Elliptic curves over the complex field: lattice-elliptic curve correspondence

Abstract: 

In this talk we discuss the approximation of transcendental numbers by algebraic numbers of given degree and bounded height. More precisely, for any real number xx, by w∗n(x)wn∗(x) we define the supremum of all positive real values ww such that the inequality

|x−a|<H(a)−w−1|x−a|<H(a)−w−1

has infinitely many solutions in algebraic real numbers aa of degree at most nn. Here H(a)H(a) means the naive height of the minimal polynomial in Z[x]Z[x] with coprime coefficients. In 1961, Wirsing asked: is it true that the quantity w∗n(x)wn∗(x) is at least n for all transcendental xx? Apart from partial results for small values of nn, this problem still remains open. Wirsing himself managed to establish the lower bound of the form w∗n(x)≥n/2+1−o(1)wn∗(x)≥n/2+1−o(1). Until recently, the only improvements to this bound were in terms of O(1)O(1). I will talk about our resent work with Schleischitz where we managed to improve the bound by a quantity of the size O(n)O(n). More precisely, we show that w∗n(x)>n/3–√wn∗(x)>n/3.

This talk is part of the online Number Theory Web Seminar, and will be streamed live on Zoom.

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Organisers:

Mike Bennett (University of British Columbia)

Philipp Habegger (University of Basel)

Alina Ostafe (UNSW Sydney)