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The Manin–Peyre conjecture is established for smooth spherical Fano threefolds of semisimple rank one and type N. Together with the previously solved case T and the toric cases, this covers all types of smooth spherical Fano threefolds. The case N features a number of structural novelties; most notably, one may lose regularity of the ambient toric variety, the height conditions may contain fractional exponents, and it may be necessary to exclude a thin subset with exceptionally many rational points from the count, as otherwise Manin's conjecture in its original form would turn out to be incorrect. [ABSTRACT FROM AUTHOR]

In this note we present a construction of an infinite family of diagonal quintic threefolds defined over \mathbb {Q} each containing infinitely many rational points. As an application, we prove that there are infinitely many quadruples B=(B_{0}, B_{1}, B_{2}, B_{3}) of co-prime integers such that for a suitable chosen integer b (depending on B), the equation B_{0}X_{0}^5+B_{1}X_{1}^5+B_{2}X_{2}^5+B_{3}X_{3}^{5}=b has infinitely many positive integer solutions. [ABSTRACT FROM AUTHOR]

We count the number of rational elliptic curves of bounded naive height that have a rational N -isogeny, for N ∈ { 2 , 3 , 4 , 5 , 6 , 8 , 9 , 12 , 16 , 18 }. For some N , this is done by generalizing a method of Harron and Snowden. For the remaining cases, we use the framework of Ellenberg, Satriano and Zureick-Brown, in which the naive height of an elliptic curve is the height of the corresponding point on a moduli stack. [ABSTRACT FROM AUTHOR]

For 0-cycles on a variety over a number field, we define an analogue of the classical descent set for rational points. This leads to, among other things, a definition of the étale-Brauer obstruction set for 0-cycles. We show that all these constructions are compatible with Suslin's singular homology of degree 0. We then transfer some tools and techniques used to study the arithmetic of rational points into the setting of 0-cycles. For example, we extend the strategy developed by Y. Liang, relating the arithmetic of rational points over finite extensions of the base field to that of 0-cycles, to torsors. We give applications of our results to study the arithmetic behaviour of 0-cycles for Enriques surfaces, torsors given by (twisted) Kummer varieties, universal torsors, and torsors under tori. [ABSTRACT FROM AUTHOR]

We establish an implication between two long-standing open problems in complex dynamics. The roots of the |$n$| th Gleason polynomial |$G_{n}\in{\mathbb{Q}}[c]$| comprise the |$0$| -dimensional moduli space of quadratic polynomials with an |$n$| -periodic critical point. |$\operatorname{Per}_{n}(0)$| is the |$1$| -dimensional moduli space of quadratic rational maps on |${\mathbb{P}}^{1}$| with an |$n$| -periodic critical point. We show that if |$G_{n}$| is irreducible over |${\mathbb{Q}}$|⁠ , then |$\operatorname{Per}_{n}(0)$| is irreducible over |${\mathbb{C}}$|⁠. To do this, we exhibit a |${\mathbb{Q}}$| -rational smooth point on a projective completion of |$\operatorname{Per}_{n}(0)$|⁠ , using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large |$n$|⁠ , |$\operatorname{Per}_{n}(0)$| itself has no |${\mathbb{Q}}$| -rational points. [ABSTRACT FROM AUTHOR]

We study the problem of determining, given an integer k, the rational solutions to C k : x 3 z + x 2 y 2 + y 3 z = k z 4 . For k ≠ 0 , the curve C k has genus 3 and its Jacobian is isogenous to the product of three elliptic curves E 1 , k , E 2 , k , E 3 , k . We explicitly determine the rational points on C k under the assumption that one of these elliptic curves has rank zero. We discuss the challenges involved in extending our result to handle all k ∈ ℚ. [ABSTRACT FROM AUTHOR]

Let r ⩾ 3 r\geqslant 3 be an integer and 푄 any positive definite quadratic form in 푟 variables. We establish asymptotic formulae with power-saving error terms for the number of rational points of bounded height on singular hypersurfaces S Q S_{Q} defined by x 3 = Q ⁢ ( y 1 , … , y r ) ⁢ z x^{3}=Q(y_{1},\dots,y_{r})z . This confirms Manin’s conjecture for any S Q S_{Q} . Our proof is based on analytic methods, and uses some estimates for character sums and moments of 퐿-functions. In particular, one of the ingredients is Siegel’s mass formula in the argument for the case r = 3 r=3 . [ABSTRACT FROM AUTHOR]

