Rational Points Research Articles

Rational points on nonlinear horocycles and pigeonhole statistics for the fractional parts of

AbstractIn this paper, we investigate pigeonhole statistics for the fractional parts of the sequence $\sqrt {n}$ . Namely, we partition the unit circle $ \mathbb {T} = \mathbb {R}/\mathbb {Z}$ into N intervals and show that the proportion of intervals containing exactly j points of the sequence $(\sqrt {n} + \mathbb {Z})_{n=1}^N$ converges in the limit as $N \to \infty $ . More generally, we investigate how the limiting distribution of the first $sN$ points of the sequence varies with the parameter $s \geq 0$ . A natural way to examine this is via point processes—random measures on $[0,\infty )$ which represent the arrival times of the points of our sequence to a random interval from our partition. We show that the sequence of point processes we obtain converges in distribution and give an explicit description of the limiting process in terms of random affine unimodular lattices. Our work uses ergodic theory in the space of affine unimodular lattices, building upon work of Elkies and McMullen [Gaps in $\sqrt {n}$ mod 1 and ergodic theory. Duke Math. J.123 (2004), 95–139]. We prove a generalisation of equidistribution of rational points on expanding horocycles in the modular surface, working instead on nonlinear horocycle sections.

ON -ACTIONS ON K3 SURFACES IN POSITIVE CHARACTERISTIC

AbstractIn characteristic $0$ , symplectic automorphisms of K3 surfaces (i.e., automorphisms preserving the global $2$ -form) and non-symplectic ones behave differently. In this paper, we consider the actions of the group schemes $\mu _{n}$ on K3 surfaces (possibly with rational double point [RDP] singularities) in characteristic p, where n may be divisible by p. We introduce the notion of symplecticness of such actions, and we show that symplectic $\mu _{n}$ -actions have similar properties, such as possible orders, fixed loci, and quotients, to symplectic automorphisms of order n in characteristic $0$ . We also study local $\mu _n$ -actions on RDPs.

Extending torsors over regular models of curves

Let R be a discrete valuation ring with field of fractions K and residue field k of characteristic $$p>0$$ . Given a commutative finite group scheme G over K and a smooth projective curve C over K with a rational point, we study the extension of pointed fppf G-torsors over C to pointed torsors over some R-regular model $${\mathcal {C}}$$ of C. We first study this problem in the category of log schemes: given a finite flat R-group scheme $${\mathcal {G}}$$ , we prove that the data of a pointed $${\mathcal {G}}$$ -log torsor over $${\mathcal {C}}$$ is equivalent to that of a morphism $${\mathcal {G}}^D \rightarrow {{\,\mathrm{Pic}\,}}^{log}_{{\mathcal {C}}/R}$$ , where $${\mathcal {G}}^D$$ is the Cartier dual of $${\mathcal {G}}$$ and $${{\,\mathrm{Pic}\,}}^{log}_{{\mathcal {C}}/R}$$ the log Picard functor. After that, we give a sufficient condition for such a log extension to exist, and then we compute the obstruction for the existence of an extension in the category of usual schemes. In a second part, we generalize a result of Chiodo (Manuscr Math 129(3):337–368, 2009) which gives a criterion for the r-torsion subgroup of the Néron model of J to be a finite flat group scheme, and we combine it with the results of the first part. Finally, we give a detailed example of extension of torsors when C is a hyperelliptic curve defined over $${\mathbb {Q}}$$ , which illustrates our techniques.

