We give effective bounds for the set quasi-integral points in orbits of non-isotrivial rational maps over function fields under some conditions, generalizing previous work of Hsia and Silverman (2011) for orbits over function fields of characteristic zero. We then use this to prove finiteness results for algebraic functions whose orbit under a rational function has multiplicative dependent elements modulo rings of $S$-integers, generalizing recent results over number fields.
In this note, basing on a certain functional equation of the dilogarithm function, we establish nontrivial lower bounds for the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers involving harmonic numbers. Then we use our estimate to derive the integrality of some sequences of rational numbers, which cannot be seen directly from their definitions.
One extends P. Deligne's notion of integrality over a finite field for a $\ell$-adic sheaf on a scheme of finite type over a local field with finite residue field. One shows that this integrality notion is preserved by $Rf_!$, as it is over a finite field after P. Deligne.
If $V$ is a smooth projective variety defined over a local field $K$ with finite residue field, so that its étale cohomology over the algebraic closure $\bar{K}$ is supported in codimension 1, then the mod $p$ reduction of a projective regular model carries a rational point. As a consequence, if the Chow group of 0-cycles of $V$ over a large algebraically closed field is trivial, then the mod $p$ reduction of a projective regular model carries a rational point.
Let $A$ be a set of $N$ vectors in ${\mathbb Z}^n$ and let $v$ be a vector in ${\mathbb C}^N$ that has minimal negative support for $A$. Such a vector $v$ gives rise to a formal series solution of the $A$-hypergeometric system with parameter $β=Av$. If $v$ lies in ${\mathbb Q}^n$, then this series has rational coefficients. Let $p$ be a prime number. We characterize those $v$ whose coordinates are rational, $p$-integral, and lie in the closed interval $[-1,0]$ for which the corresponding normalized series solution has $p$-integral coefficients. From this we deduce further integrality results f…
For the hypergeometric series with parameters from the rational fields, there is an effective criterion due to Christol to decide whether the hypergeometric series is N-integral or not. Christol criterion is a basic and vital tool in the recent striking work of Delaygue, Rivoal and Roques on the N-integrality of the hypergeometric mirror maps with rational parameters. In this paper, we develop a systematic theory on the N-integrality of the hypergeometric series with parameters from quadratic fields. We first present a detailed $p$-adic analysis to set up a criterion of the $p$-adic integralit…
In this paper, we study multiplicative dependence of values of polynomials or rational functions over a number field. As an application, we obtain new results on multiplicative dependence in the orbits of a univariate polynomial dynamical system. We also obtain a broad generalisation of the Northcott theorem replacing the finiteness of preperiodic points from a given number field by the finiteness of initial points with two multiplicatively dependent elements in their orbits.
We study multiplicative dependence between elements in orbits ofalgebraic dynamical systems over number fields modulo a finitely generated multiplicative subgroup of the field. We obtain a series of results, many of which may be viewed as a blend of Northcott's theorem on boundedness of preperiodic points and Siegel's theorem on finiteness of solutions to $S$-unit equations.
Let $A$ be a set of $N$ vectors in ${\mathbb Z}^n$ and let $v$ be a vector in ${\mathbb C}^N$ that has minimal negative support for $A$. Such a vector $v$ gives rise to a formal series solution of the $A$-hypergeometric system with parameter $β= Av$. If $v$ lies in ${\mathbb Q}^n$, then this series has rational coefficients. Let $p$ be a prime number. We characterize those $v$ whose coordinates are rational, $p$-integral, and lie in the closed interval $[-1,0]$ for which the corresponding normalized series solution has $p$-integral coefficients.
We say that a formal power series $\sum a_n z^n$ with rational coefficients is a 2-function if the numerator of the fraction $a_{n/p}-p^2 a_n$ is divisible by $p^2$ for every prime number $p$. One can prove that 2-functions with rational coefficients appear as building block of BPS generating functions in topological string theory. Using the Frobenius map we define 2-functions with coefficients in algebraic number fields. We establish two results pertaining to these functions. First, we show that the class of 2-functions is closed under the so-called framing operation (related to compositional…
We give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of said iterates. This leads to a generalisation of Gao's method for constructing elements in the finite field $\mathbb{F}_{q^n}$ whose orders are larger than any polynomial in $n$ when $n$ becomes large. Additionally, we discuss the finiteness of polynomials which translate a given finite set of polynomials to become multiplicatively dependent.
This is the second of two coupled papers estimating the mean values of multiplicative functions, of unknown support, on arithmetic progressions with large differences. Applications are made to the study of primes in arithmetic progression and to the Fourier coefficients of automorphic cusp forms.