Let E be a non-CM elliptic curve defined over \mathbb {Q}. Fix an algebraic closure {\overline {\mathbb Q}} of \mathbb {Q}. We get a Galois representation \[ \rho _E \colon {Gal}({\overline {\mathbb Q}}/\mathbb {Q})\to GL_2({\widehat {\mathbb {Z}}}) \] associated to E by choosing a system of compatible bases for the N-torsion subgroups of E({\overline {\mathbb Q}}). Associated to an open subgroup G of GL_2({\widehat {\mathbb {Z}}}) satisfying -I \in G and \det (G)={\widehat {\mathbb {Z}}}^{\times }, we have the modular curve (X_G,\pi _G) over \mathbb {Q} which loosely parametrises elliptic curves E such that the image of \rho _E is conjugate to a subgroup of G^t. In this article we give a complete classification of all such genus 0 modular curves that have a rational point. This classification is given in finitely many families. Moreover, for each such modular curve morphism \pi _G \colon X_G \to X_{GL_2({\widehat {\mathbb {Z}}})} can be explicitly computed. [ABSTRACT FROM AUTHOR]

Generalizing the problem of counting rational points on curves and surfaces over finite fields, we consider the setting of n × n matrix points with finite field entries. We obtain exact formulas for matrix point counts on elliptic curves and certain K 3 surfaces for "supersingular" primes. These exact formulas, which involve partitions of integers up to n , essentially coincide with the expected value for the number of such points. Therefore, in analogy with the Sato–Tate conjecture, it is natural to study the distribution of the deviation from the expected values for all primes. We determine the limiting distributions for elliptic curves and a family of K 3 surfaces. For non-CM elliptic curves with square-free conductor, our results are explicit. [ABSTRACT FROM AUTHOR]

Working over a base number field K , we study the attractive question of Zariski non-density for (D , S) -integral points in O f (x) the forward f -orbit of a rational point x ∈ X (K). Here, f : X → X is a regular surjective self-map for X a geometrically irreducible projective variety over K. Given a non-zero and effective f -quasi-polarizable Cartier divisor D on X and defined over K , our main result gives a sufficient condition, that is formulated in terms of the f -dynamics of D , for non-Zariski density of certain dynamically defined subsets of O f (x). For the case of (D , S) -integral points, this result gives a sufficient condition for non-Zariski density of integral points in O f (x). Our approach expands on that of Yasufuku, [13] , building on earlier work of Silverman [11]. Our main result gives an unconditional form of the main results of [13] ; the key arithmetic input to our main theorem is the Subspace Theorem of Schmidt in the generalized form that has been given by Ru and Vojta in [10] and expanded upon in [3] and [6]. [ABSTRACT FROM AUTHOR]

Let O be a Dedekind domain whose field of fractions K is a global field. Let A be a finite-dimensional separable K -algebra and let Λ be an O -order in A. Suppose that X is a Λ-lattice such that K ⊗ O X is free of some finite rank n over A. Then X contains a (non-unique) free Λ-sublattice of rank n. The main result of the present article is to show there exists such a sublattice Y such that the generalised module index [ X : Y ] O has explicit upper bounds with respect to division that are independent of X and can be chosen to satisfy certain conditions. We give examples of applications to the approximation of normal integral bases and strong Minkowski units, and to the Galois module structure of rational points over abelian varieties. [ABSTRACT FROM AUTHOR]

Conditionally on the Tate–Shafarevich and Bloch–Kato Conjectures, we give an explicit upper bound on the size of the |$p$| -adic Chabauty–Kim locus, and hence on the number of rational points, of a smooth projective curve |$X/{\mathbb{Q}}$| of genus |$g\geq 2$| in terms of |$p$|⁠ , |$g$|⁠ , the Mordell–Weil rank |$r$| of its Jacobian, and the reduction types of |$X$| at bad primes. This is achieved using the effective Chabauty–Kim method, generalizing bounds found by Coleman and Balakrishnan–Dogra using the abelian and quadratic Chabauty methods. [ABSTRACT FROM AUTHOR]

