We classify pairs $(S, γ)$, consisting of a rational elliptic surface $S$ and a Galois cover $γ$ of the base, which satisfy a condition we call $\mathcal{L}$-stability. We explain how to use the theory of Mordell-Weil lattices to compute the kernel of the restriction maps of Weil-Chatelet groups for $\mathcal{L}$-stable pairs. We also prove results about the injectivity of restriction maps of Weil-Chatelet groups for some pairs which are not $\mathcal{L}$-stable.
Châtelet surfaces provide a rich source of geometrically rational surfaces which do not always satisfy the Hasse principle. Restricting attention to a special class of Châtelet surfaces, we investigate the frequency that such counter-examples arise over the rationals.
The surface $z^2=ay^2+P(x), \, a \in k, \, P(x) \in k[x]$ is not $k$-rational, if $a \not\in k^2$ and $P(x)$ satisfies some conditions. This result essentially due to Iskovskih but his statement is in terms of algebraic geometry, and not so easy to access for the researchers of the field extension. This paper aims to give a formulation accessible more easily. A necessary and sufficient condition for $k$-rationality is given in terms of $a$ and $P$, assuming that $\ch k \not= 2$ and every irreducible component of $P(x)$ is separable over $k$.
Given an extension of number fields $E \subset F$ and a projective variety $X$ over $F$, we compare the problem of counting the number of rational points of bounded height on $X$ with that of its Weil restriction over $E$. In particular, we consider the compatibility with respect to the Weil restriction of conjectural asymptotic formulae due to Manin and others. Using our methods we prove several new cases of these conjectures. We also construct new counterexamples over every number field.
We consider the finite set of isogeny classes of $g$-dimensional abelian varieties defined over the finite field $\mathbb{F}_q$ with endomorphism algebra being a field. We prove that the class within this set whose varieties have maximal number of rational points is unique, for any prime even power $q$ big enough and verifying mild conditions. We describe its Weil polynomial and we prove that the class is ordinary and cyclic outside the primes dividing an integer that only depends on $g$. In dimension $3$, we prove that the class is ordinary and cyclic and give explicitly its Weil polynomial, …
We study Weil-etale cohomology, introduced by Lichtenbaum for varieties over finite fields. In the first half of the paper we give an explicit description of the base change from Weil-etale cohomology to etale cohomology. As a consequence, we get a long exact sequence relating Weil-etale cohomology to etale cohomology, show that for finite coefficients the cohomology theories agree, and with rational coefficients a Weil-etale cohomology group is the direct sum of two etale cohomology groups. In the second half of the paper we restrict ourselves to Weil-etale cohomology of the motivic complex. …
We use a generalised Kummer construction to realise all but one known weight four newforms with complex multiplication and rational Fourier coefficients in smooth Calabi-Yau threefolds defined over the rational numbers. The Calabi-Yau manifolds are smooth models of quotients of the Weil restrictions of elliptic curves with CM of class number three.
Let $Σ$ be a smooth projective surface, let $f' : S' \to Σ$ be a double cover of $Σ$ and let $μ: S \to S'$ be the canonical resolution. Put $f = f'\circμ$. An irreducible curve $C$ on $Σ$ is said to be a splitting curve with respect to $f$ if $f^*C$ is of the form $C^+ + C^- + E$, where $C^- = σ_f^*C^+$, $σ_f$ being the covering transformation of $f$ and all irreducible components of $E$ are contained in the exceptional set of $μ$. In this article, we show that a kind of "reciprocity" of splitting curves holds for a certain pair of curves on rational ruled surfaces. As an application, we consi…
Let $X$ be a smooth cubic hypersurface of dimension $n \ge 1$ over the rationals. It is well-known that new rational points may be obtained from old ones by secant and tangent constructions. In view of the Mordell--Weil theorem for $n=1$, Manin (1968) asked if there exists a finite set $S$ from which all other rational points can be thus obtained. We give an affirmative answer for $n \ge 48$, showing in fact that we can take the generating set $S$ to consist of just one point. Our proof makes use of a weak approximation theorem due to Skinner, a theorem of Browning, Dietmann and Heath-Brown on…
Mumford defined a rational pullback for Weil divisors on normal surfaces, which is linear, respects effectivity, and satisfies the projection formula. In higher dimensions, the existence of small resolutions of singularities precludes such general results. We single out a higher-dimensional situation that resembles the surface case and show for it that a rational pullback for Weil divisors exists, which is also linear, respects effectivity, and satisfies the projection formula.
