d) > max(1.780022, 1.66 + α 1), where d is the degree of α and 0 < α 1 < · · · < α d its conjugates. This improves previous results of Smyth, Flammang and Rhin.

There are 26 possibilities for the torsion group of elliptic curves dened over quadratic number elds. We present examples of high rank elliptic curves with given torsion group which give the current records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for Z/15Z, there exist an elliptic curve over… (More)

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