arXiv subjects

Peter Stevenhagen

Publications and source records attributed to Peter Stevenhagen.

13 recordsLinked to original sources

arXiv2023

Locally imprimitive points on elliptic curves

Under GRH, any element in the multiplicative group of a number field $K$ that is globally primitive (i.e., not a perfect power in $K^*$) is a primitive root modulo a set of primes of $K$ of positive density. For elliptic curves $E/K$ that are known to have infinitely many primes $\mathfrak p$ of cyclic reduction, possibly under GRH, a globally primitive point $P\in E(K)$ may fail to generate any of the point groups $E(k_{\mathfrak p})$. We describe this phenomenon in terms of an associated Galois representation $ρ_{E/K, P}:G_K\to\mathrm{GL}_3(\hat{\mathbf Z})$, and use it to construct non-trivial examples of global points on elliptic curves that are locally imprimitive.

math.NT↗

arXiv2022

Cyclic reduction densities for elliptic curves

For an elliptic curve $E$ defined over a number field $K$, the heuristic density of the set of primes of $K$ for which $E$ has cyclic reduction is given by an inclusion-exclusion sum $δ_{E/K}$ involving the degrees of the $m$-division fields $K_m$ of $E$ over $K$. This density can be proved to be correct under assumption of GRH. For $E$ without complex multiplication (CM), we show that $δ_{E/K}$ is the product of an explicit non-negative rational number reflecting the finite entanglement of the division fields of $E$ and a universal infinite Artin-type product. For $E$ admitting CM over $K$ by a quadratic order ${\mathcal{O}}$, we show that $δ_{E/K}$ admits a similar `factorization' in which the Artin type product also depends on ${\mathcal{O}}$. For $E$ admitting CM over $\bar K$ by an order ${\mathcal{O}}\not\subset K$, which occurs for $K={\bf Q}$, the entanglement of division fields over $K$ is non-finite. In this case we write $δ_{E/K}$ as the sum of two contributions coming from the primes of $K$ that are split and inert in ${\mathcal{O}}$. The split contribution can be dealt with by the previous methods, the inert contribution is of a different nature. We determine the ways in which the density can vanish, and provide numerical examples of the different kinds of densities.

math.NT↗

arXiv2021

Redei reciprocity, governing fields, and negative Pell

We discuss the origin, an improved definition and the key reciprocity property of the trilinear symbol introduced by Rédei in the study of 8-ranks of narrow class groups of quadratic number fields. It can be used to show that such 8-ranks are governed by Frobenius conditions on the primes dividing the discriminant, a fact used the recent work of A. Smith. In addition, we explain its impact in the progress towards proving my conjectural density for solvability of the negative Pell equation.

math.NT↗

arXiv2021

Adelic point groups of elliptic curves

We show that for an elliptic curve E defined over a number field K, the group E(A) of points of E over the adele ring A of K is a topological group that can be analyzed in terms of the Galois representation associated to the torsion points of E. An explicit description of E(A) is given, and we prove that for K of degree n, almost all elliptic curves over K have an adelic point group topologically isomorphic to a universal group depending on n. We also show that there exist infinitely many elliptic curves over K having a different adelic point group.

math.NT↗

arXiv2014

Genus-2 curves and Jacobians with a given number of points

We study the problem of efficiently constructing a curve C of genus 2 over a finite field F for which either the curve C itself or its Jacobian has a prescribed number N of F-rational points. In the case of the Jacobian, we show that any `CM-construction' to produce the required genus-2 curves necessarily takes time exponential in the size of its input. On the other hand, we provide an algorithm for producing a genus-2 curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus-2 curve having exactly 10^2014 + 9703 (prime) points, and two genus-2 curves each having exactly 10^2013 points. In an appendix we provide a complete parametrization, over an arbitrary base field k of characteristic neither 2 nor 3, of the family of genus-2 curves over k that have k-rational degree-3 maps to elliptic curves, including formulas for the genus-2 curves, the associated elliptic curves, and the degree-3 maps.