Given a polynomial system $\mathcal{F}$ over a finite field $k$ which is not necessarily of dimension zero, we consider the Weil descent $\mathcal{F}'$ of $\mathcal{F}$ over a subfield $k'$. We prove a theorem which relates the last fall degrees of $\mathcal{F}_1$ and $\mathcal{F}'_1$, where the zero set of $\mathcal{F}_1$ corresponds bijectively to the set of $k$-rational points of $\mathcal{F}$, and the zero set of $\mathcal{F}'_1$ is the set of $k'$-rational points of the Weil descent $\mathcal{F}'$. As an application we derive upper bounds on the last fall degree of $\mathcal{F}'_1$ in the…
We show that there is a bound depending only on g and [K:Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its Jacobian is at most g-3. If K = Q, an explicit bound is 8 r g + 33 (g - 1) + 1. The proof is based on Chabauty's method; the new ingredient is an estimate for the number of zeros of a logarithm in a p-adic `annulus' on the curve, which generalizes the standard bound on disks. The key observation is that for a p-adic field k, the set of k-points on C can be covered by a collection of disks and annuli …
Let $k$ be a number field and $U$ a smooth integral $k$-variety. Let $X \to U$ be an abelian scheme. We consider the set $\mathcal{R}$ of rational points $m \in U(k)$ such that the Mordell-Weil rank of the fibre $U_{m}$ is strictly bigger than the Mordell-Weil rank of the generic fibre. We prove the following results. If the $k$-variety $X$ is $k$-unirational, then $\mathcal{R}$ is dense for the Zariski topology on $U$. If $X$ is $k$-rational, then $\mathcal{R}$ is not thin in $U$. This generalizes results of Billard and of Salgado. The main idea goes back to Néron's thesis: use the generic po…
Given number fields $L \supset K$, smooth projective curves $C$ defined over $L$ and $B$ defined over $K$, and a non-constant $L$-morphism $h \colon C \to B_L$,we consider the curve $C_h$ defined over $K$ whose $K$-rational points parametrize the $L$-rational points on $C$ whose images under $h$ are defined over $K$. Our construction provides a framework which includes as a special case that used in Elliptic Curve Chabauty techniques and their higher genus versions. The set $C_h(K)$ can be infinite only when $C$ has genus at most 1; we analyze completely the case when $C$ has genus 1.
In this text we investigate the good behaviour of the elementary obstruction, introduced by Colliot-Thelene and Sansuc. This is an obstruction to the existence of a rational points on certain algebraic varieties. Assuming some conditions on the Picard group, we prove that the elementary obstruction behaves well under the Weil restriction of a variety.
Using Weil descent, we give bounds for the number of rational points on two families of curves over finite fields with a large abelian group of automorphisms: Artin-Schreier curves of the form $y^q-y=f(x)$ with $f\in\Fqr[x]$, on which the additive group $\Fq$ acts, and Kummer curves of the form $y^{\frac{q-1}{e}}=f(x)$, which have an action of the multiplicative group $\Fq^\star$. In both cases we can remove a $\sqrt{q}$ factor from the Weil bound when $q$ is sufficiently large.
Let $X$ be a curve of genus $g\geq 2$ over a number field $F$ of degree $d = [F:Q]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $\#X(F)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic geometry, known by [CHM97] to follow from the Bombieri--Lang conjecture. A related conjecture posits the existence of a uniform bound $N_{{\rm tors},\dagger}(g,d)$ on the number of geometric torsion points of the Jacobian $J$ of $X$ which lie on the image of $X$ under an Abel--Jacobi map. For fixed $X$ this quantity was conjectured to be finite by Manin--Mumfor…
Earlier it was described to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. It was shown that, except some explicitly described cases, the Alexander polynomial of an algebraic link determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere coincides with the Poincaré series of the corresponding set of curve valuations. The latter one can be defined as an integral over …
The main goal of this paper is to show that the notions of Weil and Cartier $\mathbb{Q}$-divisors coincide for $V$-manifolds and give a procedure to express a rational Weil divisor as a rational Cartier divisor. The theory is illustrated on weighted projective spaces and weighted blow-ups.
