Rational Points Research Articles

Surface bundles and the section conjecture

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of this conjecture. In so doing we produce many examples of curves satisfying the section conjecture over fields of geometric interest, and then over p-adic fields and number fields via a Chebotarev argument. We construct two Galois cohomology classes o_1 and o_2, which obstruct the existence of pi_1-sections and hence of rational points. The first is an abelian obstruction, closely related to the period of a curve and to a cohomology class on the moduli space of curves M_g studied by Morita. The second is a 2-nilpotent obstruction and appears to be new. We study the degeneration of these classes via topological techniques, and we produce examples of surface bundles over surfaces where these classes obstruct sections. We then use these constructions to produce curves over p-adic fields and number fields where each class obstructs pi_1-sections and hence rational points. Among our geometric results are a new proof of the section conjecture for the generic curve of genus g>2, and a proof of the section conjecture for the generic curve of even genus with a rational divisor class of degree one (where the obstruction to the existence of a section is genuinely non-abelian).

Local-global principle for classical groups over function fields of $p$-adic curves

Let K be a local field with residue field \kappa and F the function field of a curve over K . Let G be a connected linear algebraic group over F of classical type. Suppose \operatorname{char}(\kappa) is a good prime for G . Then we prove that projective homogeneous spaces under G over F satisfy a local-global principle for rational points with respect to discrete valuations of F . If G is a semisimple simply connected group over F , then we also prove that principal homogeneous spaces under G over F satisfy a local-global principle for rational points with respect to discrete valuations of F .

Hyperelliptic Covers of Different Degree for Elliptic Curves

In elliptic curve cryptography (ECC) and hyperelliptic curve cryptography (HECC), the size of cipher-text space defined by the cardinality of Jacobian is a significant factor to measure the security level. Counting problems on Jacobians of elliptic curve can be solved in polynomial time by Schoof–Elkies–Atkin (SEA) algorithm. However, counting problems on Jacobians of hyperelliptic curves are solved less satisfactorily than those on elliptic curves. So, we consider the construction of the cover map from the hyperelliptic curves to the elliptic curves to convert point counting problems on hyperelliptic curves to those on elliptic curves. We can also use the cover map as a kind of cover attacks. Given an elliptic curve over an extension field of degree n , one might try to use the cover attack to reduce the discrete logarithm problem (DLP) in the group of rational points of the elliptic curve to DLPs in the Jacobian of a curve of genus g ≥ n over the base field. An algorithm has been proposed for finding genus 3 hyperelliptic covers as a cover attack for elliptic curves with cofactor 2. Our algorithms are about the cover map from hyperelliptic curves of genus 2 to elliptic curves of prime order. As an application, an example of an elliptic curve whose order is a 256-bit prime vulnerable to our algorithms is given.

Rational points on cubic surfaces and AG codes from the Norm–Trace curve

In this paper we give a complete characterization of the intersections between the Norm-Trace curve over $\mathbb{F}_{q^3}$ and the curves of the form $y=ax^3+bx^2+cx+d$, generalizing a previous result by Bonini and Sala, providing more detailed information about the weight spectrum of one-point AG codes arising from such curve. We also derive, with explicit computations, some general bounds for the number of rational points on a cubic surface defined over $\mathbb{F}_{q}$.

Rational points on cubic surfaces

Rational points on cubic surfaces

Rational points on Del Pezzo surfaces of degree four

We study the distribution of the Brauer group and the frequency of the Brauer–Manin obstruction to the Hasse principle and weak approximation in a family of smooth del Pezzo surfaces of degree four over the rationals.

The Tate-Shafarevich groups of multinorm-one tori

Let k be a global field and L be a product of cyclic extensions of k . Let T be the torus defined by the multinorm equation N L / k ( x ) = 1 and let T ˆ be its character group. The Tate-Shafarevich group and the algebraic Tate-Shafarevich group of T ˆ in degree 2 give obstructions to the Hasse principle and weak approximation for rational points on principal homogeneous spaces of T . We give concrete descriptions of these groups and provide several examples.

The Brauer–Manin obstruction for constant curves over global function fields

Let 𝔽 be a finite field and C,D smooth, geometrically irreducible, proper curves over 𝔽 and set K=𝔽(D). We consider Brauer–Manin and abelian descent obstructions to the existence of rational points and to weak approximation for the curve C⊗ 𝔽 K. In particular, we show that Brauer–Manin is the only obstruction to weak approximation and the Hasse principle in the case that the genus of D is less than that of C. We also show that we can identify the points corresponding to non-constant maps D→C using Frobenius descents.