Integrality of Hecke eigenvalues and the growth of Hecke fields
Faculty of Education, Shinshu University（信州大学教育学部）；Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University（金泽大学理工学域数理系）
We study the integrality properties of the coefficients of the mirror map attached to the generalized hypergeometric function $_{n}F_{n-1}$ with rational parameters and with a maximal unipotent monodromy. We present a conjecture on the $p$-integrality of the mirror map which can be verified experimentally. We prove its consequence on the $N$-integrality of the mirror map for the particular cases $1\leq n\leq 4$. For $n=2$ we obtain the Takeuchi's classification of arithmetic triangle groups with a cusp, and for $n=4$ we prove that $14$ examples of hypergeometric Calabi-Yau equations are the fu…
Given a finitely generated multiplicative subgroup of rational numbers $Γ$, assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for average over prime numbers, powers of the order of the reduction group modulo $p$. The problem was considered in the case of rank $1$ by Pomerance and Kurlberg. In the case when $Γ$ contains only positive numbers, we give an explicit expression for the involved density in terms of an Euler product. We conclude with some numerical computations.
Let $L/K$ be a finite Galois CM-extension of number fields with Galois group $G$. In an earlier paper, the author has defined a module $SKu(L/K)$ over the center of the group ring $\mathbb Z[G]$ which coincides with the Sinnott-Kurihara ideal if $G$ is abelian and, in particular, contains many Stickelberger elements. It was shown that a certain conjecture on the integrality of $SKu(L/K)$ implies the minus part of the equivariant Tamagawa number conjecture at an odd prime $p$ for an infinite class of (non-abelian) Galois CM-extensions of number fields which are at most tamely ramified above $p$…
Using Dwork's theory, we prove a broad generalisation of his famous p-adic formal congruences theorem. This enables us to prove certain p-adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number p and not only for almost all primes. Along the way, using Christol's functions, we provide an explicit formula for the "Eisenstein constant" of any globally bounded hypergeometric series with rational parameters. As an application of these results, we obtain an arithmetic statement of a new type concerning the integrality of Tay…
Let $E_1$ and $E_2$ be elliptic curves in Legendre form with integer parameters. We show there exists a constant $C$ such that for almost all primes, for all but at most $C$ pairs of points on the reduction of $E_1 \times E_2$ modulo $p$ having equal $x$ coordinate, at least one among $P_1$ and $P_2$ has a large group order. We also show similar abundance over finite fields of elements whose images under the reduction modulo $p$ of a finite set of rational functions have large multiplicative orders
In this paper, we give a brief survey of recent developments on Noether's problem and rationality problem for multiplicative invariant fields including author's recent papers Hoshi [Hos15] about Noether's problem over Q, Hoshi, Kang and Kunyavskii [HKK13], Chu, Hoshi, Hu and Kang [CHHK15], Hoshi [Hos16] and Hoshi, Kang and Yamasaki [HKY16] about Noether's problem over C, and Hoshi, Kang and Kitayama [HKK14] and Hoshi, Kang and Yamasaki [HKY] about rationality problem for multiplicative invariant fields.
For a prime $p$ and a polynomial $F(X,Y)$ over a finite field $\mathbb{F}_p$ of $p$ elements, we give upper bounds on the number of solutions $$ F(x,y)=0, \quad x\in\mathcal{A}, \ y\in \mathcal{B}, $$ where $\mathcal{A}$ and $\mathcal{B}$ are very small intervals or subgroups. These bounds can be considered as positive characteristic analogues of the result of Bombieri and Pila (1989) on sparsity of integral points on curves. As an application we prove that distinct consecutive elements in sequences generated compositions of several rational functions are not contained in any short intervals o…
We consider the problem of counting the number of varieties in a family over $\mathbb{Q}$ with a rational point. We obtain lower bounds for this counting problem for some families over $\mathbb{P}^1$, even if the Hasse principle fails. We also obtain sharp results for some multinorm equations and for specialisations of certain Brauer group elements on higher dimensional projective spaces, where we answer some cases of a question of Serre. Our techniques come from arithmetic geometry and additive combinatorics.
