Related papers: Counting extensions of imaginary quadratic fields
The aim of this work is to analyze general infinite sums containing modified Bessel functions of the second kind. In particular we present a method for the construction of a proper asymptotic expansion for such series valid when one of the…
We consider the problem of simultaneous approximation to a number and to its square in a general framework that encompasses imaginary quadratic number fields and fields of rational functions in one variable. In this context, we construct…
We provide an asymptotic estimate for the number of rational points of bounded height on a non-singular conic over the rationals. The estimate is uniform in the coefficients of the underlying quadratic form.
We produce an infinite family of imaginary quadratic fields whose ideal class groups have $3$-rank at least $2$.
Counting integral binary quadratic forms with certain restrictions is a classical problem. In this paper, we count binary quadratic forms of fixed discriminant given restrictions on the size of their coefficients. We accomplish this by…
The theory of continued fractions of functions $ \sqrt D $ is used to give lower bound for class numbers $h(D)$ of general real quadratic function fields $K=k(\sqrt D)$ over $k={\bf F}_q(T)$. For five series of real quadratic function…
We establish an asymptotic formula for the number of integral solutions of bounded height for pairs of diagonal quartic equations in $26$ or more variables. In certain cases, pairs in $25$ variables can be handled.
We generalise remarks of Euler and of Perron by explaining how to detail all quadratic irrational integers for which the symmetric part of the period of their continued fraction expansion commences with prescribed partial quotients. The…
We present an improved algorithm for tabulating class groups of imaginary quadratic fields of bounded discriminant. Our method uses classical class number formulas involving theta-series to compute the group orders unconditionally for all…
For a given odd positive integer $n$ and an odd prime $p$, we construct an infinite family of quadruples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$, $\mathbb{Q}(\sqrt{d+4})$ and…
In this paper we give a survey of recent methods for the asymptotic and exact enumeration of number fields with given Galois group of the Galois closure. In particular, the case of fields of degree up to 4 is now almost completely solved,…
We discuss continued fractions on real quadratic number fields of class number 1. If the field has the property of being 2-stage euclidean, a generalization of the euclidean algorithm can be used to compute these continued fractions.…
We generalize the explicit quadratic Chabauty techniques for integral points on odd degree hyperelliptic curves and for rational points on genus 2 bielliptic curves to arbitrary number fields using restriction of scalars. This is achieved…
We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the…
Let $\ell$ be a prime number and $q$ be a power of $\ell$. Given an odd prime number $p$ and an imaginary quadratic extension $F$ of the rational function field $\mathbb{F}_q(T)$, let $\lambda_p(F)$ denote the Iwasawa $\lambda$-invariant of…
In this paper we extend methods of Rubin to prove the Gras conjecture for abelian extensions of a given imaginary quadratic field k and prime numbers p which divide the number of roots of unity in k.
We give defining equations for function fields over finite fields with many rational places. They are obtained from composita of quadratic extensions of the rational function field.
In this paper, we construct infinitely many quadruples of real quadratic fields whose class numbers are all divisible by $3$. To the best of our knowledge, this is the first result towards the divisibility of the class numbers of certain…
We present two distinct families of imaginary biquadratic fields, each of which contains infinitely many members, with each member having large class groups. Construction of the first family involves elliptic curves and their quadratic…
In this paper, we give a survey of the known results concerning the tensor rank of the multiplication in finite extensions of finite fields, enriched with some not published recent results as well as analyzes enhancing the qualitative…