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Peter Bruin

Publications and source records attributed to Peter Bruin.

At least 19 recordsLinked to original sources
arXiv2025

Hidden shift problem for complex functions

We study quantum algorithms for the hidden shift problem of complex scalar- and vector-valued functions on finite abelian groups. Given oracle access to a shifted function and the Fourier transform of the unshifted function, the goal is to find the hidden shift. We analyze the success probability of our algorithms when using a constant number of queries. For bent functions, they succeed with probability 1, while for arbitrary functions the success probability depends on the `bentness' of the function.

quant-ph↗
arXiv2023

The Directed Van Kampen Theorem in Lean

Directed topology is an area of mathematics with applications in concurrency. It extends the concept of a topological space by adding a notion of directedness, which restricts how paths can evolve through a space and enables thereby a faithful representation of computation with their direction. In this paper, we present a Lean formalisation of directed spaces and a Van Kampen theorem for them. This theorem allows the calculation of the homotopy type of a space by combining local knowledge the homotopy type of subspaces. With this theorem, the reasoning about spaces can be reduced to subspaces and, by representing concurrent systems as directed spaces, we can reduce the deduction of properties of a composed system to that of subsystems. The formalisation in Lean can serve to support computer-assisted reasoning about the behaviour of concurrent systems.

cs.LO↗
arXiv2023

Counting rational points on weighted projective spaces over number fields

Deng (arXiv:math/9812082) gave an asymptotic formula for the number of rational points on a weighted projective space over a number field with respect to a certain height function. We prove a generalization of Deng's result involving a morphism between weighted projective spaces, allowing us to count rational points whose image under this morphism has bounded height. This method provides a more general and simpler proof for a result of the first-named author and Najman on counting elliptic curves with prescribed level structures over number fields. We further include some examples of applications to modular curves.

math.NT↗
arXiv2022

Extensions and torsors for finite group schemes

We give an explicit description of the category of central extensions of a group scheme by a sheaf of Abelian groups. Based on this, we describe a framework for computing with central extensions of finite commutative group schemes, torsors under such group schemes and groups of isomorphism classes of these objects.

math.AG↗
arXiv2022

Counting elliptic curves with prescribed level structures over number fields

Harron and Snowden counted the number of elliptic curves over $\mathbb{Q}$ up to height $X$ with torsion group $G$ for each possible torsion group $G$ over $\mathbb{Q}$. In this paper we generalize their result to all number fields and all level structures $G$ such that the corresponding modular curve $X_G$ is a weighted projective line $\mathbb{P}(w_0,w_1)$ and the morphism $X_G\to X(1)$ satisfies a certain condition. In particular, this includes all modular curves $X_1(m,n)$ with coarse moduli space of genus $0$. We prove our results by defining a size function on $\mathbb{P}(w_0,w_1)$ following unpublished work of Deng, and working out how to count the number of points on $\mathbb{P}(w_0,w_1)$ up to size $X$.

math.NT↗
arXiv2022

Hyperelliptic modular curves $X_0(n)$ and isogenies of elliptic curves over quadratic fields

Let $n$ be an integer such that the modular curve $X_0(n)$ is hyperelliptic of genus $\ge2$ and such that the Jacobian of $X_0(n)$ has rank $0$ over $\mathbb Q$. We determine all points of $X_0(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. As a consequence, we show that up to $\overline{\mathbb Q}$-isomorphism, all but finitely many elliptic curves with $n$-isogenies over quadratic fields are in fact $\mathbb Q$-curves, and we list all exceptions. We also show that, again with finitely many exceptions up to $\overline{\mathbb Q}$-isomorphism, every $\mathbb Q$-curve $E$ over a quadratic field $K$ admitting an $n$-isogeny is $d$-isogenous, for some $d\mid n$, to the twist of its Galois conjugate by some quadratic extension $L$ of $K$; we determine $d$ and $L$ explicitly.