The classical Siciak-Zakharyuta theorem states that the Siciak-Zakharyuta function V E of a subset E of C n , also called a pluricomplex Green function or global extremal function of E, equals the logarithm of the Siciak function Φ E if E is compact. The Siciak-Zakharyuta function is defined as the upper envelope of functions in the Lelong class that are negative on E, and the Siciak function is the upper envelope of m-th roots of the modulus of polynomials p in P m (C n) of degree ≤ m such that | p | ≤ 1 on E. We generalize the Siciak-Zakharyuta theorem to the case where the polynomial space P m (C n) is replaced by P m S (C n) consisting of all polynomials with exponents restricted to sets mS, where S is a compact convex subset of R + n with 0 ∈ S . It states that if q is an admissible weight on a closed set E in C n then V E , q S = log Φ E , q S on C ∗ n if and only if the rational points in S form a dense subset of S. [ABSTRACT FROM AUTHOR]

Motivated by the classical Rodrigues' formula, we study below the root asymptotic of the polynomial sequence R [ α n ] , n , P (z) = d [ α n ] P n (z) d z [ α n ] , n = 0 , 1 , ⋯ where P(z) is a fixed univariate polynomial, α is a fixed positive number smaller than deg P , and [ α n ] stands for the integer part of α n . Our description of this asymptotic is expressed in terms of an explicit harmonic function uniquely determined by the plane rational curve emerging from the application of the saddle point method to the integral representation of the latter polynomials using Cauchy's formula for higher derivatives. As a consequence of our method, we conclude that this curve is birationally equivalent to the zero locus of the bivariate algebraic equation satisfied by the Cauchy transform of the asymptotic root-counting measure for the latter polynomial sequence. We show that this harmonic function is also associated with an abelian differential having only purely imaginary periods and the latter plane curve belongs to the class of Boutroux curves initially introduced in Bertola (Anal Math Phys 1: 167–211, 2011), Bertola and Mo (Adv Math 220(1): 154–218, 2009). As an additional relevant piece of information, we derive a linear ordinary differential equation satisfied by { R [ α n ] , n , P (z) } as well as higher derivatives of powers of more general functions. [ABSTRACT FROM AUTHOR]

By the circle method, an asymptotic formula is established for the number of integer points on certain hypersurfaces within multiprojective space. Using Möbius inversion and the modified hyperbola method, we prove the Manin–Peyre conjecture on the asymptotic behavior of the number of rational points of bounded anticanonical height for certain smooth hypersurfaces in the multiprojective space of sufficiently large dimension. [ABSTRACT FROM AUTHOR]

Let k be a field with algebraic closure k ¯ and D ⊂ P k ¯ 1 a reduced, effective divisor of degree n ≥ 3 , write k D for the field of moduli of D. A. Marinatto proved that when n is odd, or n = 4 , D descends to a divisor on P k D 1. We analyze completely the problem of when D descends to a divisor on a smooth, projective curve of genus 0 on k D , possibly with no rational points. In particular, we study the remaining cases n ≥ 6 even, and we obtain conceptual proofs of Marinatto's results and of a theorem by B. Huggins about the field of moduli of hyperelliptic curves. [ABSTRACT FROM AUTHOR]

Over the moduli space of pointed smooth algebraic curves, the projectivized k th Hodge bundle is the space of k -canonical divisors. The incidence loci are defined by requiring the k -canonical divisors to have prescribed multiplicities at the marked points. We compute the classes of the closure of the incidence loci in the projectivized k th Hodge bundle over the moduli space of curves with rational tails. The classes are expressed as a linear combination of tautological classes indexed by decorated stable graphs with coefficients enumerating appropriate weightings. As a consequence, we obtain an explicit expression for some relations in tautological rings of moduli of curves with rational tails. [ABSTRACT FROM AUTHOR]