We evaluate in closed form series of the type $\sum u(n) R(n)$, where $(u(n))_n$ is a strongly $B$-multiplicative sequence and $R(n)$ a (well-chosen) rational function. A typical example is: $$ \sum_{n \geq 1} (-1)^{s_2(n)} \frac{4n+1}{2n(2n+1)(2n+2)} = -\frac{1}{4} $$ where $s_2(n)$ is the sum of the binary digits of the integer $n$. Furthermore closed formulas for series involving automatic sequences that are not strongly $B$-multiplicative, such as the regular paperfolding and Golay-Shapiro-Rudin sequences, are obtained; for example, for integer $d \geq 0$: $$ \sum_{n \geq 0} \frac{v(n)}{(n…
In this paper, we give a new proof and an extension of the following result of Bézivin. Let $f:\B{N}\to K$ be a multiplicative function taking values in a field $K$ of characteristic 0 and write $F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$ for its generating series. Suppose that $F(z)$ is algebraic over $K(z)$. Then either there is a natural number $k$ and a periodic multiplicative function $χ(n)$ such that $f(n)=n^k χ(n)$ for all $n$, or $f(n)$ is eventually zero. In particular, $F(z)$ is either transcendental or rational. For $K=\B{C}$, we also prove that if $F(z)$ is a $D$-finite generating seri…
We prove a stratification result for certain families of $n$-dimensional (complete algebraic) multiplicative character sums. The character sums we consider are sums of products of $r$ multiplicative characters evaluated at rational functions, and the families (with $nr$ parameters) are obtained by allowing each of the $r$ rational functions to be replaced by an "offset", i.e. a translate, of itself. For very general such families, we show that the stratum of the parameter space on which the character sum has maximum weight $n+j$ has codimension at least $j\lfloor(r-1)/2(n-1)\rfloor$ for $1\le …
In this note we prove that for every integer $d \geq 1$, there exists an explicit constant $B_d$ such that the following holds. Let $K$ be a number field of degree $d$, let $q > \max\{d-1,5\}$ be any rational prime that is totally inert in $K$ and $E$ any elliptic curve defined over $K$ such that $E$ has potentially multiplicative reduction at the prime $\mathfrak q$ above $q$. Then for every rational prime $p> B_d$, $E$ has an irreducible mod $p$ Galois representation. This result has Diophantine applications within the "modular method". We present one such application in the form of an Asymp…
Mahler series with multiplicative coefficient sequences
University of Waterloo（滑铁卢大学）；Faculty of Mathematics and Physics (FMF) University of Ljubljana（卢布尔雅那大学数学与物理学院）；Institute of Mathematics, Physics and Mechanics (IMFM)（数学、物理与力学研究所）
The exceptional zero conjecture relates the first derivative of the $p$-adic $L$-function of a rational elliptic curve with split multiplicative reduction at $p$ to its complex $L$-function. Teitelbaum formulated an analogue of Mazur and Tate's refined (multiplicative) version of this conjecture for elliptic curves over the rational function field $\FQ(T)$ with split multiplicative reduction at two places $\fp$ and $\infty$, avoiding the construction of a $\fp$-adic $L$-function. This article proves Teitelbaum's conjecture up to roots of unity by developing Darmon's theory of double integrals …
Let $K$ be a field of characteristic 0, $f:\mathbb{N}\to K$ be a multiplicative function, and $F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$ be algebraic over $K(z)$. Then either there is a natural number $k$ and a periodic multiplicative function $χ(n)$ such that $f(n)=n^k χ(n)$ for all $n$, or $f(n)$ is eventually zero. In particular, the generating function of a multiplicative function $f:\mathbb{N}\to K$ is either transcendental or rational.
For a large prime $p$, a rational function $ψ\in F_p(X)$ over the finite field $F_p$ of $p$ elements, and integers $u$ and $H\ge 1$, we obtain a lower bound on the number consecutive values $ψ(x)$, $x = u+1, \ldots, u+H$ that belong to a given multiplicative subgroup of $F_p^*$.
We say a power series $\sum_{n=0}^\infty a_n q^n$ is multiplicative if the function $n\mapsto a_n/a_1$ ($n\ge 1$) is so. In this paper, we consider multiplicative power series $f$ such that $f^2$ is also multiplicative. We find various solutions for which $f$ is a rational function or a theta series and prove that the complete set of solutions is the locus of a (probably reducible) affine variety over C. The precise determination of this variety is a finite computational problem but seems to be outside the reach of current computer algebra systems. The proof of the theorem depends on a bound o…
The 3x+1 semigroup is the multiplicative semigroup generated by the rational numbers of form (2k+1)/(3k+2) for non-negative k, together with 2. This semigroup encodes backward iteration under the 3x+1 map, and the 3x+1 conjecture implies that it contains every positive integer. We prove this is the case, and show that this semigroup consists of all positive rational numbers a/b such that 3 does not divide b.
