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We study the generalized Fermat equation
$x^{2}+y^{3}=z^{p}$
, to be solved in coprime integers, where
$p\geqslant 7$
is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve
$X(p)$
. We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic
$p$
-torsion modules. Using these criteria we produce the minimal list of twists of
$X(p)$
that have to be considered, based on local information at 2 and 3; this list depends on
$p\hspace{0.2em}{\rm mod}\hspace{0.2em}24$
. We solve the equation completely when
$p=11$
, which previously was the smallest unresolved
$p$
. One new ingredient is the use of the ‘Selmer group Chabauty’ method introduced by the third author, applied in an elliptic curve Chabauty context, to determine relevant points on
$X_{0}(11)$
defined over certain number fields of degree 12. This result is conditional on the generalized Riemann hypothesis, which is needed to show correctness of the computation of the class groups of five specific number fields of degree 36. We also give some partial results for the case
$p=13$
. The source code for the various computations is supplied as supplementary material with the online version of this article.
Given systems of two (inhomogeneous) quadratic equations in four variables, it is known that the Hasse principle for integral points may fail. Sometimes this failure can be explained by some integral Brauer–Manin obstruction. We study the existence of a non-trivial algebraic part of the Brauer group for a family of such systems and show that the failure of the integral Hasse principle due to an algebraic Brauer–Manin obstruction is rare, as for a generic choice of a system the algebraic part of the Brauer-group is trivial. We use resolvent constructions to give quantitative upper bounds on the number of exceptions.
Let
$G$
be a connected linear algebraic group over a number field
$k$
. Let
$U{\hookrightarrow}X$
be a
$G$
-equivariant open embedding of a
$G$
-homogeneous space
$U$
with connected stabilizers into a smooth
$G$
-variety
$X$
. We prove that
$X$
satisfies strong approximation with Brauer–Manin condition off a set
$S$
of places of
$k$
under either of the following hypotheses:
(i)
$S$
is the set of archimedean places;
(ii)
$S$
is a non-empty finite set and
$\bar{k}^{\times }=\bar{k}[X]^{\times }$
.
The proof builds upon the case
$X=U$
, which has been the object of several works.
Let
$K$
be an algebraic number field. A cuboid is said to be
$K$
-rational if its edges and face diagonals lie in
$K$
. A
$K$
-rational cuboid is said to be perfect if its body diagonal lies in
$K$
. The existence of perfect
$\mathbb{Q}$
-rational cuboids is an unsolved problem. We prove here that there are infinitely many distinct cubic fields
$K$
such that a perfect
$K$
-rational cuboid exists; and that, for every integer
$n\geq 2$
, there is an algebraic number field
$K$
of degree
$n$
such that there exists a perfect
$K$
-rational cuboid.
Building on recent work of Bhargava, Elkies and Schnidman and of Kriz and Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.
Let
$C\in \mathbb{Z}[x_{1},\ldots ,x_{n}]$
be a cubic form. Assume that
$C$
splits into four forms. Then
$C(x_{1},\ldots ,x_{n})=0$
has a non-trivial integer solution provided that
$n\geqslant 10$
.
We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what is computed to the cohomologically defined Selmer groups. Selmer group computations have been practical for many Jacobians of curves over
$\mathbb{Q}$
of genus up to 2 since the 1990s, but our approach is the first to be practical for general curves of genus 3. We show that our approach succeeds on some genus 3 examples defined by polynomials with small coefficients.
A conjecture of Scharaschkin and Skorobogatov states that there is a Brauer–Manin obstruction to the existence of rational points on a smooth geometrically irreducible curve over a number field. In this paper, we verify the Scharaschkin–Skorobogatov conjecture for explicit families of generalized Mordell curves. Our approach uses standard techniques from the Brauer–Manin obstruction and the arithmetic of certain threefolds.
The second author has recently introduced a new class of
$L$
-series in the arithmetic theory of function fields over finite fields. We show that the values at one of these
$L$
-series encode arithmetic information of a generalization of Drinfeld modules defined over Tate algebras that we introduce (the coefficients can be chosen in a Tate algebra). This enables us to generalize Anderson’s log-algebraicity theorem and an analogue of the Herbrand–Ribet theorem recently obtained by Taelman.
For
$n=2$
the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For
$n=p$
,
$p$
any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its
$(2p+1)$
th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).
Assuming the Tate conjecture and the computability of étale cohomology with finite coefficients, we give an algorithm that computes the Néron–Severi group of any smooth projective geometrically integral variety, and also the rank of the group of numerical equivalence classes of codimension
$p$
cycles for any
$p$
.
