Residual Galois representations of elliptic curves with image contained in the normaliser of a non-split Cartan

It is known that if $p>37$ is a prime number and $E/\mathbb{Q}$ is an elliptic curve without complex multiplication, then the image of the mod $p$ Galois representation $$\bar{\rho}_{E,p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}(E[p])$$ of $E$ is either the whole of $\operatorname{GL}(E[p])$, or is \emph{contained} in the normaliser of a non-split Cartan subgroup of $\operatorname{GL}(E[p])$. In this paper, we show that when $p>1.4\times 10^7$, the image of $\bar{\rho}_{E,p}$ is either $\operatorname{GL}(E[p])$, or the \emph{full} normaliser of a non-split Cartan subgroup. We use this to show the following result, partially settling a question of Najman. For $d\geq 1$, let $I(d)$ denote the set of primes $p$ for which there exists an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication admitting a degree $p$ isogeny defined over a number field of degree $\leq d$. We show that, for $d\geq 1.4\times 10^7$, we have $$ I(d)=\{p\text{ prime}:p\leq d-1\}. $

Torsion of Q-curves over quadratic fields

We determine all the possible torsion groups of Q-curves overquadratic fields and determine which groups appear finitely andwhich appear infinitely often

Millimeter wave carbon nanotube gas sensor

This Letter reports experimental observations regarding the significant changes in the transmission modulus and phase of the propagating microwave signals up to 110 GHz in a micromachined coplanar waveguide supported on a dielectric membrane with a thickness of 1.4 m filled with a mixture of carbon nanotubes when exposed to nitrogen gas. These large shifts of amplitude and phase of microwave signals due to gas absorption represent the experimental basis on which a miniature wireless gas sensor could be implemented

Torsion of \Q-curves over quadratic fields

We find all the possible torsion groups of \Q-curves over quadratic fields and determine which groups appear finitely and which appear infinitely often.Comment: 11 pages, to appear in Math. Res. Letter

Experimental determination of microwave attenuation and electrical permittivity of double-walled carbon nanotubes

The attenuation and the electrical permittivity of the double-walled carbon nanotubes (DWCNTs) were determined in the frequency range of 1–65 GHz. A micromachined coplanar waveguide transmission line supported on a Si membrane with a thickness of 1.4 µm was filled with a mixture of DWCNTs. The propagation constants were then determined from the S parameter measurements. The DWCNTs mixture behaves like a dielectric in the range of 1–65 GHz with moderate losses and an abrupt change of the effective permittivity that is very useful for gas sensor detection. ©2006 American Institute of Physic

Explicit height estimates for CM curves of genus 2

In this paper we make explicit the constants of Habegger and Pazuki's work from 2017 on bounding the discriminant of cyclic Galois CM fields corresponding to genus 2 curves with CM by them and potentially good reduction outside a predefined set of primes. We also simplify some of the arguments

Torsion bounds for a fixed abelian variety and varying number field

Let $A$ be an abelian variety defined over a number field $K$. For a finite extension $L/K$, the cardinality of the group $A(L)_{\operatorname{tors}}$ of torsion points in $A(L)$ can be bounded in terms of the degree $[L:K]$. We study the smallest real number $\beta_A$ such that for any finite extension $L/K$ and $\varepsilon>0$, we have $|A(L)_{\operatorname{tors}}| \leq C \cdot [L:K]^{\beta_A+\varepsilon}$, where the constant $C$ depends only on $A$ and $\varepsilon$ (and not $L$). Assuming the Mumford-Tate conjecture for $A$, we will show that $\beta_A$ agrees with the conjectured value of Hindry and Ratazzi.Comment: 29 pages, comments very welcome