Mahler measures and pullbacks of isogenies

Specialeforsvar ved Leonard Grohmann

Titel: Mahler measures and pullbacks of isogenies

 Abstract: We will prove some equalities between Mahler measures of multivariable polynomials. First, we will prove an identity between a Mahler measures and an $L$-function using complex analysis. Afterwards, we will define basic definitions such as elliptic curves, Weierstrass equations, divisors of a curve and isogenies between curves. Elliptic curves over $\R$ and $\C$ will be considered. Two isogenies of degree 2 and 3 will be introduced, and the existence of the dual isogeny will be proven. By using pullbacks of isogenies and the theory of regulators, we will prove for \(g(\alpha)=m((1+x)(1+y)(x+y)-\alpha xy)\) and \(n(\alpha)=m(x^5y^5+1-\alpha xy), \alpha\in\Z,\) that \( g(p)=\frac{4}{3}n\left(\frac{p-2}{p^{1/3}}\right)+\frac{1}{3}n\left(\frac{p+4}{p^{2/3}}\right) \) holds for $|p|>8.$

Vejleder: Fabien Pazuki
Censor:   Simon Kristensen, Aarhus Universitet