We investigate sections of the arithmetic fundamental group $\pi _1(X)$ where X is either a smooth affinoid p-adic curve , or a formal germ of a p-adic curve , and prove that they can be lifted (unconditionally) to sections of cuspidally abelian Galois groups. As a consequence, if X admits a compactification Y , and the exact sequence of $\pi _1(X)$ splits , then $\text {index} (Y)=1$. We also exhibit a necessary and sufficient condition for a section of $\pi _1(X)$ to arise from a rational point of Y. One of the key ingredients in our investigation is the fact, we prove in this paper in case X is affinoid, that the Picard group of X is finite. [ABSTRACT FROM AUTHOR]

In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. "Classroom" is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science. [ABSTRACT FROM AUTHOR]

We give a general criterion for Zariski degeneration of integral points in the complement of a divisor D$D$ with n$n$ components in a variety of dimension n$n$ defined over Q$\mathbb {Q}$ or over a quadratic imaginary field. The key condition is that the intersection of the components of D$D$ is not well approximated by rational points, and we discuss several cases where this assumption is satisfied. We also prove a greatest common divisor (GCD) bound for algebraic points in varieties, which can be of independent interest. [ABSTRACT FROM AUTHOR]

We establish higher dimensional versions of a recent theorem by Chen and Haynes [Int. J. Number Theory 19 (2023), 1405–1413] on the expected value of the smallest denominator of rational points in a randomly shifted interval of small length, and of the closely related 1977 Kruyswijk–Meijer conjecture recently proved by Balazard and Martin [Bull. Sci. Math. 187 (2023), Paper No. 103305]. We express the distribution of smallest denominators in terms of the void statistics of multidimensional Farey fractions and prove convergence of the distribution function and certain finite moments. The latter was previously unknown even in the one‐dimensional setting. We furthermore obtain a higher dimensional extension of Kargaev and Zhigljavsky's work on moments of the distance function for the Farey sequence [J. Number Theory 65 (1997), 130–149] as well as new results on pigeonhole statistics. [ABSTRACT FROM AUTHOR]

Given a perfect field $k$ with algebraic closure $\overline {k}$ and a variety $X$ over $\overline {k}$ , the field of moduli of $X$ is the subfield of $\overline {k}$ of elements fixed by field automorphisms $\gamma \in \operatorname {Gal}(\overline {k}/k)$ such that the Galois conjugate $X_{\gamma }$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset \overline {k}$ such that $X$ descends to $k'$. In this paper, we extend the formalism and define the field of moduli when $k$ is not perfect. Furthermore, Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and prove that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher-dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As a consequence, we prove that a variety $X$ of dimension $d$ with a smooth marked point $p$ such that $\operatorname {Aut}(X,p)$ is finite, étale and of degree prime to $d!$ is defined over its field of moduli. [ABSTRACT FROM AUTHOR]

The decreasing norm-trace codes are evaluation codes defined by a set of monomials closed under divisibility and the rational points of the extended norm-trace curve. In particular, the decreasing norm-trace codes contain the one-point algebraic geometry (AG) codes over the extended norm-trace curve. We use Gröbner basis theory and find the indicator functions on the rational points of the curve to determine the basic parameters of the decreasing norm-trace codes: length, dimension, and minimum distance. We also obtain their dual codes. We give conditions for a decreasing norm-trace code to be a self-orthogonal or a self-dual code. We provide a linear exact repair scheme to correct single erasures for decreasing norm-trace codes, which applies to higher rate codes than the scheme developed by Jin et al. (IEEE Trans Inf Theory 64(2):900–908, 2018) when applied to the one-point AG codes over the extended norm-trace curve. [ABSTRACT FROM AUTHOR]

We prove that del Pezzo surfaces of degree 2 over a field k$k$ satisfy weak weak approximation if k$k$ is a number field and the Hilbert property if k$k$ is Hilbertian of characteristic zero, provided that they contain a k$k$‐rational point lying neither on any 4 of the 56 exceptional curves nor on the ramification divisor of the anticanonical morphism. This builds upon results of Manin, Salgado–Testa–Várilly‐Alvarado, and Festi–van Luijk on the unirationality of such surfaces, and upon work of the first two authors verifying weak weak approximation under the assumption of a conic fibration. [ABSTRACT FROM AUTHOR]