In this paper we consider ordinary elliptic curves over global function fields of characteristic
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$
. We present a method for performing a descent by using powers of the Frobenius and the Verschiebung. An examination of the local images of the descent maps together with a duality theorem yields information about the global Selmer groups. Explicit models for the homogeneous spaces representing the elements of the Selmer groups are given and used to construct independent points on the elliptic curve. As an application we use descent maps to prove an upper bound for the naive height of an
$S$
-integral point on
$A$
. To illustrate our methods, a detailed example is presented.
Given an intersection of two quadrics
$X\subset { \mathbb{P} }^{m- 1} $
, with
$m\geq 9$
, the quantitative arithmetic of the set
$X( \mathbb{Q} )$
is investigated under the assumption that the singular locus of
$X$
consists of a pair of conjugate singular points defined over
$ \mathbb{Q} (i)$
.
Let
$J$
be an abelian variety and
$A$
be an abelian subvariety of
$J$
, both defined over
$Q$
. Let
$x$
be an element of
${{H}^{1}}\left( Q,\,A \right)$
. Then there are at least two definitions of
$x$
being visible in
$J$
: one asks that the torsor corresponding to
$x$
be isomorphic over
$Q$
to a subvariety of
$J$
, and the other asks that
$x$
be in the kernel of the natural map
$
{{H}^{1}}\left( Q,\,A \right)\,\to \,{{H}^{1}}\left( \text{Q},\,J \right)$
. In this article, we clarify the relation between the two definitions.
Let X be a smooth projective variety over a finite field . We discuss the unramified cohomology group H3nr(X, ℚ/ℤ(2)). Several conjectures put together imply that this group is finite. For certain classes of threefolds, H3nr(X, ℚ/ℤ(2)) actually vanishes. It is an open question whether this holds for arbitrary threefolds. For a threefold X equipped with a fibration onto a curve C, the generic fibre of which is a smooth projective surface V over the global field (C), the vanishing of H3nr(X, ℚ/ℤ(2)) together with the Tate conjecture for divisors on X implies a local-global principle of Brauer–Manin type for the Chow group of zero-cycles on V. This sheds new light on work started thirty years ago.
Frey and Jarden asked if any abelian variety over a number field
$K$
has the infinite Mordell–Weil rank over the maximal abelian extension
${{K}^{\text{ab}}}$
. In this paper, we give an affirmative answer to their conjecture for the Jacobian variety of any smooth projective curve
$C$
over
$K$
such that
$\sharp C\left( {{K}^{\text{ab}}} \right)\,=\,\infty $
and for any abelian variety of
$\text{G}{{\text{L}}_{2}}$
-type with trivial character.
We study the arithmetic of abelian varieties over where is an arbitrary field. The main result relates Mordell–Weil groups of certain Jacobians over to homomorphisms of other Jacobians over . Our methods also yield completely explicit points on elliptic curves with unbounded rank over and a new construction of elliptic curves with moderately high rank over .
We associate to certain filtrations of a graded linear series of a big line bundle a concave function on its Okounkov body, whose law with respect to the Lebesgue measure describes the asymptotic distribution of the jumps of the filtration. As a consequence, we obtain a Fujita-type approximation theorem in this general filtered setting. We then specialize these results to the filtrations by minima in the usual context of Arakelov geometry (and for more general adelically normed graded linear series), thereby obtaining in a simple way a natural construction of an arithmetic Okounkov body, the existence of the arithmetic volume as a limit and an arithmetic Fujita approximation theorem for adelically normed graded linear series. We also obtain an easy proof of the existence of the sectional capacity previously obtained by Lau, Rumely and Varley.
We discuss the Mordell–Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the Mordell–Weil sieve algorithm and discuss its efficiency.
Let f1,…,fg∈ℂ(z) be rational functions, let Φ=(f1,…,fg) denote their coordinate-wise action on (ℙ1)g, let V ⊂(ℙ1)g be a proper subvariety, and let P be a point in (ℙ1)g(ℂ). We show that if 𝒮={n≥0:Φn(P)∈V (ℂ)} does not contain any infinite arithmetic progressions, then 𝒮 must be a very sparse set of integers. In particular, for any k and any sufficiently large N, the number of n≤N such that Φn(P)∈V (ℂ) is less than log kN, where log k denotes the kth iterate of the log function. This result can be interpreted as an analogue of the gap principle of Davenport–Roth and Mumford.