math.NT↗
arXiv2020

Fields of definition of elliptic curves with prescribed torsion

We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to $C_{16}$, as well as all elliptic curves over cubic fields with a subgroup isomorphic to $C_2\times C_{14}$, are base changes of elliptic curves defined over $\Q$. We obtain these results by studying geometric properties of modular curves and maps between modular curves, and then obtaining a modular description of these curves and maps.

math.NT↗
arXiv2018

On quantum computation of Kloosterman sums

We give two quantum algorithms for computing (twisted) Kloosterman sums attached to a finite field $\mathbf{F}$ of $q$ elements. The first algorithm computes a quantum state containing, as its coefficients with respect to the standard basis, all Kloosterman sums for $\mathbf{F}$ twisted by a given multiplicative character, and runs in time polynomial in $\log q$. The second algorithm computes a single Kloosterman sum to a prescribed precision, and runs in time quasi-linear in $\sqrt{q}$.

quant-ph↗
arXiv2018

Reductions of points on algebraic groups, II

Let $A$ be the product of an abelian variety and a torus over a number field $K$, and let $m$ be a positive integer. If $α\in A(K)$ is a point of infinite order, we consider the set of primes $\mathfrak p$ of $K$ such that the reduction $(α\bmod \mathfrak p)$ is well defined and has order coprime to $m$. This set admits a natural density, which we are able to express as a finite sum of products of $\ell$-adic integrals, where $\ell$ varies in the set of prime divisors of $m$. We deduce that the density is a rational number, whose denominator is bounded (up to powers of $m$) in a very strong sense. This extends the results of the paper "Reductions of points on algebraic groups" by Davide Lombardo and the second author, where the case $m$ prime is established.

math.NT↗
arXiv2017

Dual pairs of algebras and finite commutative group schemes

We introduce a category of dual pairs of finite locally free algebras over a ring. This gives an efficient way to represent finite locally free commutative group schemes. We give a number of algorithms to compute with dual pairs of algebras, and we apply our results to Galois representations on finite Abelian groups.

math.NT↗
arXiv2017

On $L$-functions of quadratic $\mathbb{Q}$-curves

Let $K$ be a quadratic number field of discriminant $Δ_K$, let $E$ be a $\mathbb Q$-curve without CM completely defined over $K$ and let $ω_E$ be an invariant differential on $E$. Let $L(E,s)$ be the $L$-function of $E$. In this setting, it is known that $L(E,s)$ possesses an analytic continuation to $\mathbb C$. The period of $E$ can be written (up to a power of $2$) as the product of the Tamagawa numbers of $E$ with $Ω_E/\sqrt{|Δ_K|}$, where $Ω_E$ is a quantity, independent of $ω_E$, which encodes the real periods of $E$ when $K$ is real and the covolume of the period lattice of $E$ when $K$ is imaginary. In this paper we compute, under the generalized Manin conjecture, an effective nonzero integer $Q=Q(E,ω_E)$ such that if $L(E,1)\neq 0$ then $L(E,1)\cdot Q\cdot\sqrt{|Δ_K|}/Ω_E$ is an integer. Computing $L(E,1)$ up to sufficiently high precision, our result allows us to prove that $L(E,1)=0$ whenever this is the case and to compute the $L$-ratio $L(E,1)\cdot\sqrt{|Δ_K|}/Ω_E$ when $L(E,1)\neq 0$. An important ingredient is an algorithm to compute a newform $f$ of weight $2$ level $Γ_1(N)$ such that $L(E,s)=L(f,s)\cdot L({}^{σ\!} f,s)$, for ${}^{σ\!} f$ the unique Galois conjugate of $f$. As an application of these results, we verify the validity of the weak BSD conjecture for some $\mathbb Q$-curves of rank $2$ and we will compute the $L$-ratio of a curve of rank $0$.