On homogeneous spaces with finite anti-solvable stabilizers

We say that a group is anti-solvable if all of its composition factors are non-abelian. We consider a particular family of anti-solvable finite groups containing the simple alternating groups for $n\neq 6$ and all 26 sporadic simple groups. We prove that, if $K$ is a perfect field and $X$ is a homogeneous space of a smooth algebraic $K$-group $G$ with finite geometric stabilizers lying in this family, then $X$ is dominated by a $G$-torsor. In particular, if $G=\mathrm{SL}_n$, all such homogeneous spaces have rational points.

Metrical theorems for unconventional height functions

In this paper we consider the simultaneous approximation of real points by rational points with the error of approximation given by the functions of ‘non-standard’ heights. We prove analogues of Khintchine and Jarník-Besicovitch theorems for this setting, thus answering some questions raised by Fishman and Simmons (2017).

Divisibility by 2 on quartic models of elliptic curves and rational Diophantine D(q)-quintuples

Let C be a smooth genus one curve described by a quartic polynomial equation over the rational field \({{\mathbb {Q}}}\) with \(P\in C({{\mathbb {Q}}})\). We give an explicit criterion for the divisibility-by-2 of a rational point on the elliptic curve (C, P). This provides an analogue to the classical criterion of the divisibility-by-2 on elliptic curves described by Weierstrass equations. We employ this criterion to investigate the question of extending a rational D(q)-quadruple to a quintuple. We give concrete examples to which we can give an affirmative answer. One of these results implies that although the rational \( D(16t+9) \)-quadruple \(\{t, 16t+8,2 25t+14, 36t+20 \}\) can not be extended to a polynomial \( D(16t+9) \)-quintuple using a linear polynomial, there are infinitely many rational values of t for which the aforementioned rational \( D(16t+9) \)-quadruple can be extended to a rational \( D(16t+9) \)-quintuple. Moreover, these infinitely many values of t are parametrized by the rational points on a certain elliptic curve of positive Mordell–Weil rank.

Density of rational points on a family of del Pezzo surfaces of degree one

Let k be an infinite field of characteristic 0, and X a del Pezzo surface of degree d with at least one k-rational point. Various methods from algebraic geometry and arithmetic statistics have shown the Zariski density of the set X(k) of k-rational points in X for d≥2 (under an extra condition for d=2), but fail to work in generality when the degree of X is 1, leaving a large class of del Pezzo surfaces for which the question of density of rational points is still open. In this paper, we prove the Zariski density of X(k) when X has degree 1 and is represented in the weighted projective space P(2,3,1,1) with coordinates x,y,z,w by an equation of the form y2=x3+az6+bz3w3+cw6 for a,b,c∈k with a,c non-zero, under the condition that the elliptic surface obtained by blowing up the base point of the anticanonical linear system |−KX| contains a smooth fiber above a point in P1∖{(1:0),(0:1)} with positive rank over k. When k is of finite type over Q, this condition is sufficient and necessary.

On abelian covers of the projective line with fixed gonality and many rational points

A smooth geometrically connected curve over the finite field [Formula: see text] with gonality [Formula: see text] has at most [Formula: see text] rational points. Faber and Grantham conjectured that there exist curves of every sufficiently large genus with gonality [Formula: see text] that achieve this bound. In this paper, we show that this bound can be achieved for an infinite sequence of genera using abelian covers of the projective line. We also argue that abelian covers will not suffice to prove the full conjecture.

Counting multiplicities in a hypersurface over a finite field

In this paper, we consider a problem of counting multiplicities. We fix a counting function of multiplicity of rational points in a hypersurface of a projective space over a finite field, and we give an upper bound for the sum with respect to this counting function in terms of the degree of the hypersurface and the dimension and the cardinality of the finite field. This upper bound gives a description of the complexity of the singular locus of this hypersurface. In order to obtain this upper bound, we introduce a notion called an intersection tree by intersection theory. We construct a sequence of intersections, such that the multiplicity of a singular rational point is equal to that of one of the irreducible components in these intersections. The multiplicities of these irreducible components constructed above are bounded by their multiplicities in the intersection tree.

Geometric consistency of Manin's conjecture

We conjecture that the exceptional set in Manin's conjecture has an explicit geometric description. Our proposal includes the rational point contributions from any generically finite map with larger geometric invariants. We prove that this set is contained in a thin subset of rational points, verifying that there is no counterexample to Manin's conjecture which arises from an incompatibility of geometric invariants.

