Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCounting Number Fields in Fibers
128
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let X be a projective curve over Q and t a non-constant Q-rational function on X of degree n>1. For every integer a pick a points P(a) on X such that t(P(a))=a. Dvornicich and Zannier (1994) proved that for large N the field Q(P(1), ..., P(N)) is of degree at least exp(cN/log N) over Q, where c>0 depends only on X and t. In this note we extend this result, replacing Q by an arbitrary number field.
rate research
Read More
Let X be an algebraic curve over Q and t a non-constant Q-rational function on X such that Q(t) is a proper subfield of Q(X). For every integer n pick a point P_n on X such that t(P_n)=n. We conjecture that, for large N, among the number fields Q(P_1), ..., Q(P_N) there are at least cN distinct. We prove this conjecture in the special case when t defines a geometrically abelian covering of the projective line, and the critical values of t are all rational. This implies, in particular, that our conjecture follows from a famous conjecture of Schinzel.
For a cubic algebraic extension $K$ of $mathbb{Q}$, the behavior of the ideal counting function is considered in this paper. Let $a_{K}(n)$ be the number of integral ideals of the field $K$ with norm $n$. An asymptotic formula is given for the sum $$ sumlimits_{n_{1}^2+n_{2}^2leq x}a_{K}(n_{1}^2+n_{2}^2). $$
We explore whether a root lattice may be similar to the lattice $mathscr O$ of integers of a number field $K$ endowed with the inner product $(x, y):={rm Trace}_{K/mathbb Q}(xcdottheta(y))$, where $theta$ is an involution of $K$. We classify all pairs $K$, $theta$ such that $mathscr O$ is similar to either an even root lattice or the root lattice $mathbb Z^{[K:mathbb Q]}$. We also classify all pairs $K$, $theta$ such that $mathscr O$ is a root lattice. In addition to this, we show that $mathscr O$ is never similar to a positive-definite even unimodular lattice of rank $leqslant 48$, in particular, $mathscr O$ is not similar to the Leech lattice. In appendix, we give a general cyclicity criterion for the primary components of the discriminant group of $mathscr O$.
Let X be a projective curve defined over Q and t a non-constant Q-rational function on X of degree at least 2. For every integer n pick a point P_n on X such that t(P_n)=n. A result of Dvornicich and Zannier implies that, for large N, among the number fields Q(P_1),...,Q(P_N) there are at least cN/log N distinct, where c>0. We prove that there are at least N/(log N)^{1-c} distinct fields, where c>0.
Consider an algebraic number field, $K$, and its ring of integers, $mathcal{O}_K$. There exists a smallest $B_K>1$ such that for any $x>1$ we can find a prime ideal, $mathfrak{p}$, in $mathcal{O}_K$ with norm $N(mathfrak{p})$ in the interval $[x,B_Kx]$. This is a generalization of Bertrands postulate to number fields, and in this paper we produce bounds on $B_K$ in terms of the invariants of $K$ from an effective prime ideal theorem due to Lagarias and Odlyzko. We also show that a bound on $B_K$ can be obtained from an asymptotic estimate for the number of ideals in $mathcal{O}_K$ less than $x$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
