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Aigner showed in 1934 that nontrivial quadratic solutions to
$x^4 + y^4 = 1$
exist only in
$\mathbb Q(\sqrt {-7})$
. Following a method of Mordell, we show that nontrivial quadratic solutions to
$x^4 + 2^ny^4 = 1$
arise from integer solutions to the equations
$X^4 \pm 2^nY^4 = Z^2$
investigated in 1853 by V. A. Lebesgue.
We show, under some natural restrictions, that some semigroup orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime
$p$
, extending previous work of Shparlinski [‘Multiplicative orders in orbits of polynomials over finite fields’, Glasg. Math. J.60(2) (2018), 487–493].
Let
$X$
be a finite-dimensional connected compact abelian group equipped with the normalized Haar measure
$\unicode[STIX]{x1D707}$
. We obtain the following mean ergodic theorem over ‘thin’ phase sets. Fix
$k\geq 1$
and, for every
$n\geq 1$
, let
$A_{n}$
be a subset of
$\mathbb{Z}^{k}\cap [-n,n]^{k}$
. Assume that
$(A_{n})_{n\geq 1}$
has
$\unicode[STIX]{x1D714}(1/n)$
density in the sense that
$\lim _{n\rightarrow \infty }(|A_{n}|/n^{k-1})=\infty$
. Let
$T_{1},\ldots ,T_{k}$
be ergodic automorphisms of
$X$
. We have
for any
$f_{1},\ldots ,f_{k}\in L_{\unicode[STIX]{x1D707}}^{\infty }$
. When the
$T_{i}$
are ergodic epimorphisms, the same conclusion holds under the further assumption that
$A_{n}$
is a subset of
$[0,n]^{k}$
for every
$n$
. The density assumption on the
$A_{i}$
is necessary. Immediate applications include certain Poincaré style recurrence results.
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.
Using work of the first author [S. Bettin, High moments of the Estermann function. Algebra Number Theory47(3) (2018), 659–684], we prove a strong version of the Manin–Peyre conjectures with a full asymptotic and a power-saving error term for the two varieties respectively in
$\mathbb{P}^{2}\times \mathbb{P}^{2}$
with bihomogeneous coordinates
$[x_{1}:x_{2}:x_{3}],[y_{1}:y_{2},y_{3}]$
and in
$\mathbb{P}^{1}\times \mathbb{P}^{1}\times \mathbb{P}^{1}$
with multihomogeneous coordinates
$[x_{1}:y_{1}],[x_{2}:y_{2}],[x_{3}:y_{3}]$
defined by the same equation
$x_{1}y_{2}y_{3}+x_{2}y_{1}y_{3}+x_{3}y_{1}y_{2}=0$
. We thus improve on recent work of Blomer et al [The Manin–Peyre conjecture for a certain biprojective cubic threefold. Math. Ann.370 (2018), 491–553] and provide a different proof based on a descent on the universal torsor of the conjectures in the case of a del Pezzo surface of degree 6 with singularity type
$\mathbf{A}_{1}$
and three lines (the other existing proof relying on harmonic analysis by Chambert-Loir and Tschinkel [On the distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math.148 (2002), 421–452]). Together with Blomer et al [On a certain senary cubic form. Proc. Lond. Math. Soc.108 (2014), 911–964] or with work of the second author [K. Destagnol, La conjecture de Manin pour une famille de variétés en dimension supérieure. Math. Proc. Cambridge Philos. Soc.166(3) (2019), 433–486], this settles the study of the Manin–Peyre conjectures for this equation.
We establish asymptotic formulae for the number of
$k$
-free values of square-free polynomials
$F(x_{1},\ldots ,x_{n})\in \mathbb{Z}[x_{1},\ldots ,x_{n}]$
of degree
$d\geqslant 2$
for any
$n\geqslant 1$
, including when the variables are prime, as long as
$k\geqslant (3d+1)/4$
. This generalizes a work of Browning.
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.
We shall show that, for any positive integer D > 0 and any primes p1, p2, the diophantine equation x2 + D = 2sp1kp2l has at most 63 integer solutions (x, k, l, s) with x, k, l ≥ 0 and s ∈ {0, 2}.
Let
$\mathbf{f}=(f_{1},\ldots ,f_{R})$
be a system of polynomials with integer coefficients in which the degrees need not all be the same. We provide sufficient conditions for which the system of equations
$f_{j}(x_{1},\ldots ,x_{n})=0~(1\leqslant j\leqslant R)$
satisfies a general local to global type statement, and has a solution where each coordinate is prime. In fact we obtain the asymptotic formula for number of such solutions, counted with a logarithmic weight, under these conditions. We prove the statement via the Hardy–Littlewood circle method. This is a generalization of the work of Cook and Magyar [‘Diophantine equations in the primes’, Invent. Math.198 (2014), 701–737], where they obtained the result when the polynomials of
$\mathbf{f}$
all have the same degree. Hitherto, results of this type for systems of polynomial equations involving different degrees have been restricted to the diagonal case.