Let S be a smooth cubic surface over a field K. It is well-known that new K-rational points may be obtained from old ones by secant and tangent constructions. A Mordell-Weil generating set is a subset B of S(K) of minimal cardinality which generates S(K) via successive secant and tangent constructions. Let r(S,K) be the cardinality of such a Mordell-Weil generating set. Manin posed what is known as the Mordell-Weil problem for cubic surfaces: if K is finitely generated over its prime subfield then r(S,K) is finite. In this paper, we prove a special case of this conjecture. Namely, if S contain…
We compare the asymptotic grows of the number of rational points on modular varieties of D-elliptic sheaves over finite fields to the grows of their Betti numbers as the degree of the level tends to infinity. This is a generalization to higher dimensions of a well-known result for modular curves. As a consequence of the main result, we also produce a new asymptotically optimal sequence of curves.
A central problem in Diophantine geometry is to uniformly bound the number of $K$-rational points on a smooth curve $X/K$ in terms of $K$ and its genus $g$. A recent paper by Stoll proved uniform bounds for the number of $K$-rational points on a hyperelliptic curve $X$ provided that the rank of the Jacobian of $X$ is at most $g - 3$. Katz, Rabinoff and Zureick-Brown generalized his result to arbitrary curves satisfying the same rank condition. In this paper, we prove conditional uniform bounds on the number of rational points on the symmetric square of $X$ outside its algebraic special set, pr…
We continue work of Gekeler and others on elliptic curves over ${\mathbb F}_q(T)$ with conductor $\infty\cdot{\mathfrak n}$ where ${\mathfrak n}\in{\mathbb F}_q[T]$ has degree 3. Because of the Frobenius isogeny there are infinitely many curves in each isogeny class, and we discuss in particular which of these curves is the strong Weil curve with respect to the uniformization by the Drinfeld modular curve $X_0({\mathfrak n})$. As a corollary we obtain that the strong Weil curve $E/{\mathbb F}_q(T)$ always gives a rational elliptic surface over $\bar{{\mathbb F}_q}$.
The aim of this paper is to study Weil divisors on a singular rational normal scroll X. In particular the author describes explicitly the group of divisorial sheaves associated to Weil divisors on X, via the direct image of the Picard group of the canonical resolution of X, which is very well known. The analysis naturally splits in two cases, depending on the codimension of the vertex of X (>2 or =2). When the codimension of the vertex is >2, then the two groups are isomorphic and there is a natural intersection form in X inherited from the one in the canonical resolution. When the codimension…
Drawing the secant through two rational points of a cubic surface we can get the third one. Is the set of rational points finitely generated? We discuss some numerical data and prove a finite generation statement with respect to a modified composition law.
Let $A$ be a simple abelian variety of dimension $g$ over the field $\mathbb{F}_q$. The paper provides improvements on the Weil estimates for the size of $A(\mathbb{F}_q)$. For an arbitrary value of $q$ we prove $(\lfloor(\sqrt{q}-1)^2 \rfloor + 1)^g \leqslant A(\mathbb{F}_q) \leqslant (\lceil(\sqrt{q}+1)^2 \rceil - 1)^{g}$ holds with finitely many exceptions. We compute improved bounds for various small values of $q$. For instance, the Weil bounds for $q=3,4$ give a trivial estimate $A(\mathbb{F}_q) \geqslant 1$; we prove $A(\mathbb{F}_3) \geqslant 1.359^g$ and $A(\mathbb{F}_4) \geqslant 2.27…
We introduce the use of $p$-descent techniques for elliptic surfaces over a perfect field of characteristic not $2$ or $3$. Under mild hypotheses, we obtain an upper bound for the rank of a non-constant elliptic surface. When $p=2$, this bound is an arithmetic refinement of a well-known geometric bound for the rank deduced from Igusa's inequality. This answers a question raised by Ulmer. We give some applications to rank bounds for elliptic surfaces over the rational numbers.
PhD dissertation consists in three lines of investigation involving rational elliptic surfaces, namely 1) a study of conic bundles on these surfaces; 2) an investigation of the possible intersection numbers of two sections and 3) a theorem regarding the set of fibers whose Mordell-Weil ranks are at least 3 units higher than the generic rank of the elliptic fibration.
If E is a non-isotrivial elliptic curve over a global function field F of odd characteristic we show that certain Mordell-Weil groups of E have 1-dimensional eigenspace relative to a fixed complex ring class character provided that the projection onto this eigenspace of a suitable Drinfeld-Heegner point is nonzero. This represents the analogue in the function field setting of a theorem for rational elliptic curves due to Bertolini and Darmon, and at the same time is a generalization of the main result proved by Brown in his monograph on Heegner modules. As in the number field case, our proof e…
Let $E$ be an elliptic surface over the curve $C$, defined over a number field $k$, let $P$ be a section of $E$, and let $\ell$ be a rational prime. For any non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of (algebraic) degree at most $D$ over $k$, such that $\ell^n Q=P_t$, for some $n\geq 1$. The bound obtained depends only on $\ell$, the surface and section in question, $D$, and the degree $[k(t):k]$; that is, it is uniform across all fibres of bounded degree. In special cases, we obtain more specific, in some instances sharp, bounds.