Coefficient Systems on the Bruhat-Tits Building and Pro-𝑝 Iwahori-Hecke Modules

Let G G be the group of rational points of a split connected reductive group over a nonarchimedean local field of residue characteristic p p . Let I I be a pro- p p Iwahori subgroup of G G and let R R be a commutative quasi-Frobenius ring. If H = R [ I ∖ G / I ] H=R[I\backslash G/I] denotes the pro- p p Iwahori-Hecke algebra of G G over R R we clarify the relation between the category of H H -modules and the category of G G -equivariant coefficient systems on the semisimple Bruhat-Tits building of G G . If R R is a field of characteristic zero this yields alternative proofs of the exactness of the Schneider-Stuhler resolution and of the Zelevinski conjecture for smooth G G -representations generated by their I I -invariants. In general, it gives a description of the derived category of H H -modules in terms of smooth G G -representations and yields a functor to generalized ( φ , Γ ) (\varphi ,\Gamma ) -modules extending the constructions of Colmez, Schneider and Vignéras.

Computing rational powers of monomial ideals

This paper concerns fractional powers of monomial ideals. Rational powers of a monomial ideal generalize the integral closure operation as well as recover the family of symbolic powers. They also highlight many interesting connections to the theory of convex polytopes. We provide multiple algorithms for computing the rational powers of a monomial ideal. We also introduce a mild generalization allowing real powers of monomial ideals. An important result is that given any monomial ideal I, the function taking a real number to the corresponding real power of I is a step function which is left continuous and has rational discontinuity points.

Canonical heights on Pell conics over number fields

In "Higher descent on Pell conics. III. The first 2-descent", Lemmermeyer introduced the canonical heights on the groups of rational points on Pell conics, which are analogues of the canonical heights on elliptic curves. In this paper, we generalize this: We introduce the canonical heights on the groups of $\overline{\mathbb{Q}}$-rational points on Pell conics over number fields.

Degenerated Calabi–Yau varieties with infinite components, moduli compactifications, and limit toroidal structures

For any degenerating Calabi–Yau family, we introduce a new limit space which we call galaxy, whose dense subspace is the disjoint union of countably infinite open Calabi–Yau varieties, parametrized by the rational points of the Kontsevich–Soibelman’s essential skeleton, while dominated by the Huber adification over the Puiseux series field. Other topics include: projective limits of toroidal compactifications (Sect. 3), locally modelled on limit toric varieties (Sect. 2.4), the way to attach a tropicalized family to a given Calabi–Yau family (Sect. 4), which are weakly related to each other.

Saturation and vanishing ideals

We consider an homogeneous ideal I in the polynomial ring \(S=K[x_1,\dots ,\) \(x_m]\) over a finite field \(K={\mathbb {F}}_q\) and the finite set of projective rational points \({{\mathbb {X}}}\) that it defines in the projective space \({{\mathbb {P}}}^{m-1}\). We concern ourselves with the problem of computing the vanishing ideal \(I({{\mathbb {X}}})\). This is usually done by adding the equations of the projective space \(I({{\mathbb {P}}}^{m-1})\) to I and computing the radical. We give an alternative and more efficient way using the saturation with respect to the homogeneous maximal ideal.

Congruences of elliptic curves arising from nonsurjective mod 𝑁 Galois representations

We study N N -congruences between quadratic twists of elliptic curves. If N N has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular curves in question correspond to the normaliser of a Cartan subgroup of GL 2 ⁡ ( Z / N Z ) \operatorname {GL}_2(\mathbb {Z}/N\mathbb {Z}) . By computing explicit models for these double covers we find all pairs ( N , r ) (N, r) such that there exist infinitely many j j -invariants of elliptic curves E / Q E/\mathbb {Q} which are N N -congruent with power r r to a quadratic twist of E E . We also find an example of a 48 48 -congruence over Q \mathbb {Q} . We make a conjecture classifying nontrivial ( N , r ) (N,r) -congruences between quadratic twists of elliptic curves over Q \mathbb {Q} . Finally, we give a more detailed analysis of the level 15 15 case. We use elliptic Chabauty to determine the rational points on a modular curve of genus 2 2 whose Jacobian has rank 2 2 and which arises as a double cover of the modular curve X ( n s 3 + , n s 5 + ) X(\mathrm {ns\,} 3^+, \mathrm {ns\,} 5^+) . As a consequence we obtain a new proof of the class number 1 1 problem.