math.NT↗
arXiv2017

Strongly modular models of $\mathbb Q$-curves

Let $E$ be a $\mathbb Q$-curve without complex multiplication. We address the problem of deciding whether $E$ is geometrically isomorphic to a strongly modular $\mathbb Q$-curve. We show that the question has a positive answer if and only if $E$ has a model that is completely defined over an abelian number field. Next, if $E$ is completely defined over a quadratic or biquadratic number field $L$, we classify all strongly modular twists of $E$ over $L$ in terms of the arithmetic of $L$. Moreover, we show how to determine which of these twists come, up to isogeny, from a subfield of $L$.

math.NT↗
arXiv2015

A criterion to rule out torsion groups for elliptic curves over number fields

We present a criterion for proving that certain groups of the form $\mathbb Z/m\mathbb Z\oplus\mathbb Z/n\mathbb Z$ do not occur as the torsion subgroup of any elliptic curve over suitable (families of) number fields. We apply this criterion to eliminate certain groups as torsion groups of elliptic curves over cubic and quartic fields. We also use this criterion to give the list of all torsion groups of elliptic curves occurring over a specific cubic field and over a specific quartic field.

math.NT↗
arXiv2014

Polynomial bounds for Arakelov invariants of Belyi curves

We explicitly bound the Faltings height of a curve over Q polynomially in its Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings' delta invariant and the self-intersection of the dualizing sheaf. Our results allow us to explicitly bound Arakelov invariants of modular curves, Hurwitz curves and Fermat curves in terms of their genus. Moreover, as an application, we show that the Couveignes-Edixhoven-Bruin algorithm to compute coefficients of modular forms for congruence subgroups of SL2(Z) runs in polynomial time under the Riemann hypothesis for zeta-functions of number fields. This was known before only for certain congruence subgroups. Finally, we use our results to prove a conjecture of Edixhoven, de Jong and Schepers on the Faltings height of a cover of the projective line with fixed branch locus. Our proof uses Merkl's method for bounding Arakelov-Green's functions to deal with archimedean contributions. In the appendix, Peter Bruin proves an explicit version of Merkl's results.

math.AG↗
arXiv2012

Bornes optimales pour la différence entre la hauteur de Weil et la hauteur de Néron-Tate sur les courbes elliptiques sur \Qbar

We give an algorithm that, given an elliptic curve $E$ over $\Qbar$ in Weierstraß form, computes the infimum and supremum of the difference between the na\"ıve and canonical height functions on $E(\Qbar)$. ----- Nous donnons un algorithme qui, étant donnée une courbe elliptique $E$ sur $\Qbar$ sous la forme de Weierstraß, calcule l'infimum et le supremum de la différence entre la hauteur na\"ıve et la hauteur canonique sur $E(\Qbar)$.

math.NT↗
arXiv2012

The growth of the rank of Abelian varieties upon extensions

We study the growth of the rank of elliptic curves and, more generally, Abelian varieties upon extensions of number fields. First, we show that if $L/K$ is a finite Galois extension of number fields such that $\Gal(L/K)$ does not have an index 2 subgroup and $A/K$ is an Abelian variety, then $\rk A(L)-\rk A(K)$ can never be 1. We obtain more precise results when $\Gal(L/K)$ is of odd order, alternating, $\SL_2(\F_p)$ or $\PSL_2(\F_p)$. This implies a restriction on $\rk E(K(E[p]))-\rk E(K(ζ_p))$ when $E/K$ is an elliptic curve whose mod $p$ Galois representation is surjective. Similar results are obtained for the growth of the rank in certain non-Galois extensions. Second, we show that for every $n\ge2$ there exists an elliptic curve $E$ over a number field $K$ such that $\Q\otimes_\Q\Res_{K/\Q} E$ contains a number field of degree $2^n$. We ask whether every elliptic curve $E/K$ has infinite rank over $K\Q(2)$, where $\Q(2)$ is the compositum of all quadratic extensions of $\Q$. We show that if the answer is yes, then for any $n\ge2$, there exists an elliptic curve $E/K$ admitting infinitely many quadratic twists whose rank is a positive multiple of $2^n$.

math.NT↗