Given a rational elliptic surface X over an algebraically closed field, we investigate whether a given natural number k can be the intersection number of two sections of X. If not, we say that k a gap number. We try to answer when gap numbers exist, how they are distributed and how to identify them. We use Mordell-Weil lattices as our main tool, which connects the investigation to the classical problem of representing integers by positive-definite quadratic forms.
Let $K$ be a field finitely generated over the field of rational numbers, $K(c)$ the extension of $K$ obtained by adjoining all roots of unity, $L$ an infinite Galois extension of $K$, $X$ an abelian variety defined over $K$. We prove that under certain conditions on $X$ and $K$ the existence of infinitely many L-rational points of finite order on $X$ implies that the intersection of $L$ and $K(c)$ has infinite degree over $K$.
This is a textbook on arithmetic geometry with special regard to unramified Brauer groups of algebraic varieties. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, arithmetic and geometry of quadrics, Weil restriction of scalars, algebraic tori, an example of a stably rational non-rational variety, Brauer-Manin obstruction. All material is split into locally trivial problems with detailed hints.
I provide a systematic construction of points (defined over number fields) on Legendre elliptic curves over $\mathbb{Q}$: for any odd integer $n\geq 3$ my method constructs $n$ points on the Legendre curve and I show that rank of the subgroup of the Mordell-Weil group they generate is $n$ if $n\geq 7$. I also show that every elliptic curve over any number field admits similar type of points after a finite base extension.
In this paper, we study rational sections of the relative Picard scheme of a linear system on a smooth projective variety. We prove that if the linear system is basepoint-free and the locus of non-integral divisors has codimension at least two, then all rational sections of the relative Picard scheme come from restrictions of line bundles on the variety. As a consequence, we describe the group of sections of the Hitchin fibration for moduli spaces of Higgs bundles on curves.
To a pair of elliptic curves, one can naturally attach two K3 surfaces: the Kummer surface of their product and a double cover of it, called the Inose surface. They have prominently featured in many interesting constructions in algebraic geometry and number theory. There are several more associated elliptic K3 surfaces, obtained through base change of the Inose surface; these have been previously studied by Kuwata. We give an explicit description of the geometric Mordell-Weil groups of each of these elliptic surfaces in the generic case (when the elliptic curves are non-isogenous). In the non-…
We show that the Mordell Weil rank of an isotrivial abelian variety with a cyclic holonomy depends only on the fundamental group of the complement to the discriminant provided the discriminant has singularities in the introduced here CM class. This class of singularities includes all unibranched plane curves singularities. As a corollary we give a family of simple Jacobians over field of rational functions in two variable for which the Mordell Weil rank is arbitrary large.
In studying rational points on elliptic K3 surfaces of the form $f(t)y^2=g(x)$, where $f,g$ are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell-Weil rank. We prove a necessary and sufficient condition for the Zariski density of rational points by using this condition, and we relate it to the Hilbert property. Applying to surfaces of Cassels-Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology.
In this paper, we introduce the terminology of matroids into the study of Zariski-pairs related to rational elliptic surfaces, aiming to simplify the presentation and arguments involved. As an application, we provide new examples of Zariski $N$-ples of relatively low degree. Namely we show that a Zariski 102-ple of degree 18 exists.
We give a systematic method to calculate some homological data from the global monodromy of a topological elliptic surface. We apply this method to the cases 1) the transcendental lattice of an extremal elliptic K3 surface, 2) the torsion part of Mordell-Weil group of a general elliptic surface, and 3) the Mordell-Weil lattice of a rational elliptic surface. We present possible list of global monodromies of rational elliptic surface and extremal elliptic $K3$ surface which reproduce to the classification obtained by Oguiso-Shioda and Shimada-Zhang respectively.
We prove that when all hyperelliptic curves of genus $n\geq 1$ having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2. The method of Chabauty can then be used to obtain an effective bound on the number of rational points on most of these hyperelliptic curves; for example, we show that a majority of hyperelliptic curves of genus $n\geq 3$ with a rational Weierstrass point have fewer than 20 rational points.