Rational points on quadratic elliptic surfaces

We consider elliptic surfaces whose coefficients are degree 2 polynomials in a variable T. It was recently shown, Kollár and Mella (Amer J Math 139(4):915–936 2017), that for infinitely many rational values of T the resulting elliptic curves have rank at least 1. We prove that the Mordell–Weil rank of each such elliptic surface is at most 6 over \({{\mathbb {Q}}}\). In fact, we show that the Mordell–Weil rank of these elliptic surfaces is controlled by the number of zeros of a certain polynomial over \({{\mathbb {Q}}}\).

The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3

We show that the average size of the $2$-Selmer group of the family of Jacobians of non-hyperelliptic genus-$3$ curves with a marked rational hyperflex point, when ordered by a natural height, is bounded above by $3$. We achieve this by interpreting $2$-Selmer elements as integral orbits of a representation associated with a stable $\mathbb{Z}/2\mathbb{Z}$-grading on the Lie algebra of type $E_6$ and using Bhargava's orbit-counting techniques. We use this result to show that the marked point is the only rational point for a positive proportion of curves in this family. The main novelties are the construction of integral representatives using certain properties of the compactified Jacobian of the simple curve singularity of type $E_6$, and a representation-theoretic interpretation of a Mumford theta group naturally associated to our family of curves.

Rational points on fibrations with few non-split fibres

Abstract We revisit the abstract framework underlying the fibration method for producing rational points on the total space of fibrations over the projective line. By fine-tuning its dependence on external arithmetic conjectures, we render the method unconditional when the degree of the non-split locus is ≤ 2 {\leq 2} , as well as in various instances where it is 3. We are also able to obtain improved results in the regime that is conditionally accessible under Schinzel’s hypothesis, by incorporating into it, for the first time, a technique due to Harari for controlling the Brauer–Manin obstruction in families.

On the number of rational points on a class of singular hypersurfaces

On the number of rational points on a class of singular hypersurfaces

The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace

We present the formula for the number of monic irreducible polynomials of degree n over the finite field \({\mathbb {F}}_q\) where the coefficients of \(x^{n-1}\) and x vanish for \(n\ge 3\). In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements \(a\in {\mathbb {F}}_{q^n}\) for which Trace\((a)=0\) and Trace\((a^{-1})=0\).

Explicit Tamagawa numbers for certain algebraic tori over number fields

Given a number field extension K / k K/k with an intermediate field K + K^+ fixed by a central element of Gal ⁡ ( K / k ) \operatorname {Gal}(K/k) of prime order p p , there exists an algebraic torus over k k whose rational points are elements of K × K^\times sent to k × k^\times by the norm map N K / K + N_{K/K^+} . The goal is to compute the Tamagawa number such a torus explicitly via Ono’s formula that expresses it as a ratio of cohomological invariants. A fairly complete and detailed description of the cohomology of the character lattice of such a torus is given when K / k K/k is Galois. Partial results including the numerator of Ono’s formula are given when the extension is not Galois, or more generally when the torus is defined by an étale algebra. We also present tools developed in SageMath for this purpose, allowing us to build and compute the cohomology and explore the local-global principles for such an algebraic torus. Particular attention is given to the case when [ K : K + ] = 2 [K:K^+]=2 and K K is a CM-field. This case corresponds to maximal tori in G S p 2 n \mathrm {GSp}_{2n} , and most examples will be in that setting. This is motivated by the application to abelian varieties over finite fields and the Hasse principle for bilinear forms.