We study the unirationality property of an algebraic variety X (over C ) versus the so-called stable birational infinite transitivity of X. We show that in the case when X is a smooth quartic hypersurface, these two notions do not coincide. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to derive effective discrepancy estimates for the distribution of rational points on general semisimple algebraic group varieties, in general families of subsets and at arbitrarily small scales. We establish mean-square, almost sure and uniform estimates for the discrepancy with explicit error bounds. We also prove an analogue of W. Schmidt's theorem, which establishes effective almost sure asymptotic counting of rational solutions to Diophantine inequalities in the Euclidean space. We formulate and prove a version of it for rational points on the group variety, with an effective bound which in some instances can be expected to be the best possible. [ABSTRACT FROM AUTHOR]

Let $F$ be a separable integral binary form of odd degree $N \geq 5$. A result of Darmon and Granville known as 'Faltings plus epsilon' implies that the degree- $N$ superelliptic equation $y^2 = F(x,z)$ has finitely many primitive integer solutions. In this paper, we consider the family $\mathscr {F}_N(f_0)$ of degree- $N$ superelliptic equations with fixed leading coefficient $f_0 \in \mathbb {Z} \smallsetminus \pm \mathbb {Z}^2$ , ordered by height. For every sufficiently large $N$ , we prove that among equations in the family $\mathscr {F}_N(f_0)$ , more than $74.9\,\%$ are insoluble, and more than $71.8\,\%$ are everywhere locally soluble but fail the Hasse principle due to the Brauer–Manin obstruction. We further show that these proportions rise to at least $99.9\,\%$ and $96.7\,\%$ , respectively, when $f_0$ has sufficiently many prime divisors of odd multiplicity. Our result can be viewed as a strong asymptotic form of 'Faltings plus epsilon' for superelliptic equations and constitutes an analogue of Bhargava's result that most hyperelliptic curves over $\mathbb {Q}$ have no rational points. [ABSTRACT FROM AUTHOR]

Let K be an imaginary quadratic field, and fix an odd prime p that does not divide the class number of K , and is excluded from a finite set of primes depending on K. In this paper we prove that Iwasawa's λ -invariant for the cyclotomic Z p -extension of K is greater than 1 if and only if the number of points on a certain reduced elliptic curve is divisible by p 2. [ABSTRACT FROM AUTHOR]

In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme $G$ over a valuation ring $V$ with fraction field $K$ , a $G$ -torsor over $V$ is trivial if it is trivial over $K$. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thélène–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank-one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using extension properties of reflexive sheaves on formal power series over valuation rings and patching of torsors, we prove a variant of Nisnevich's purity conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we consider the affine variety codes obtained evaluating the polynomials b y = a k x k + ... + a 1 x + a 0 , b , a i ∈ F q r , at the affine F q r -rational points of the Norm-Trace curve. In particular, we investigate the weight distribution and the set of minimal codewords. Our approach, which uses tools of algebraic geometry, is based on the study of the absolute irreducibility of certain algebraic varieties. [ABSTRACT FROM AUTHOR]

Let |$k$| be a field, |$X$| a variety with tame quotient singularities, and |$\tilde {X}\to X$| a resolution of singularities. Any smooth rational point |$x\in X(k)$| lifts to |$\tilde {X}$| by the Lang–Nishimura theorem, but if |$x$| is singular this might be false. For certain types of singularities, the rational point is guaranteed to lift, though; these are called singularities of type |$\textrm {R}$|⁠. This concept has applications in the study of the fields of moduli of varieties and yields an enhanced version of the Lang–Nishimura theorem where the smoothness assumption is relaxed. We classify completely the tame quotient singularities of type |$\textrm {R}$| in dimension |$2$|⁠ ; in particular, we show that every non-cyclic tame quotient singularity in dimension |$2$| is of type |$\textrm {R}$|⁠ , and most cyclic singularities are of type |$\textrm {R}$| too. [ABSTRACT FROM AUTHOR]