Unirationality of RDP Del Pezzo surfaces of degree 2

We study unirationality of a Del Pezzo surface of degree two over a given (non algebraically closed) field, under the assumption that it admits at least one rational double point over an algebraic closure of the base field. As corollaries of our main results, we find that over a finite field, it is unirational if the cardinality of the field is greater than or equal to nine and we also find that over an infinite field, which is not necessarily perfect, it is unirational if and only if the rational points are Zariski dense over the field.

A distanced perspective reduces negative affective responses through rational view in recalling and writing angry experience.

Previous research demonstrates that self-distancing helps in regulating negative emotions. Furthermore, adopting a distanced perspective when referring to the self has been shown to be a simple and effective way to regulate emotion. Moreover, previous research has demonstrated several mechanisms whereby the distanced perspective eventually leads to the decrease in negative emotions. Building on this literature, the present research proposed that a rational point of view induced by adopting a distanced perspective would play a critical role in this process. The results from two studies supported the proposition. Specifically, in recalling (Study 1) and writing (Study 2) about anger-provoking events, those who adopted a distanced perspective were more likely to take a rational point of view when reflecting on the event than did those who adopted a self-immersed perspective. Furthermore, such differences in the rational perspective were associated with the corresponding differences in negative affect.

Rational points on algebraic curves in infinite towers of number fields

We study a natural question in the Iwasawa theory of algebraic curves of genus \(>1\). Fix a prime number p. Let X be a smooth, projective, geometrically irreducible curve defined over a number field K of genus \(g>1\), such that the Jacobian of X has good ordinary reduction at the primes above p. Fix an odd prime p and for any integer \(n>1\), let \(K_n^{(p)}\) denote the degree-\(p^n\) extension of K contained in \(K(\mu _{p^{\infty }})\). We prove explicit results for the growth of \(\#X(K_n^{(p)})\) as \(n\rightarrow \infty \). When the Jacobian of X has rank zero and the associated adelic Galois representation has big image, we prove an explicit condition under which \(X(K_{n}^{(p)})=X(K)\) for all n. This condition is illustrated through examples. We also prove a generalization of Imai’s theorem that applies to abelian varieties over arbitrary pro-p extensions.

On monomial generalized almost perfect nonlinear functions

Generalized almost perfect nonlinear (GAPN) functions are a generalization of APN functions to finite fields of odd characteristic p introduced in 2017 by Kuroda and Tsujie. In this paper we deal with GAPN functions of monomial type. To this aim, we connect the GAPN property for a monomial function over Fpn to the existence of suitable rational points of an algebraic curve defined over Fpn. We give necessary conditions for a monomial function to be GAPN, providing the converse of recent results by Özbudak and Sălăgean and by Zha, Hu and Zhang.

On the symplectic type of isomorphisms of the 𝑝-torsion of elliptic curves

Let p ≥ 3 p \geq 3 be a prime. Let E / Q E/\mathbb {Q} and E ′ / Q E’/\mathbb {Q} be elliptic curves with isomorphic p p -torsion modules E [ p ] E[p] and E ′ [ p ] E’[p] . Assume further that either (i) every G Q G_\mathbb {Q} -modules isomorphism ϕ : E [ p ] → E ′ [ p ] \phi : E[p] \to E’[p] admits a multiple λ ⋅ ϕ \lambda \cdot \phi with λ ∈ F p × \lambda \in \mathbb {F}_p^\times preserving the Weil pairing; or (ii) no G Q G_\mathbb {Q} -isomorphism ϕ : E [ p ] → E ′ [ p ] \phi : E[p] \to E’[p] preserves the Weil pairing. This paper considers the problem of deciding if we are in case (i) or (ii). Our approach is to consider the problem locally at a prime ℓ ≠ p \ell \neq p . Firstly, we determine the primes ℓ \ell for which the local curves E / Q ℓ E/\mathbb {Q}_\ell and E ′ / Q ℓ E’/\mathbb {Q}_\ell contain enough information to decide between (i) or (ii). Secondly, we establish a collection of criteria, in terms of the standard invariants associated to minimal Weierstrass models of E / Q ℓ E/\mathbb {Q}_\ell and E ′ / Q ℓ E’/\mathbb {Q}_\ell , to decide between (i) and (ii). We show that our results give a complete solution to the problem by local methods away from p p . We apply our methods to show the non-existence of rational points on certain hyperelliptic curves of the form y 2 = x p − ℓ y^2 = x^p - \ell and y 2 = x p − 2 ℓ y^2 = x^p - 2\ell where ℓ \ell is a prime; we also give incremental results on the Fermat equation x 2 + y 3 = z p x^2 + y^3 = z^p . As a different application, we discuss variants of a question raised by Mazur concerning the existence of symplectic isomorphisms between the p p -torsion of two non-isogenous elliptic curves defined over Q \mathbb {Q} .