Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNumber fields with prescribed norms
85
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 efficiently generate elliptic curves with nontrivial N-torsion by searching for affine points on X_1(N)(F_q), and we give a fast method for generating curves with (or without) a point of order 4N using X_1(2N).
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 place of E dividing v, we show that any field embedding E_S to bar{E_u} has a dense image. The unramified outside S number fields we use are cut out from the l-adic cohomology of the simple Shimura varieties studied by Kottwitz and Harris-Taylor. The main ingredients of the proof are then the local Langlands correspondence for GL_n, the main global theorem of Harris-Taylor, and the construction of automorphic representations with prescribed local behaviours. We explain how stronger results would follow from the knowledge of some expected properties of Siegel modular forms, and we discuss the case of the Galois group of a maximal algebraic extension of Q unramified outside a single prime p and infinity.
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 respect to majorization (of the eigenvalues of the frame operators of the completed sequence). Indeed, we construct (in terms of a fast algorithm) a vector - that depends on the eigenvalues of the frame operator of the initial sequence $cF_0$ and the sequence of prescribed norms - that is a minimum for majorization among all eigenvalues of frame operators of completions with prescribed norms. Then, using the eigenspaces of the frame operator of the initial sequence $cF_0$ we describe the frame operators of all optimal completions for majorization. Hence, the concrete optimal completions with prescribed norms can be obtained using recent algorithmic constructions related with the Schur-Horn theorem. The well known relation between majorization and tracial inequalities with respect to convex functions allow to describe our results in the following equivalent way: given a finite sequence of vectors $mathcal F_0$ in $C^d$ we show that the completions with prescribed norms that minimize the convex potential induced by a strictly convex function are structural minimizers, in the sense that they do not depend on the particular choice of the convex potential.
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 $cal F=(cal F_0,cal G)$ obtained by appending a sequence $cal G={g_i}_{iinmathbb{I}_k}$ of vectors in $mathbb C^d$ such that $|g_i|^2=a_i$ for $iinmathbb{I}_k$, and endow the set of completions with the metric $d(cal F,tilde {mathcal F}) =max{ ,|g_i-tilde g_i|: iinmathbb{I}_k}$ where $tilde {cal F}=(cal F_0,,tilde {cal G})$. In this context we show that local minimizers on the set of completions of a convex potential $text{P}_varphi$, induced by a strictly convex function $varphi$, are also global minimizers. In case that $varphi(x)=x^2$ then $text{P}_varphi$ is the so-called frame potential introduced by Benedetto and Fickus, and our work generalizes several well known results for this potential. We show that there is an intimate connection between frame completion problems with prescribed norms and frame operator distance (FOD) problems. We use this connection and our results to settle in the affirmative a generalized version of Strawns conjecture on the FOD.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
