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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.
We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers
$\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$
of
$\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$
, where
$\unicode[STIX]{x1D6E5}$
is a finite-index subgroup of
$\text{SL}(n+1,\mathbb{Z})$
. These subsets can be obtained by projecting to the hyperplane
$\{(x_{1},\ldots ,x_{n+1})\in \mathbb{R}^{n+1}:x_{n+1}=1\}$
sets of the form
$\mathbf{A}=\bigcup _{j=1}^{J}\mathbf{a}_{j}\unicode[STIX]{x1D6E5}$
, where for all
$j$
,
$\mathbf{a}_{j}$
is a primitive lattice point in
$\mathbb{Z}^{n+1}$
. Our method involves applying the equidistribution of expanding horospheres in quotients of
$\text{SL}(n+1,\mathbb{R})$
developed by Marklof and Strömbergsson, and more precisely understanding how the full Farey sequence distributes in
$\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$
when embedded on expanding horospheres as done in previous work by Marklof. For each of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erdős–Szüsz–Turán and Kesten. We also prove that Marklof’s result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form
$\mathbf{A}$
.
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.
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.
Such a sequence is eventually periodic and we denote by
$P(n)$
the maximal period of such sequences for given
$n$
. We prove a new upper bound in the case where
$n$
is a power of a prime
$p\equiv 5\hspace{0.6em}({\rm mod}\hspace{0.2em}8)$
for which
$2$
is a primitive root and the Pellian equation
$x^{2}-py^{2}=-4$
has no solutions in odd integers
$x$
and
$y$
.
Let
$D$
be a positive nonsquare integer,
$p$
a prime number with
$p\nmid D$
and
$0<\unicode[STIX]{x1D70E}<0.847$
. We show that there exist effectively computable constants
$C_{1}$
and
$C_{2}$
such that if there is a solution to
$x^{2}+D=p^{n}$
with
$p^{n}>C_{1}$
, then for every
$x>C_{2}$
with
$x^{2}+D=p^{n}m$
we have
$m>x^{\unicode[STIX]{x1D70E}}$
. As an application, we show that for
$x\neq \{5,1015\}$
, if the equation
$x^{2}+76=101^{n}m$
holds, then
$m>x^{0.14}$
.
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 generalize work by Bourgain and Kontorovich [On the local-global conjecture for integral Apollonian gaskets, Invent. Math. 196 (2014), 589–650] and Zhang [On the local-global principle for integral Apollonian 3-circle packings, J. Reine Angew. Math. 737, (2018), 71–110], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group
${\mathcal{A}}\leqslant \text{PSL}_{2}(K)$
satisfying certain conditions, where
$K$
is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof is that
${\mathcal{A}}$
possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in
$\operatorname{PSL}_{2}({\mathcal{O}}_{K})$
containing a Zariski dense subgroup of
$\operatorname{PSL}_{2}(\mathbb{Z})$
.
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.
In this paper we show that a polynomial equation admits infinitely many prime-tuple solutions, assuming only that the equation satisfies suitable local conditions and the polynomial is sufficiently non-degenerate algebraically. Our notion of algebraic non-degeneracy is related to the
$h$
-invariant introduced by W. M. Schmidt. Our results prove a conjecture by B. Cook and Á. Magyar for hypersurfaces of degree 3.
We prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed m there are at most
${\cal O}_{\epsilon }(n^{{3}/{5}+\epsilon })$
solutions of
${m}/{n} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$
. This improves upon a result of Browning and Elsholtz (2011) and extends a result of Elsholtz and Tao (2013) who proved this when m=4 and n is a prime. Moreover, there exists an algorithm finding all solutions in expected running time
${\cal O}_{\epsilon }(n^{\epsilon }({n^3}/{m^2})^{{1}/{5}})$
, for any
$\epsilon \gt 0$
. We also improve a bound on the maximum number of representations of a rational number as a sum of k unit fractions. Furthermore, we also improve lower bounds. In particular, we prove that for given
$m \in {\open N}$
in every reduced residue class e mod f there exist infinitely many primes p such that the number of solutions of the equation
${m}/{p} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$
is
$\gg _{f,m} \exp (({5\log 2}/({12\,{\rm lcm} (m,f)}) + o_{f,m}(1)) {\log p}/{\log \log p})$
. Previously, the best known lower bound of this type was of order
$(\log p)^{0.549}$
.
We show that Hermite’s theorem fails for every integer
$n$
of the form
$3^{k_{1}}+3^{k_{2}}+3^{k_{3}}$
with integers
$k_{1}>k_{2}>k_{3}\geqslant 0$
. This confirms a conjecture of Brassil and Reichstein. We also obtain new results for the relative Hermite–Joubert problem over a finitely generated field of characteristic 0.
We improve some previously known deterministic algorithms for finding integer solutions
$x,y$
to the exponential equation of the form
$af^{x}+bg^{y}=c$
over finite fields.
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
$\mathbf{f}=(f_{1},\ldots ,f_{R})$
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.
The cardinality of the set of
$D\leqslant x$
for which the fundamental solution of the Pell equation
$t^{2}-Du^{2}=1$
is less than
$D^{1/2+\unicode[STIX]{x1D6FC}}$
with
$\unicode[STIX]{x1D6FC}\in [\frac{1}{2},1]$
is studied and certain lower bounds are obtained, improving previous results of Fouvry by introducing the
$q$
-analogue of van der Corput method to algebraic exponential sums with smooth moduli.
with positive integers
$q,p\geq 2$
and
$\gcd (F_{n},F_{m})=1$
, where
$F_{k}$
is a Fibonacci number. We obtain results for
$q=2$
or
$q$
an odd prime with
$q\equiv 3\;(\text{mod}\;4),3<q<1087$
, and complete solutions for
$q=3$
.
We study almost prime solutions of systems of Diophantine equations in the Birch setting. Previous work shows that there exist integer solutions of size
$B$
with each component having no prime divisors below
$B^{1/u}$
, where
$u$
equals
$c_{0}n^{3/2}$
,
$n$
is the number of variables and
$c_{0}$
is a constant depending on the degree and the number of equations. We improve the polynomial growth
$n^{3/2}$
to the logarithmic
$(\log n)(\log \log n)^{-1}$
. Our main new ingredients are the generalization of the Brüdern–Fouvry vector sieve in any dimension and the incorporation of smooth weights into the Davenport–Birch version of the circle method.
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
$c\geq 2$
be a positive integer. Terai [‘A note on the Diophantine equation
$x^{2}+q^{m}=c^{n}$
’, Bull. Aust. Math. Soc.90 (2014), 20–27] conjectured that the exponential Diophantine equation
$x^{2}+(2c-1)^{m}=c^{n}$
has only the positive integer solution
$(x,m,n)=(c-1,1,2)$
. He proved his conjecture under various conditions on
$c$
and
$2c-1$
. In this paper, we prove Terai’s conjecture under a wider range of conditions on
$c$
and
$2c-1$
. In particular, we show that the conjecture is true if
$c\equiv 3\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$
and
$3\leq c\leq 499$
.