We prove a conjecture of Kleinbock which gives a clear-cut classification of all extremal affine subspaces of . We also give an essentially complete classification of all Khintchine type affine subspaces, except for some boundary cases within two logarithmic scales. More general Jarník type theorems are proved as well, sometimes without the monotonicity of the approximation function. These results follow as consequences of our novel estimates for the number of rational points close to an affine subspace in terms of diophantine properties of its defining matrix. Our main tool is the multidimensional large sieve inequality and its dual form. [ABSTRACT FROM AUTHOR]

For a general cubic four-fold |$X\subset {\mathbb {P}}^5$| with Fano scheme of lines |$F$|⁠ , we prove a number of properties of the universal family of lines |$I\to F$| and various subloci. We first describe the moduli and ramification theory of the genus four fibration |$p:I\to X$| and explore its relation to a birational model of |$F$| in |$I$|⁠. The main part of the paper is devoted to describing the locus |$V\subset F$| of triple lines, i.e. the fixed locus of the Voisin map |$\phi :F\dashrightarrow F$|⁠ , in particular proving it is an irreducible projective singular surface of class |$21\textrm {c}_2({\mathcal {U}}_F)$| and detailing its intersection with the locus |$S$| of second type lines. A consequence of the analysis of the singularities of |$V$| is a geometric proof of the fact that if |$X$| is very general, then the number of singular (necessarily 1-nodal) rational curves in |$F$| of primitive class is 3780. [ABSTRACT FROM AUTHOR]

We show that plane Cremona groups over finite fields embed as dense subgroups into Neretin groups, that is, groups of almost automorphisms of rooted trees. We also show that if the finite base field has even characteristic and contains at least four elements, then the permutations induced by birational transformations on rational points of regular projective surfaces are even. In a second part, we construct explicit locally compact CAT(0) cube complexes, on which Neretin groups act properly. This allows us to recover in a unified way various results on Neretin groups such as that they are of type |$F_{\infty }$|⁠. We also prove a new fixed-point theorem for CAT(0) cube complexes without infinite cubes and use it to deduce a regularization theorem for plane Cremona groups over finite fields. [ABSTRACT FROM AUTHOR]

The stochastic partial differential equation (SPDE) approach is widely used for modeling large spatial datasets. It is based on representing a Gaussian random field u on R d as the solution of an elliptic SPDE L β u = W where L is a second-order differential operator, 2 β ∈ N is a positive parameter that controls the smoothness of u and W is Gaussian white noise. A few approaches have been suggested in the literature to extend the approach to allow for any smoothness parameter satisfying β > d / 4 . Even though those approaches work well for simulating SPDEs with general smoothness, they are less suitable for Bayesian inference since they do not provide approximations which are Gaussian Markov random fields (GMRFs) as in the original SPDE approach. We address this issue by proposing a new method based on approximating the covariance operator L − 2 β of the Gaussian field u by a finite element method combined with a rational approximation of the fractional power. This results in a numerically stable GMRF approximation which can be combined with the integrated nested Laplace approximation (INLA) method for fast Bayesian inference. A rigorous convergence analysis of the method is performed and the accuracy of the method is investigated with simulated data. Finally, we illustrate the approach and corresponding implementation in the R package rSPDE via an application to precipitation data which is analyzed by combining the rSPDE package with the R-INLA software for full Bayesian inference. for this article are available online. [ABSTRACT FROM AUTHOR]

We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose C and C' are curves over a finite field K, with K-rational base points P and P', and let D and D' be the pullbacks (via the Abel–Jacobi map) of the multiplication-by-2 maps on their Jacobians. We say that (C,P) and (C',P') are doubly isogenous if Jac(C) and Jac(C') are isogenous over K and Jac(D) and Jac(D') are isogenous over K. For curves of genus 2 whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than naïve heuristics predict, and we provide an explanation for this phenomenon. [ABSTRACT FROM AUTHOR]

