On the automorphisms of the non-split Cartan modular curves of prime level

We study the automorphisms of the non-split Cartan modular curves $X_{ns}(p)$ of prime level $p$. We prove that if $p\geq 37$ all the automorphisms preserve the cusps. Furthermore, if $p\equiv 1\text{ mod }12$ and $p\neq 13$, the automorphism group is generated by the modular involution given by the normalizer of a non-split Cartan subgroup of $\text{GL}_2(\mathbb F_p)$. We also prove that for every $p\geq 37$ such that $X_{ns}(p)$ has a CM rational point, the existence of an exceptional rational automorphism would give rise to an exceptional rational point on the modular curve $X_{ns}^+(p)$ associated to the normalizer of a non-split Cartan subgroup of $\text{GL}_2(\mathbb F_p)$

Automorphisms of Cartan modular curves of prime and composite level

We study the automorphisms of modular curves associated to Cartan subgroups of $\mathrm{GL}_2(\mathbb Z/n\mathbb Z)$ and certain subgroups of their normalizers. We prove that if $n$ is large enough, all the automorphisms are induced by the ramified covering of the complex upper half-plane. We get new results for non-split curves of prime level $p\ge 13$: the curve $X_{\text{ns}}^+(p)$ has no non-trivial automorphisms, whereas the curve $X_{\text{ns}}(p)$ has exactly one non-trivial automorphism. Moreover, as an immediate consequence of our results we compute the automorphism group of $X_0^*(n):=X_0(n)/W$, where $W$ is the group generated by the Atkin-Lehner involutions of $X_0(n)$ and $n$ is a large enough square.Comment: 31 pages, 2 tables. Some proofs rely on MAGMA scripts available at https://github.com/guidoshore/automorphisms_of_Cartan_modular_curve

The automorphism group of the non-split Cartan modular curve of level 11

We derive equations for the modular curve $X_{ns}(11)$ associated to a non-split Cartan subgroup of $\,\mathrm{GL}_2(\mathbf{F}_{11})$. This allows us to compute the automorphism group of the curve and show that it is isomorphic to Klein's four group

Modular Curves with many Points over Finite Fields

We compute the number of points over finite fields of some classes of modular curves, namely $X_0(N)$, $X_0^+(N)$, without using explicit equations. In this way we could improve many lower bounds for the maximum number of points of a curve over finite fields

Marginalization using the metric of the likelihood

Although the likelihood function is normalizeable with respect to the data there is no guarantee that the same holds with respect to the model parameters. This may lead to singularities in the expectation value integral of these parameters, especially if the prior information is not sufficient to take care of finite integral values. However, the problem may be solved by obeying the correct Riemannian metric imposed by the likelihood. This will be demonstrated for the example of the electron temperature evaluation in hydrogen plasmas.Comment: 8 pages, 2 figures, Presented at the MaxEnt 2000 conference in Gif-sur-Yvette/Pari

Securing Household Income among Small-scale Farmers in Kakamega District: Possibilities and Limitations of Diversification

In the debate of sustainable rural livelihoods, diversification is seen as a way to secure incomes and to increase food security. On the basis of a data set on income security, this paper analyses to what extent this applies to small-scale farmers in Kakamega District, Kenya. Using the sustainable rural livelihoods approach, this paper draws the conclusion, that (1) diversification in agricultural production is not sufficient for securing rural livelihoods in Kakamega District; (2) a sufficient income diversification depends heavily on requirements like access to education, infrastructure, as well as investment capital; and (3) small-scale farmers in Kakamega District in most cases lack these requirements, therefore not being able to achieve secure household incomes or increased food security.In der Diskussion um nachhaltige ländliche Existenzgrundlagen wird Diversifizierung als ein Weg zur Erreichung von Einkommens- und Nahrungssicherheit angesehen. Auf der Grundlage eines Fragebogens zu Einkommenssicherheit analysiert dieser Beitrag, in welchem Umfang dies auf Kleinbauern im Distrikt Kakamega, Kenia zutrifft. Anhand des sustainable rural livelihood approach kommt der Beitrag zu folgenden Ergebnissen: 1) Die Diversifizierung der landwirtschaftlichen Produktion im Distrikt Kakamega reicht nicht aus, um sichere ländliche Existenzgrundlagen zu erhalten. 2) Erfolgreiche Einkommensdiversifizierung hängt sehr stark vom Zugang der Haushalte zu Bildung, Infrastruktur und Investitionskapital ab. 3) Kleinbauern im Untersuchungsgebiet stehen diese Grundlagen sehr selten zur Verfügung. Daher sind Einkommenssicherheit und ausreichende Nahrungssicherheit oft nicht gegeben