We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work with p-adic methods to obtain a lower bound of the form $cD^{n/4 - \varepsilon}$ on the degree of the splitting field of $P^{\circ n}(z)=P^{\circ n}(α)$, where $P$ is a polynomial of degree $D\geq 2$ over a number field, $P^{\circ n}$ is its $n$-th iterate and $c$ depends effectively on $P, α$ and $\varepsilon$. Our $c$ is positive for each algebraic $α$ fo…
An elliptic divisibility sequence is an integer recurrence sequence associated to an elliptic curve over the rationals together with a rational point on that curve. In this paper we present a higher-dimensional analogue over arbitrary base fields. Suppose E is an elliptic curve over a field K, and P_1, ..., P_n are points on E defined over K. To this information we associate an n-dimensional array of values in K satisfying a nonlinear recurrence relation. Arrays satisfying this relation are called elliptic nets. We demonstrate an explicit bijection between the set of elliptic nets and the set …
Mahler's measure is generalized to create the class of {\it multiplicative distance functions}. These functions measure the complexity of polynomials based on the location of their zeros in the complex plane. Following work of S.-J. Chern and J. Vaaler in \cite{chern-vaaler}, we associate to each multiplicative distance function two families of analytic functions which encode information about its range on \C[x] and \R[x]. These {\it moment functions} are Mellin transforms of distribution functions associated to the multiplicative distance function and demonstrate a great deal of arithmetic st…
Motivated by some algorithmic problems, we give lower bounds on the size of the multiplicative groups containing rational function images of low-dimensional affine subspaces of a finite field~$\mathbb{F}_{q^n}$ considered as a linear space over a subfield $\mathbb{F}_q$. We apply this to the recently introduced algorithmic problem of identity testing of "hidden" polynomials $f$ and $g$ over a high degree extension of a finite field, given oracle access to $f(x)^e$ and $g(x)^e$
In this paper, we consider the mean value of the product of two real valued multiplicative functions with shifted arguments. The functions $F$ and $G$ under consideration are close to two nicely behaved functions $A$ and $B$, such that the average value of $A(n-h)B(n)$ over any arithmetic progression is only dependent on the common difference of the progression. We use this method on the problem of finding mean value of $K(N)$, where $K(N)/\log N$ is the expected number of primes such that a random elliptic curve over rationals has $N$ points when reduced over those primes.
Let $F(x,y)$ be a polynomial over the rationals. We show that if $F$ is not an expander (over the rationals) then it has a special multiplicative or additive form. For example if $F$ is a homogeneous non-expander polynomial then $F(x,y)=c(x+ay)^α$ or $F(x,y)=c(xy)^α.$ This is an extension of an earlier result of Elekes and Rónyai who described the structure of two-variate polynomials which are not expanders over the reals.
We extend a bound of Roche-Newton, Shparlinski and Winterhof which says any subset of a finite field can be decomposed into two disjoint subset $\cU$ and $\cV$ of which the additive energy of $\cU$ and $f(\cV)$ are small, for suitably chosen rational functions $f$. We extend the result by proving equivalent results over multiplicative energy and the additive and multiplicative energy hybrids.
We construct classes of permutation polynomials over F_{Q^2} by exhibiting classes of low-degree rational functions over F_{Q^2} which induce bijections on the set of (Q+1)-th roots of unity in F_{Q^2}. As a consequence, we prove two conjectures about permutation trinomials from a recent paper by Tu, Zeng, Hu and Li.
Using Weil descent, we give bounds for the number of rational points on two families of curves over finite fields with a large abelian group of automorphisms: Artin-Schreier curves of the form $y^q-y=f(x)$ with $f\in\Fqr[x]$, on which the additive group $\Fq$ acts, and Kummer curves of the form $y^{\frac{q-1}{e}}=f(x)$, which have an action of the multiplicative group $\Fq^\star$. In both cases we can remove a $\sqrt{q}$ factor from the Weil bound when $q$ is sufficiently large.
We present a dynamical proof of the well-known fact that the Neron-Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field k of transcendence degree 1 over an algebraically closed field K of characteristic 0. More generally, we investigate the mechanism for which the local canonical height for a rational function f defined over k can take irrational values (at points in a local completion of k), providing examples in all degrees greater than 1. Building on Kiwi's classification of non-archimedean Julia sets for quadratic maps…
Let a and b be non-zero rational numbers that are multiplicatively independent. We study the natural density of the set of primes p for which the subgroup of the multiplicative group of the finite field with p elements generated by (a\mod p) contains (b\mod p). It is shown that, under assumption of the generalized Riemann hypothesis (GRH), this density exists and equals a positive rational multiple of the universal constant S=\prod_{p prime}(1-p/(p^3-1)). An explicit value of the density is given under mild conditions on a and b. This extends and corrects earlier work of P.J. Stephens (1976). …
Let $K$ be an algebraically closed field of characteristic zero and let $G$ be a finitely generated subgroup of the multiplicative group of $K$. We consider $K$-valued sequences of the form $a_n:=f(φ^n(x_0))$, where $φ\colon X\to X$ and $f\colon X\to\mathbb{P}^1$ are rational maps defined over $K$ and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $φ$ and $f$. Many classical sequences from number theory and algebraic combinatorics fall under this dynamical framework, and we show that the set of $n$ for which $a_n\in G$ is a finite union of arithmetic progressions al…