Let f_t(z)=z^2+t. For any z\in \mathbb {Q}, let S_z be the collection of t\in \mathbb {Q} such that z is preperiodic for f_t. In this article, assuming a well-known conjecture of Flynn, Poonen, and Schaefer [Duke Math. J. 90 (1997), pp. 435–463], we prove a uniform result regarding the size of S_z over z\in \mathbb {Q}. In order to prove it, we need to determine the set of rational points on a specific non-hyperelliptic curve C of genus 4 defined over \mathbb {Q}. We use Chabauty's method, which requires us to determine the Mordell-Weil rank of the Jacobian J of C. We give two proofs that the rank is 1: an analytic proof, which is conditional on the BSD rank conjecture for J and some standard conjectures on L-series, and an algebraic proof, which is unconditional, but relies on the computation of the class groups of two number fields of degree 12 and degree 24, respectively. We finally combine the information obtained from both proofs to provide a numerical verification of the strong BSD conjecture for J. [ABSTRACT FROM AUTHOR]

Let G be a group. Let X be an algebraic group over an algebraically closed field K. Denote by A = X.K/the set of rational points of X. We study algebraic group cellular AG → AG whose local defining map is induced by a homomorphism of algebraic groups τ: XM → X, where M is a finite memory. When G is sofic and K is uncountable, we show that if - is post-surjective, then it is weakly pre-injective. Our result extends the dual version of Gottschalk's conjecture for finite alphabets proposed by Capobianco, Kari, and Taati. When G is amenable, we prove that if τ is surjective, then it is weakly pre-injective, and conversely, if - is pre-injective, then it is surjective. Hence, we obtain a complete answer to a question of Gromov on the Garden of Eden theorem in the case of algebraic group cellular automata. [ABSTRACT FROM AUTHOR]

In this paper, we study rational Collet–Eckmann maps for which the Julia set is not the whole sphere and for which the critical points are recurrent at a slow rate. In families where the orders of the critical points are fixed, we prove that such maps are Lebesgue density points of hyperbolic maps. In particular, if all critical points are simple, these maps are Lebesgue density points of hyperbolic maps in the full space of rational maps of any degree d⩾2$d \geqslant 2$. [ABSTRACT FROM AUTHOR]

Let C be a curve defined over a number field k. We say a closed point x ∈ C of degree d is isolated if it does not belong to an infinite family of degree d points parametrized by the projective line or a positive rank abelian subvariety of the curve's Jacobian. Building on work of Bourdon et al. (Adv Math 357(33):106824, 2019), we characterize elliptic curves with rational j-invariant which give rise to an isolated point of odd degree on X 1 (N) / Q for some positive integer N. [ABSTRACT FROM AUTHOR]

Building on work by Chabauty from 1941, Coleman proved in 1985 an explicit bound for the number of rational points of a curve C of genus g ≥ 2 defined over a number field F , with Jacobian of rank at most g − 1 . Namely, in the case F = Q , if p > 2 g is a prime of good reduction, then the number of rational points of C is at most the number of F p -points plus a contribution coming from the canonical class of C . We prove a result analogous to Coleman's bound in the case of a hyperbolic surface X over a number field, embedded in an abelian variety A of rank at most one, under suitable conditions on the reduction type at the auxiliary prime. This provides the first extension of Coleman's explicit bound beyond the case of curves. The main innovation in our approach is a new method to study the intersection of a p -adic analytic subgroup with a subvariety of A by means of overdetermined systems of differential equations in positive characteristic. [ABSTRACT FROM AUTHOR]

In this paper, we derive from deep results due to Clozel and Ullmo a sharp density result of Noether-Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular, we obtain the density of Hilbert schemes of points on projective K3 surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of the Mori cone of curves and of the associated extremal birational contractions, and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is isomorphic to a Hilbert scheme of two points on a K3 surface. We also discuss Voisin's conjecture on the existence of coisotropic subvarieties on IHS varieties and relate it to a stronger statement on Noether-Lefschetz loci in their moduli spaces. [ABSTRACT FROM AUTHOR]

