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We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for $100%$ of $G$-extensions of $k$, when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.
We present a method for constructing optimized equations for the modular curve X_1(N) using a local search algorithm on a suitably defined graph of birationally equivalent plane curves. We then apply these equations over a finite field F_q to efficie
Let E/F be a CM field split above a finite place v of F, let l be a rational prime number which is prime to v, and let S be the set of places of E dividing lv. If E_S denotes a maximal algebraic extension of E unramified outside S, and if u is a plac
Given a finite sequence of vectors $mathcal F_0$ in $C^d$ we characterize in a complete and explicit way the optimal completions of $mathcal F_0$ obtained by adding a finite sequence of vectors with prescribed norms, where optimality is measured with
In this paper we study the problem of how to determine all elliptic curves defined over an arbitrary number field $K$ with good reduction outside a given finite set of primes $S$ of $K$ by solving $S$-unit equations. We give examples of elliptic curves over $mathbb Q$ and quadratic fields.
Let $mathcal F_0={f_i}_{iinmathbb{I}_{n_0}}$ be a finite sequence of vectors in $mathbb C^d$ and let $mathbf{a}=(a_i)_{iinmathbb{I}_k}$ be a finite sequence of positive numbers. We consider the completions of $cal F_0$ of the form $cal F=(cal F_0,cal