In the $p^n$-th cyclotomic field ${\mathbb{Q}}_{p^n},p$ a prime number, $n\in \mathbb{N}$, the prime $p$ is totally ramified and the only ideal above $p$ is generated by ${\omega }_n={\zeta }_{p^n}-1$, with the primitive $p^n$-th root of unity ${\zeta }_{p^n}=e^{\frac{2\pi i}{p^n}}$. Moreover these numbers represent a norm coherent set, i.e. ${\mathbf{\text{N}}}_{{\mathbb{Q}}_{p^{n+1}}}/{\mathbb{Q}}_{p^n}\left({\omega }_{n+1}\right)={\omega }_n$. It is the aim of this article to establish a similar result for the ray class field $K_{{𝔭}^n}$ of conductor ${𝔭}^n$ over an imaginary quadratic number field $K$ where ${𝔭}^n$ is the power of a prime ideal in $K$. Therefore the exponential function has to be replaced by a suitable elliptic...

For a prime $p$ and positive integers $\ell \<k\<h\<p$ with $d=(h,k,\ell ,p-1)$, we show that $M$, the number of simultaneous solutions $x,y,z,w$ in ${\mathbb{Z}}_p^{*}$ to $x^h+y^h=z^h+w^h$, $x^k+y^k=z^k+w^k$, $x^{\ell }+y^{\ell }=z^{\ell }+w^{\ell }$, satisfies $${\displaystyle M\le 3d^2{(p-1)}^2+25hk\ell (p-1).}$$ When $hk\ell =o\left(p{d}^2\right)$ we obtain a precise asymptotic count on $M$. This leads to the new twisted exponential sum bound $${\displaystyle \left|\sum _{x=1}^{p-1}\chi \left(x\right)e^{2\pi if\left(x\right)/p}\right|\le 3^{\frac{1}{4}}{d}^{\frac{1}{2}}{p}^{\frac{7}{8}}+\sqrt{5}{\left(hk\ell \right)}^{\frac{1}{4}}{p}^{\frac{5}{8}},}$$ for trinomials $f=a{x}^h+b{x}^k+c{x}^{\ell }$, and to results on the average size of such sums.

Let $K$ be a field of degree $n$ over ${\mathbf{Q}}_p$, the field of rational $p$-adic numbers, say with residue degree $f$, ramification index $e$ and differential exponent $d$. Let $O$ be the ring of integers of $K$ and $\mathit{P}$ its unique prime ideal. The trace and norm maps for $K/{\mathbf{Q}}_p$ are denoted $Tr$ and $N$, respectively. Fix $q=p^r$, a power of a prime $p$, and let $\eta$ be a numerical character defined modulo $q$ and of order $o\left(\eta \right)$. The character $\eta$ extends to the ring of $p$-adic integers ${\mathbb{Z}}_p$ in the natural way; namely $\eta \left(u\right)=\eta \left(\tilde{u}\right)$, where $\tilde{u}$ denotes the residue...

We provide a generalization of Scholz’s reciprocity law using the subfields $K_{2^{t-1}}$ and $K_{2^t}$ of $\mathbb{Q}\left({\zeta }_p\right)$, of degrees $2^{t-1}$ and $2^t$ over $\mathbb{Q}$, respectively. The proof requires a particular choice of primitive element for $K_{2^t}$ over $K_{2^{t-1}}$ and is based upon the splitting of the cyclotomic polynomial ${\Phi }_p\left(x\right)$ over the subfields.

A well known theorem of Mestre and Schoof implies that the order of an elliptic curve $E$ over a prime field ${𝔽}_q$ can be uniquely determined by computing the orders of a few points on $E$ and its quadratic twist, provided that $q\>229$. We extend this result to all finite fields with $q\>49$, and all prime fields with $q\>29$.