Mahler measure of 3D Landau–Ginzburg potentials

We express the Mahler measures of 23 families of Laurent polynomials in terms of Eisenstein–Kronecker series. These Laurent polynomials arise as Landau–Ginzburg potentials on Fano 3-folds, sixteen of which define K ⁢ 3 {K3} hypersurfaces of generic Picard rank 19, and the rest are of generic Picard rank less than 19. We relate the Mahler measure at each rational singular moduli to the value at 3 of the L-function of some weight-3 newform. Moreover, we find ten exotic relations among the Mahler measures of these families.

Singular units and isogenies between CM elliptic curves

In this note, we will apply the results of Gross–Zagier, Gross–Kohnen–Zagier and their generalizations to give a short proof that the differences of singular moduli are not units. As a consequence, we obtain a result on isogenies between reductions of CM elliptic curves.

Completions and algebraic formulas for the coefficients of Ramanujan’s mock theta functions

We present completions of mock theta functions to harmonic weak Maass forms of weight $$\nicefrac {1}{2}$$ 1 2 and algebraic formulas for the coefficients of mock theta functions. We give several harmonic weak Maass forms of weight $$\nicefrac {1}{2}$$ 1 2 that have mock theta functions as their holomorphic part. Using these harmonic weak Maass forms and the Millson theta lift, we compute finite algebraic formulas for the coefficients of the appearing mock theta functions in terms of traces of singular moduli.

Separating singular moduli and the primitive element problem

We prove that ${|x-y|\ge 800X^{-4}}$, where $x$ and $y$ are distinct singular moduli of discriminants not exceeding $X$. We apply this result to the ‘primitive element problem’ for two singular moduli. In a previous article, Faye and Riffaut show that the number field ${{\mathbb{Q}}}(x,y)$, generated by two distinct singular moduli $x$ and $y$, is generated by ${x-y}$ and, with some exceptions, by ${x+y}$ as well. In this article we fix a rational number ${\alpha \ne 0,\pm 1}$ and show that the field ${{\mathbb{Q}}}(x,y)$ is generated by ${x+\alpha y}$, with a few exceptions occurring when $x$ and $y$ generate the same quadratic field over ${{\mathbb{Q}}}$. Together with the above-mentioned result of Faye and Riffaut, this generalizes a theorem due to Allombert et al. (2015) about solutions of linear equations in singular moduli.