arXiv · 1401.7598
This is a survey of some of Erd\H os's work on bases in additive number theory.
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Melvyn B. Nathanson. 2014-01-29. Paul Erd\H os and additive bases. https://arxiv.org/abs/1401.7598
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Let L be the transfer operator associated with the Gauss' continued fraction map, known also as the Gauss-Kuzmin-Wirsing operator, acting on the Banach space. In this work we prove a two-term asymptotic formula for the eigenvalues of L, show their algebraic simplicity, sign alternation pattern, and decrease in absolute value. This settles, in a stronger form, the conjectures of D. Mayer and G. Roepstorff (1988), A.J. MacLeod (1992), Ph. Flajolet and B. Vallee (1995), also supported by several other authors. Further, we find an exact series for the eigenvalues, which also gives the canonical decomposition of trace formulas due to D. Mayer (1976) and K.I. Babenko (1978). This crystallizes the contribution of each individual eigenvalue in the trace formulas.
Let $ρ$ be a smooth $n$-dimensional representation of $\mathcal{G}_{\mathbb{Q}_p}$ over $\overline{\mathbb{F}_p}$. When $ρ$ is generic and a good conjugate, the article "Conjectures and results on modular representations of $\mathrm{GL}_n(K)$ for a $p$-adic field $K$", by Breuil-Herzig-Hu-Morra-Schraen, introduces the notion of an admissible representation $Π$ of $\mathrm{GL}_n(\mathbb{Q}_p)$ compatible with $ρ$. In loc. cit., the five authors also question whether there exists some $Π$ compatible with $ρ$ from which Zábrádi's functor $\mathbf{V}_Δ$ recovers a specific representation $\overline{L}^{\boxtimes}(ρ)$ of $\mathcal{G}_{\mathbb{Q}_p}^{n-1}$, constructed from $ρ$. We give a range of results about how badly $\mathbf{V}_Δ(Π)$ behaves for an arbitrary $Π$ satisfying some weaker compatibilities with $ρ$. In particular, when $ρ$ is reducible and $n\geq 3$, no representation $Π$ compatible with $\widetilde{P}_ρ$ can satisfy $\mathbf{V}_Δ(Π)\simeq \overline{L}^{\boxtimes}(ρ)$.
We show that up to isomorphism there are exactly twenty pairs $(C,E)$, where $C$ is a genus-$2$ curve over ${\mathbf C}$, where $E$ is an elliptic curve over ${\mathbf C}$, and where for every integer $n>1$ there is a map of degree $n$ from $C$ to $E$. We also show that for every genus-$2$ curve $C$, there is an integer $n$ with $1 < n \le 59$ such that there is no minimal degree-$n$ map from $C$ to an elliptic curve.