We describe two constructions of elliptic K3 surfaces starting from the Kummer surface of the Jacobian of a genus 2 curve. These parallel the base-change constructions of Kuwata for the Kummer surface of a product of two elliptic curves. One of these also involves the analogue of an Inose fibration. We use these methods to provide explicit examples of elliptic K3 surfaces over the rationals of geometric Mordell-Weil rank 15.
Let $\mathbb{F}_q$ be a finite field with $q=p^n$ elements. In this paper, we study the number of $\mathbb{F}_q$-rational points on the affine hypersurface $\mathcal X$ given by $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$, where $b\in\mathbb{F}_q^*$. A classic well-konwn result from Weil yields a bound for such number of points. This paper presents necessary and sufficient conditions for the maximality and minimality of $\mathcal X$ with respect to Weil's bound.
Let $p$ be a prime, let $V/\mathbb{F}_p$ be an absolutely irreducible affine variety inside the affine $r$-space. In this paper, we consider the problem of how often a box $B$ will contain the expected number of points. In particular, we give a lower bound on the volume of $B$ that guarantees almost all translations of $B$ in the $r$-space contain the expected number of points. This shows that the Weil estimate holds in smaller regions in an "almost all" sense.
Let V be a plane smooth cubic curve over a finitely generated field k. The Mordell-Weil theorem for V states that there is a finite subset P \subset V(k) such that the whole V(k) can be obtained from P by drawing secants and tangents through pairs of previously constructed points and consecutively adding their new intersection points with V. In this paper we present numerical data regarding the analogous statement for cubic surfaces. For the surfaces examined, we also test Manin's conjecture relating the asymptotics of rational points of bounded height on a Fano variety with the rank of the Pi…
We study the arithmetic of the twist of the Fermat quartic defined by $X^4 + Y^4 + Z^4 = 0$ which has no $\mathbb{Q}$-rational point. We calculate the Mordell--Weil group of the Jacobian variety explicilty. We show that the degree $0$ part of the Picard group is a free $\mathbb{Z}/2\mathbb{Z}$-module of rank $2$, whereas the Mordell--Weil group is a free $\mathbb{Z}/2\mathbb{Z}$-module of rank $3$. Thus the relative Brauer group is non-trivial. We also show that this quartic violates the local-global property for linear determinantal representations.
Let $K$ be a field finitely generated over ${\Q}$, and $A$ an Abelian variety defined over $K$. Then by the Mordell-Weil Theorem, the set of rational points $A(K)$ is a finitely-generated Abelian group. In this paper, assuming Tate's Conjecture on algebraic cycles, we prove a limit formula for the Mordell-Weil rank of an arbitrary family of Abelian varieties $A$ over a number field $k$; this is the Abelian fibration analogue of the Nagao formula for elliptic surfaces $E$, originally conjectured by Nagao, and proven by Rosen and Silverman to be equivalent to Tate's Conjecture for $E$. We also g…
Consider an ordinary elliptic curve $E_b\!: y^2 = x^3 - b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_{\!q}$ such that $\sqrt[3]{b} \notin \mathbb{F}_{\!q}$. This article tries to resolve the problem of constructing a rational $\mathbb{F}_{\!q}$-curve on the Kummer surface of the direct product $E_b \!\times\! E_b^\prime$, where $E_b^\prime$ is the quadratic $\mathbb{F}_{\!q}$-twist of $E_b$. More precisely, we propose to search such a curve among infinite order $\mathbb{F}_{\!q}$-sections of some elliptic surface of $j=0$, analyzing its Mordell--Weil group. Unfortunately, we prove…
We consider a rational surface with a relatively minimal fibration. Picard number of a such fibred surface is bounded in terms of the genus of a general fibre. When Picard number is the maximum for any given genus, we characterize a such fibred surface whose Mordell-Weil group is trivial by singular fibres. Furthermore, we describe the defining equation explicitly.
We consider relatively minimal fibrations of curves of genus two on rational surfaces whose Picard numbers are not maximal. By birational morphisms, such fibred surfaces are interpreted as pencils of plane curves. We show that only four are canonical, among a variety of possible models. For each canonical pencil, we give an example with trivial Mordell-Weil group.
In this work we use arithmetic, geometric, and combinatorial techniques to compute the cohomology of Weil divisors of a special class of normal surfaces, the so-called rational ruled toric surfaces. These computations are used to study the topology of cyclic coverings of such surfaces ramified along Q-normal crossing divisors.