Via correspondence theorems, rational log Gromov–Witten invariants of the plane can be computed in terms of tropical geometry. For many cases, there exists a range of algorithms to compute tropically: for instance, there are (generalised) lattice path counts and floor diagram techniques. So far, the cases for which there exist algorithms do not extend to non‐stationary rational descendant log Gromov–Witten invariants, that is, those where Psi‐conditions do not have to be matched up with the evaluation of a point. The case of rational descendant log Gromov–Witten invariants satisfying point conditions (without Psi‐conditions) and one Psi‐condition of any power combined with a line plays a particularly important role, because it shows up in mirror symmetry as contributions to coefficients of the J$J$‐function. We provide recursive formulas to compute those numbers via tropical methods. Our method is inspired by the tropical proof of the WDVV equations. We also extend our study to counts involving two lines, both paired up with a Psi‐condition, appearing with power 1. [ABSTRACT FROM AUTHOR]

Let α be a non-zero algebraic unit which is not a root of unity and K be a number field of degree d over Q . In this paper, we prove the following: Let P be a prime ideal of O K which lies above a rational odd prime p such that P O K (α) = P 1 e 1 P 2 e 2 ⋯ P g e g , where max 1 ≤ i ≤ g { e i } ≤ p and e 1 + ⋯ + e g = [ K (α) : K ]. Then h (α) ≥ c , where c > 0 is an effectively computable constant depending only on p and [ K : Q ] = d. This generalizes a result of Petsche. Also, we prove the following: Let P be a prime ideal of O K which lies above 2 such that P O K (α) = P 1 e 1 P 2 e 2 ⋯ P r e r , where e 1 + ⋯ + e r = [ K (α) : K ]. Then M (α) ≥ C (K) , where C (K) > 1 is a constant which depends only on [ K : Q ] = d. [ABSTRACT FROM AUTHOR]

Let 픽 q be the finite field of odd characteristic p with q elements ( q = p n , n ∈ ℕ ) and let 픽 q * represent the set of nonzero elements of 픽 q . By making use of the Smith normal form of exponent matrices, we obtain an explicit formula for the number of rational points on the variety defined by the following system of equations over 픽 q : { ∑ i = 1 r a i (1) ⁢ x 1 e i ⁢ 1 (1) ⁢ ⋯ ⁢ x n e i ⁢ n (1) = b 1 , ∑ j ′ = 0 t - 1 ∑ i ′ = 1 r j ′ + 1 - r j ′ a r j ′ + i ′ (2) ⁢ x 1 e r j ′ + i ′ , 1 (2) ⁢ ⋯ ⁢ x n j ′ + 1 e r j ′ + i ′ , n j ′ + 1 (2) = b 2 , where b i ∈ 픽 q ( i = 1 , 2 ), t ∈ ℕ , 0 = n 0 < n 1 < n 2 < ⋯ < n t , n k - 1 < n ≤ n k for some 1 ≤ k ≤ t , 0 = r 0 < r 1 < r 2 < ⋯ < r t , a i (1) ∈ 픽 q * for i ∈ { 1 , ... , r } , a i ′ (2) ∈ 픽 q * for i ′ ∈ { 1 , ... , r t } , and the exponent of each variable is a positive integer. This generalizes the results obtained previously by Wolfmann, Sun, Cao, and others. Our result also gives a partial answer to an open problem raised by Hu, Hong and Zhao [S. N. Hu, S. F. Hong and W. Zhao, The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 2015, 135–153]. [ABSTRACT FROM AUTHOR]

In recent years, many useful applications of the polynomial method have emerged in finite geometry. Indeed, algebraic curves, especially those defined by Rédei-type polynomials, are powerful in studying blocking sets. In this paper, we reverse the engine and study when blocking sets can arise from rational points on plane curves over finite fields. We show that irreducible curves of low degree cannot provide blocking sets and prove more refined results for cubic and quartic curves. On the other hand, using tools from number theory, we construct smooth plane curves defined over F p of degree at most 4 p 3 / 4 + 1 whose points form blocking sets. [ABSTRACT FROM AUTHOR]

In this manuscript, we prove several common fixed point theorems for generalized rational-type contraction mappings under several conditions in the context of double-controlled metric spaces. Further, we utilize a double-controlled metric space equipped with a graph to prove rational-type common fixed point theorems. Furthermore, we establish non-trivial examples to show the validity of the main results. These results improve and generalize already known results. At the end, we solve the Fredholm-type integral equation by utilizing the main results. [ABSTRACT FROM AUTHOR]