math.NT↗

arXiv2013

Imaginary quadratic fields with isomorphic abelian Galois groups

In 1976, Onabe discovered that, in contrast to the Neukirch-Uchida results that were proved around the same time, a number field $K$ is not completely characterized by its absolute abelian Galois group $A_K$. The first examples of non-isomorphic $K$ having isomorphic $A_K$ were obtained on the basis of a classification by Kubota of idele class character groups in terms of their infinite families of Ulm invariants, and did not yield a description of $A_K$. In this paper, we provide a direct `computation' of the profinite group $A_K$ for imaginary quadratic $K$, and use it to obtain many different $K$ that all have the same minimal absolute abelian Galois group.

math.NT↗

arXiv2008

Computational class field theory

Class field theory furnishes an intrinsic description of the abelian extensions of a number field that is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such extensions.

math.NT↗

arXiv2008

Abelian Varieties with Prescribed Embedding Degree

We present an algorithm that, on input of a CM-field $K$, an integer $k\ge1$, and a prime $r \equiv 1 \bmod k$, constructs a $q$-Weil number $π\in Ø_K$ corresponding to an ordinary, simple abelian variety $A$ over the field $\F$ of $q$ elements that has an $\F$-rational point of order $r$ and embedding degree $k$ with respect to $r$. We then discuss how CM-methods over $K$ can be used to explicitly construct $A$.

math.NT↗

arXiv2007

Constructing elliptic curves of prime order

We present a very efficient algorithm to construct an elliptic curve E and a finite field F such that the order of the point group E(F) is a given prime number N. Heuristically, this algorithm only takes polynomial time Otilde((\log N)^3), and it is so fast that it may profitably be used to tackle the related problem of finding elliptic curves with point groups of prime order of prescribed size. We also discuss the impact of the use of high level modular functions to reduce the run time by large constant factors and show that recent gonality bounds for modular curves imply limits on the time reduction that can be obtained.

math.NT↗

arXiv2005

Constructing elliptic curves in almost polynomial time

We present an algorithm that, on input of a positive integer N together with its prime factorization, constructs a finite field F and an elliptic curve E over F for which E(F) has order N. Although it is unproved that this can be done for all N, a heuristic analysis shows that the algorithm has an expected run time that is polynomial in 2^omega(N) log N, where omega(N) is the number of distinct prime factors of N. In the cryptographically relevant case where N is prime, an expected run time O((log N)^{4+epsilon}) can be achieved. We illustrate the efficiency of the algorithm by constructing elliptic curves with point groups of order N=10^2004 and N=nextprime(10^{2004})=10^{2004}+4863.

math.NT↗

arXiv2003

Principal moduli and class fields

We study the values taken by Gamma_0(n) modular functions at elliptic points of order 2 for the Fricke extension that lie outside Gamma_0(n). In the case of a principal modulus (`Hauptmodul') for Gamma_0(n) or its Fricke extension, we determine the class fields generated by these values.

math.NT↗

arXiv1999

Prime divisors of the Lagarias sequence

For integer a let us consider the sequence X_a={x_0,x_1,x_2,...} defined by x_0=a, x_1=1 and, for n>=1, x_{n+1}=x_n+x_{n-1}. We say that a prime p divides X_a if p divides at least one term of the sequence. It is easy to see that every prime p divides X_1, the sequence of Fibonacci numbers. Lagarias, using a technique involving the computation of degrees of various Kummerian extensions first employed by Hasse, showed in 1985 that X_2, the set of primes dividing some Lucas number has natural density 2/3 and posed as a challenge finding the density of prime divisors of X_3. In this paper we resolve this challenge, assuming GRH, by showing that the density of X_3 equals 1573727S/1569610, with S the so called Stephens constant. This is the first example of a `non-torsion' second order recurrent sequence with irreducible recurrence relation for which we can determine the associated density of prime divisors.

math.NT↗

arXiv1999

A two variable Artin conjecture

Let a and b be non-zero rational numbers that are multiplicatively independent. We study the natural density of the set of primes p for which the subgroup of the multiplicative group of the finite field with p elements generated by (a\mod p) contains (b\mod p). It is shown that, under assumption of the generalized Riemann hypothesis (GRH), this density exists and equals a positive rational multiple of the universal constant S=\prod_{p prime}(1-p/(p^3-1)). An explicit value of the density is given under mild conditions on a and b. This extends and corrects earlier work of P.J. Stephens (1976). Our result, in combination with earlier work of the second author, allows us to deduce that any second order linear recurrence with reducible characteristic polynomial having integer elements, has a positive density of prime divisors (under GRH).

math.NT↗