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The sequence of prime numbers p for which a variety over ℚ has no p-adic point plays a fundamental role in arithmetic geometry. This sequence is deterministic, however, we prove that if we choose a typical variety from a family then the sequence has random behaviour. We furthermore prove that this behaviour is modelled by a random walk in Brownian motion. This has several consequences, one of them being the description of the finer properties of the distribution of the primes in this sequence via the Feynman–Kac formula.
We show that any smooth projective cubic hypersurface of dimension at least 29 over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley.
We include an appendix in which we highlight some slight modifications to a recent result of Papanikolopoulos and Siksek. It follows that the set of rational points on smooth projective cubic hypersurfaces of dimension at least 29 is generated via secant and tangent constructions from just a single point.
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.
We provide evidence for this conclusion: given a finite Galois cover
$f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$
of group
$G$
, almost all (in a density sense) realizations of
$G$
over
$\mathbb{Q}$
do not occur as specializations of
$f$
. We show that this holds if the number of branch points of
$f$
is sufficiently large, under the abc-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of
$\mathbb{Q}$
of given group and bounded discriminant. This widely extends a result of Granville on the lack of
$\mathbb{Q}$
-rational points on quadratic twists of hyperelliptic curves over
$\mathbb{Q}$
with large genus, under the abc-conjecture (a diophantine reformulation of the case
$G=\mathbb{Z}/2\mathbb{Z}$
of our result). As a further evidence, we exhibit a few finite groups
$G$
for which the above conclusion holds unconditionally for almost all covers of
$\mathbb{P}_{\mathbb{Q}}^{1}$
of group
$G$
. We also introduce a local–global principle for specializations of Galois covers
$f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$
and show that it often fails if
$f$
has abelian Galois group and sufficiently many branch points, under the abc-conjecture. On the one hand, such a local–global conclusion underscores the ‘smallness’ of the specialization set of a Galois cover of
$\mathbb{P}_{\mathbb{Q}}^{1}$
. On the other hand, it allows to generate conditionally ‘many’ curves over
$\mathbb{Q}$
failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.
Given a finite group
$\text{G}$
and a field
$K$
, the faithful dimension of
$\text{G}$
over
$K$
is defined to be the smallest integer
$n$
such that
$\text{G}$
embeds into
$\operatorname{GL}_{n}(K)$
. We address the problem of determining the faithful dimension of a
$p$
-group of the form
$\mathscr{G}_{q}:=\exp (\mathfrak{g}\otimes _{\mathbb{Z}}\mathbb{F}_{q})$
associated to
$\mathfrak{g}_{q}:=\mathfrak{g}\otimes _{\mathbb{Z}}\mathbb{F}_{q}$
in the Lazard correspondence, where
$\mathfrak{g}$
is a nilpotent
$\mathbb{Z}$
-Lie algebra which is finitely generated as an abelian group. We show that in general the faithful dimension of
$\mathscr{G}_{p}$
is a piecewise polynomial function of
$p$
on a partition of primes into Frobenius sets. Furthermore, we prove that for
$p$
sufficiently large, there exists a partition of
$\mathbb{N}$
by sets from the Boolean algebra generated by arithmetic progressions, such that on each part the faithful dimension of
$\mathscr{G}_{q}$
for
$q:=p^{f}$
is equal to
$fg(p^{f})$
for a polynomial
$g(T)$
. We show that for many naturally arising
$p$
-groups, including a vast class of groups defined by partial orders, the faithful dimension is given by a single formula of the latter form. The arguments rely on various tools from number theory, model theory, combinatorics and Lie theory.
The Chabauty–Kim method allows one to find rational points on curves under certain technical conditions, generalising Chabauty’s proof of the Mordell conjecture for curves with Mordell–Weil rank less than their genus. We show how the Chabauty–Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a non-abelian generalisation of Coleman’s effective Chabauty theorem.
We consider the problem of counting the number of rational points of bounded height in the zero-loci of Brauer group elements on semi-simple algebraic groups over number fields. We obtain asymptotic formulae for the counting problem for wonderful compactifications using the spectral theory of automorphic forms. Applications include asymptotic formulae for the number of matrices over
$\mathbb{Q}$
whose determinant is a sum of two squares. These results provide a positive answer to some cases of a question of Serre concerning such counting problems.
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.
In a previous article, we proved that Shimura curves have no points rational over number fields under a certain assumption. In this article, we give another criterion of the nonexistence of rational points on Shimura curves and obtain new counterexamples to the Hasse principle for Shimura curves. We also prove that such counterexamples obtained from the above results are accounted for by the Manin obstruction.
We prove analogs of the Bezout and the Bernstein–Kushnirenko–Khovanskii theorems for systems of algebraic differential conditions over differentially closed fields. Namely, given a system of algebraic conditions on the first
$l$
derivatives of an
$n$
-tuple of functions, which admits finitely many solutions, we show that the number of solutions is bounded by an appropriate constant (depending singly-exponentially on
$n$
and
$l$
) times the volume of the Newton polytope of the set of conditions. This improves a doubly-exponential estimate due to Hrushovski and Pillay. We illustrate the application of our estimates in two diophantine contexts: to counting transcendental lattice points on algebraic subvarieties of semi-abelian varieties, following Hrushovski and Pillay; and to counting the number of intersections between isogeny classes of elliptic curves and algebraic varieties, following Freitag and Scanlon. In both cases we obtain bounds which are singly-exponential (improving the known doubly-exponential bounds) and which exhibit the natural asymptotic growth with respect to the degrees of the equations involved.
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.
A strong quantitative form of Manin’s conjecture is established for a certain variety in biprojective space. The singular integral in an application of the circle method involves the third power of the integral sine function and is evaluated in closed form.
Consider a system of polynomials in many variables over the ring of integers of a number field
$K$
. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety
$X\subseteq \mathbb{P}_{K}^{m}$
satisfies the Hasse principle, weak approximation, and the Manin–Peyre conjecture if only its dimension is large enough compared to its degree. This generalizes work of Skinner, who considered the case where all polynomials have the same degree, and recent work of Browning and Heath-Brown, who considered the case where
$K=\mathbb{Q}$
. Our main tool is Skinner’s number field version of the Hardy–Littlewood circle method. As a by-product, we point out and correct an error in Skinner’s treatment of the singular integral.
Given a family of varieties
$X\rightarrow \mathbb{P}^{n}$
over a number field, we determine conditions under which there is a Brauer–Manin obstruction to weak approximation for 100% of the fibres which are everywhere locally soluble.
For any number field we calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field.
We show that the restriction to square-free numbers of the representation function attached to a norm form does not correlate with nilsequences. By combining this result with previous work of Browning and the author, we obtain an application that is used in recent work of Harpaz and Wittenberg on the fibration method for rational points.
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.
We prove an analog of the Yomdin–Gromov lemma for
$p$
-adic definable sets and more broadly in a non-Archimedean definable context. This analog keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of
$p$
-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over
$\mathbb{C}(\!(t)\!)$
, in analogy to results by Pila and Wilkie, and by Bombieri and Pila, respectively. Along the way we prove, for definable functions in a general context of non-Archimedean geometry, that local Lipschitz continuity implies piecewise global Lipschitz continuity.
We study the arithmetic of a family of non-hyperelliptic curves of genus 3 over the field
$\mathbb{Q}$
of rational numbers. These curves are the nearby fibers of the semi-universal deformation of a simple singularity of type
$E_{6}$
. We show that average size of the 2-Selmer sets of these curves is finite (if it exists). We use this to show that a positive proposition of these curves (when ordered by height) has integral points everywhere locally, but no integral points globally.