Periods of orbits modulo primes

Let S be a monoid of endomorphisms of a quasiprojective variety V defined over a global field K. We prove a lower bound for the size of the reduction modulo places of K of the orbit of any point α∈V(K) under the action of the endomorphisms from S. We also prove a similar result in the context of Drinfeld modules. Our results may be considered as dynamical variants of Artin's primitive root conjecture.     

Some curious congruences modulo primes

Let n be a positive odd integer and let p>n+1 be a prime. We mainly derive the following congruence:

     

Factoring polynomials modulo special primes

We consider the problem of factoring polynomials overGF(p) for those prime numbersp for which all prime factors ofp– 1 are small. We show that if we have a primitivet-th root of unity for every primet dividingp– 1 then factoring polynomials overGF(p) can be done in deterministic polynomial time.Research partially supported by Hungarian National Foundation for Scientific Research, Grant 1812.     

Lacunary eta-quotients modulo powers of primes

The Ramanujan Journal - An integral power series is called lacunary modulo M if almost all of its coefficients are divisible by M. Motivated by the parity problem for the partition function, Gordon...     

Polynomial products modulo primes and applications

Monatshefte für Mathematik - For any polynomial $$P(x)in {mathbb {Z}}[x],$$ we study arithmetic dynamical systems generated by $$displaystyle {F_P(n)=mathop {prod nolimits _{kle...     

Congruences of multipartition functions modulo powers of primes

Let p _r (n) denote the number of r-component multipartitions of n, and let S _γ,λ be the space spanned by η(24z) ^γ ?(24z), where η(z) is the Dedekind’s eta function and ?(z) is a holomorphic modular form in \(M_{\lambda}(\mathrm{SL}_{2}(\mathbb{Z}))\) . In this paper, we show that the generating function of \(p_{r}(\frac{m^{k} n +r}{24})\) with respect to n is congruent to a function in the space S _γ,λ modulo m ^k . As special cases, this relation leads to many well known congruences including the Ramanujan congruences of p(n) modulo 5,7,11 and Gandhi’s congruences of p ₂(n) modulo 5 and p ₈(n) modulo 11. Furthermore, using the invariance property of S _γ,λ under the Hecke operator \(T_{\ell^{2}}\) , we obtain two classes of congruences pertaining to the m ^k -adic property of p _r (n).     

Maximum zero strings of Bell numbers modulo primes

For any prime p, the sequence of Bell exponential numbers B_n is shown to have p ? 1 consecutive values congruent to zero (mod p), beginning with B_m, where $\text{m ≡ 1 ?}\text{(p}{}^{\mathrm{p}}\text{? 1)}\text{(p ? 1)}{}^2$ ($\text{mod}\text{(p}{}^{\mathrm{p}}\text{? 1)}\text{(p ? 1)}$). This is an improvement over previous results on the maximal strings of zero residues of the Bell numbers. Similar results are obtained for the sequence of generalized Bell numbers A_n generated by .     

Distribution of a certain partition function modulo powers of primes

In this paper, we study a certain partition function a(n) defined by Σ _n≥0 a(n)q ⁿ := Π _n=1(1 − q ⁿ )⁻¹(1 − q ²ⁿ )⁻¹. We prove that given a positive integer j ≥ 1 and a prime m ≥ 5, there are infinitely many congruences of the type a(An + B) ≡ 0 (mod m ^j ). This work is inspired by Ono’s ground breaking result in the study of the distribution of the partition function p(n).     

Hyperbolic-parabolic deformations of rational maps

We develop a Thurston-like theory to characterize geometrically finite rational maps, and then apply it to study pinching and plumbing deformations of rational maps. We show that under certain conditions the pinching path converges uniformly and the quasiconformal conjugacy converges uniformly to a semi-conjugacy from the original map to the limit. Conversely, every geometrically finite rational map with parabolic points is the landing point of a pinching path for any prescribed plumbing combinatorics.     

On the tautological ring of a Jacobian modulo rational equivalence

We consider the Chow ring with rational coefficients of the Jacobian of a curve. Assume D is a divisor in a base point free of the curve such that the canonical divisor K is a multiple of the divisor D. We find relations between tautological cycles. We give applications for curves having a degree d covering of whose ramification points are all of order d, and then for hyperelliptic curves.      

Conjugacies between rational maps and extremal quasiconformal maps

We show that two rational maps which are -quasiconformally combinatorially equivalent are -quasiconformally conjugate. We also study the relationship between the boundary dilatation of a combinatorial equivalence and the dilatation of a conjugacy.

     

Powers of rationals modulo 1 and rational base number systems

A new method for representing positive integers and real numbers in a rational base is considered. It amounts to computing the digits from right to left, least significant first. Every integer has a unique expansion. The set of expansions of the integers is not a regular language but nevertheless addition can be performed by a letter-to-letter finite right transducer. Every real number has at least one such expansion and a countable infinite number of them have more than one. We explain how these expansions can be approximated and characterize the expansions of reals that have two expansions. The results that we derive are pertinent on their own and also as they relate to other problems in combinatorics and number theory. A first example is a new interpretation and expansion of the constant K(p) from the so-called “Josephus problem.” More important, these expansions in the base allow us to make some progress in the problem of the distribution of the fractional part of the powers of rational numbers. Work partially supported by the CNRS/JSPS contract 13 569, and by the “ACI Nouvelles Interfaces des Mathématiques”, contract 04 312.     

Regular versus continuous rational maps

We prove that a continuous map from a compact nonsingular real algebraic variety X into the unit 2-sphere can be approximated by regular maps if and only if it is homotopic to a continuous map which is regular in the complement of a Zariski closed subvariety A of X of codimension at least 3. The assumption on the codimension of A is essential.     

Foliations invariant by rational maps

We give a classification of pairs ${(\mathcal{F}, \phi)}$ where ${\mathcal{F}}$ is a holomorphic foliation on a projective surface and ${\phi}$ is a non-invertible dominant rational map preserving ${\mathcal{F}}$ .     

Evaluation maps in rational homotopy

In the rational category of nilpotent complexes, let E be an H-space acting on a space X. With mild hypotheses we show that the action on the base point factors through a map Γ_E:S_E→X, where S_E is a finite product of odd-dimensional spheres and Γ_E is a homotopy monomorphism. Among others, the following consequences are obtained: if and only if is essential and if and only if X satisfies a strong splitting condition.