𝜇_{𝑝}- and 𝛼_{𝑝}-actions on K3 surfaces in characteristic 𝑝

We consider μ p \mu _p - and α p \alpha _p -actions on RDP K3 surfaces (K3 surfaces with rational double point (RDP) singularities allowed) in characteristic p > 0 p > 0 . We study possible characteristics, quotient surfaces, and quotient singularities. It turns out that these properties of μ p \mu _p - and α p \alpha _p -actions are analogous to those of Z / l Z \mathbb {Z}/l\mathbb {Z} -actions (for primes l ≠ p l \neq p ) and Z / p Z \mathbb {Z}/p\mathbb {Z} -quotients respectively. We also show that conversely an RDP K3 surface with a certain configuration of singularities admits a μ p \mu _p - or α p \alpha _p - or Z / p Z \mathbb {Z}/p\mathbb {Z} -covering by a “K3-like” surface, which is often an RDP K3 surface but not always, as in the case of the canonical coverings of Enriques surfaces in characteristic 2 2 .

Building cryptographic schemes based on elliptic curves over rational numbers

The possibility of using elliptic curves over the rational field of non-zero ranks in cryptographic schemes is studied. For the first time, the construction of cryptosystems is proposed the security of which is based on the complexity of solving the knapsack problem on elliptic curves over rational numbers of non-zero ranks. A new approach to the use of elliptic curves for cryptographic schemes is proposed. A few experiments have been carried out to estimate the heights characteristic of points on elliptic curves of infinite order. A model of a cryptosystem resistant to computations on a quantum computer and based on rational points of an infinite order curve is proposed. A study of the security and effectiveness of the proposed scheme has been carried out. An attack on the secret search in such a cryptosystem is implemented and it is shown that the complexity of the attack is exponential. The proposed solution can be applied in the construction of real cryptographic schemes as well as cryptographic protocols.

Simultaneous rational periodic points of degree-2 rational maps

Let S be the collection of quadratic polynomial maps, and degree 2-rational maps whose automorphism groups are isomorphic to C2 defined over the rational field. Assuming standard conjectures of Poonen and Manes on the period length of a periodic point under the action of a map in S, we give a complete description of triples (f1,f2,p) such that p is a rational periodic point for both fi∈S, i=1,2. We also show that no more than three quadratic polynomial maps can possess a common periodic point over the rational field. In addition, under these hypotheses we show that two nonzero rational numbers a,b are periodic points of the map ϕt1,t2(z)=t1z+t2/z for infinitely many nonzero rational pairs (t1,t2) if and only if a2=b2.

Torsion Subgroups of Modular Jacobian Varieties Over Splitting Fields of Cuspidal Subgroups

Abstract For any positive integer $N$, let $J_0(N)$ be the Jacobian variety of the modular curve $X_0(N)$ over ${\mathbb {Q}}$ and ${\mathcal {C}}_N$ its cuspidal subgroup. Let $F_N$ denote the splitting field of ${\mathcal {C}}_N$, which is the smallest number field whose absolute Galois group acts trivially on ${\mathcal {C}}_N$. Let ${\mathcal {J}}_N=J_0(N)(F_{N})_{\textrm {tor}}$ be the torsion subgroup of the group of $F_N$-rational points on $J_0(N)$. We prove that ${\mathcal {J}}_N$ coincides with ${\mathcal {C}}_N$ outside $6N[F_N:{\mathbb {Q}}]$.

P∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^\\infty $$\\end{document}-Selmer groups and rational points on CM elliptic curves

Let E/Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E/{\\mathbb {Q}}$$\\end{document} be a CM elliptic curve and p a prime of good ordinary reduction for E. We show that if Selp∞(E/Q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Sel}_{p^\\infty }(E/{\\mathbb {Q}})$$\\end{document} has Zp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}_p$$\\end{document}-corank one, then E(Q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E({\\mathbb {Q}})$$\\end{document} has a point of infinite order. The non-torsion point arises from a Heegner point, and thus ords=1L(E,s)=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{ord}\\,}}_{s=1}L(E,s)=1$$\\end{document}, yielding a p-converse to a theorem of Gross–Zagier, Kolyvagin, and Rubin in the spirit of [49, 54]. For p>3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>3$$\\end{document}, this gives a new proof of the main result of [12], which our approach extends to all primes. The approach generalizes to CM elliptic curves over totally real fields [4].