Let
$s\geqslant 3$
be a fixed positive integer and let
$a_{1},\ldots ,a_{s}\in \mathbb{Z}$
be arbitrary. We show that, on average over
$k$
, the density of numbers represented by the degree
$k$
diagonal form
For any odd prime
$\ell$
, let
$h_{\ell }(-d)$
denote the
$\ell$
-part of the class number of the imaginary quadratic field
$\mathbb{Q}(\sqrt{-d})$
. Nontrivial pointwise upper bounds are known only for
$\ell =3$
; nontrivial upper bounds for averages of
$h_{\ell }(-d)$
have previously been known only for
$\ell =3,5$
. In this paper we prove nontrivial upper bounds for the average of
$h_{\ell }(-d)$
for all primes
$\ell \geqslant 7$
, as well as nontrivial upper bounds for certain higher moments for all primes
$\ell \geqslant 3$
.
Schmidt [‘Integer points on curves of genus 1’, Compos. Math.81 (1992), 33–59] conjectured that the number of integer points on the elliptic curve defined by the equation
$y^{2}=x^{3}+ax^{2}+bx+c$
, with
$a,b,c\in \mathbb{Z}$
, is
$O_{\unicode[STIX]{x1D716}}(\max \{1,|a|,|b|,|c|\}^{\unicode[STIX]{x1D716}})$
for any
$\unicode[STIX]{x1D716}>0$
. On the other hand, Duke [‘Bounds for arithmetic multiplicities’, Proc. Int. Congress Mathematicians, Vol. II (1998), 163–172] conjectured that the number of algebraic number fields of given degree and discriminant
$D$
is
$O_{\unicode[STIX]{x1D716}}(|D|^{\unicode[STIX]{x1D716}})$
. In this note, we prove that Duke’s conjecture for quartic number fields implies Schmidt’s conjecture. We also give a short unconditional proof of Schmidt’s conjecture for the elliptic curve
$y^{2}=x^{3}+ax$
.
with
$1\leqslant x_{i},y_{i}\leqslant X\;(1\leqslant i\leqslant s)$
. By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for
$I_{s,k,r}(X)$
for
$1\leqslant r\leqslant k-1$
. In particular, when
$s,k\in \mathbb{N}$
satisfy
$k\geqslant 3$
and
$1\leqslant s\leqslant (k^{2}-1)/2$
, we establish the essentially diagonal behaviour
$I_{s,k,1}(X)\ll X^{s+\unicode[STIX]{x1D700}}$
.
We improve the known upper bound for the number of Diophantine
$D(4)$
-quintuples by using the most recent methods that were developed in the
$D(1)$
case. More precisely, we prove that there are at most
$6.8587\times 10^{29}$
$D(4)$
-quintuples.
An asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy-Littlewood method, and along the way we establish two new results on Weyl sums that are of some independent interest.
We classify generically transitive actions of semi-direct products on ℙ2. Motivated by the program to study the distribution of rational points on del Pezzo surfaces (Manin's conjecture), we determine all (possibly singular) del Pezzo surfaces that are equivariant compactifications of homogeneous spaces for semi-direct products .
A conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin’s conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin’s conjecture over imaginary quadratic fields
$K$
for the quartic del Pezzo surface
$S$
of singularity type
${\boldsymbol{A}}_{3}$
with five lines given in
${\mathbb{P}}_{K}^{4}$
by the equations
${x}_{0}{x}_{1}-{x}_{2}{x}_{3}={x}_{0}{x}_{3}+{x}_{1}{x}_{3}+{x}_{2}{x}_{4}=0$
.
Let
$Q(N;q,a)$
be the number of squares in the arithmetic progression
$qn+a$
, for
$n=0$
,
$1,\ldots,N-1$
, and let
$Q(N)$
be the maximum of
$Q(N;q,a)$
over all non-trivial arithmetic progressions
$qn + a$
. Rudin’s conjecture claims that
$Q(N)=O(\sqrt{N})$
, and in its stronger form that
$Q(N)=Q(N;24,1)$
if
$N\ge 6$
. We prove the conjecture above for
$6\le N\le 52$
. We even prove that the arithmetic progression
$24n+1$
is the only one, up to equivalence, that contains
$Q(N)$
squares for the values of
$N$
such that
$Q(N)$
increases, for
$7\le N\le 52$
(
$N=8,13,16,23,27,36,41$
and
$52$
).
Nous démontrons, sous la forme forte conjecturée par Peyre, la conjecture de Manin pour les surfaces de Châtelet dont les équations sont du type
${y}^{2} + {z}^{2} = P(x, 1)$
, où
$P$
est une forme binaire quartique à coefficients entiers irréductible sur
$ \mathbb{Q} [i] $
ou produit de deux formes quadratiques à coefficients entiers irréductibles sur
$ \mathbb{Q} [i] $
. De plus, nous fournissons une estimation explicite du terme d’erreur de la formule asymptotique sous-jacente. Cela finalise essentiellement la validation de la conjecture de Manin pour l’ensemble des surfaces de Châtelet. La preuve s’appuie sur deux méthodes nouvelles, concernant, du part, les estimations en moyenne d’oscillations locales de caractères sur les diviseurs, et, d’autre part, les majorations de certaines fonctions arithmétiques de formes binaires.