We introduce the notion of "quasi-symmetric" polynomials, which is a generalization of the notion of symmetry, and is particularly suited to the setting of polynomial rings over finite fields. The properties of this new class of functions are studied. It is then shown that quasi-symmetric polynomials are very useful in splitting rational places in extensions of function fields over finite fields and thus may be used in constructing extensions with many rational places.
The notion of symmetry in polynomial rings with several indeterminates is generalized to polynomial rings over finite fields. Families of extensions of the projective line over a finite field of constants possessing this property are explicitly constructed. These extensions exhibit complete splitting of all finite rational places. Subcovers of these extensions are also explicitly described. New examples of function fields attaining the Oesterle bounds are obtained. These constructions are compared with class field theoretic constructions achieving similar splitting of rational places. A genera…
Let $Σ$ be a smooth projective surface, let $f' : S' \to Σ$ be a double cover of $Σ$ and let $μ: S \to S'$ be the canonical resolution. Put $f = f'\circμ$. An irreducible curve $C$ on $Σ$ is said to be a splitting curve with respect to $f$ if $f^*C$ is of the form $C^+ + C^- + E$, where $C^- = σ_f^*C^+$, $σ_f$ being the covering transformation of $f$ and all irreducible components of $E$ are contained in the exceptional set of $μ$. In this article, we show that a kind of "reciprocity" of splitting curves holds for a certain pair of curves on rational ruled surfaces. As an application, we consi…
We analyze the probability that, for a fixed finite set of primes S, a random, monic, degree n polynomial f(x) with integer coefficients in a box of side B around 0 satisfies: (i) f(x) is irreducible over the rationals, with splitting field over the rationals having Galois group $S_n$; (ii) the polynomial discriminant Disc(f) is relatively prime to all primes in S; (iii) f(x) has a prescribed splitting type at each prime p in S. The limit probabilities as $B \to \infty$ are described in terms of values of a one-parameter family of measures on $S_n$, called splitting measures, with parameter $z…
The Duffin--Schaeffer Conjecture answers a question on how well one can approximate irrationals by rational numbers in reduced form (an imposed condition) where the accuracy of the approximation depends on the rational number. It can be viewed as an analogue to Khintchine's Theorem with the added restriction of only allowing rationals in reduced form. Other conditions such as numerator or denominator a prime, a square-free integer, or an element of a particular arithmetic progression, etc. have also been imposed and analogues of Khintchine's Theorem studied. We prove versions of Khintchine's T…
We prove that the density of polynomials $P(x)=\sum_{i=0}^n a_n x^n$ over a local field $K$ generating an étale extension with specified splitting type is a rational function in terms of the size of the residue field of $K$ in the case where the splitting type is tame. Moreover, we give a computable recursive formula for these densities and compute the asymptotics of this density as the size of the residue field tends to infinity.
Let Lambda be a numerical semigroup. Assume there exists an algebraic function field over GF(q) in one variable which possesses a rational place that has Lambda as its Weierstrass semigroup. We ask the question as to how many rational places such a function field can possibly have and we derive an upper bound in terms of the generators of Lambda and q. Our bound is an improvement to a bound by Lewittes which takes into account only the multiplicity of Lambda and q. From the new bound we derive significant improvements to Serre's upper bound in the cases q=2, 3 and 4. We finally show that Lewit…
Consider an absolutely simple abelian variety A defined over a number field K. For most places v of K, we study how the reduction A_v of A modulo v splits up to isogeny. Assuming the Mumford-Tate conjecture for A and possibly increasing K, we will show that A_v is isogenous to the m-th power of an absolutely simple abelian variety for all places v of K away from a set of density 0, where m is an integer depending only on the endomorphism ring End(A_Kbar). This proves many cases, and supplies justification, for a conjecture of Murty and Patankar. Under the same assumptions, we will also describ…
Given a number field $k$ and a finite $k$-group $G$, the Tame Approximation Problem for $G$ asks whether the restriction map $H^1(k,G)\to\prod_{v\inΣ}H^1(k_v,G)$ is surjective for every finite set of places $Σ\subseteqΩ_k$ disjoint from $\text{Bad}_G$, where $\text{Bad}_G$ is the finite set of places that either divides the order of $G$ or ramifies in the minimal extension splitting $G$. In this paper we prove that the set $\text{Bad}_G$ is "sharp". To achieve this we prove that there are finite abelian $k$-groups $A$ where the map $H^1(k,A)\to\prod_{v\inΣ_0}H^1(k_v,A)$ is not surjective in a …
Let $K$ denote a number field and a finite set $S$ of places of $K$ and $ϕ:\PP^n\rightarrow\PP^n$ be rational morphism defined over $K$. The main result of this paper proves that there are only finitely many twists of $ϕ$ defined over $K$ which have good reduction at all places outside $S$. This answers a question of Silverman in the affirmative.
Let $K$ be a number field and $ϕ\in K(z)$ a rational function. Let $S$ be the set of all archimedean places of $K$ and all non-archimedean places associated to the prime ideals of bad reduction for $ϕ$. We prove an upper bound for length of finite orbits of $ϕ$ in $\mathbb{P}_1(K)$ depending only on the cardinality of $S$.
Certain towers of function fields with complete splitting of rational places at each stage are constructed. Also, families oof towers with positive N/g ratios are described.
We partially answer, in terms of monodromy, Murty and Patankar's question: Given an absolutely simple abelian variety over a number field, does it have simple specializations at a set of places of positive Dirichlet density? The answer is based on the classification of pairs (G,V) consisting of a semi-simple algebraic group G over a non-archimedean local field and an absolutely irreducible representation V of G such that G admits a maximal torus acting irreducibly on V.
Ideas and techniques from Khare's and Wintenberger's article on the proof of Serre's conjecture for odd conductors are used to establish that for a fixed prime l infinitely many of the groups PSL_2(F_{l^r}) (for r running) occur as Galois groups over the rationals such that the corresponding number fields are unramified outside a set consisting of l, the infinite place and only one other prime.
Given a number field $K$ and a finite set $S$ of places of $K$, the first main result of this paper shows that the quadratic rational maps $ϕ:{\mathbb P}^1\to{\mathbb P}^1$ defined over $K$ which have good reduction at all places outside $S$ comprise a Zariski-dense subset of the moduli space ${\mathcal M}_2$ parametrizing all isomorphism classes of quadratic rational maps. We then consider quadratic rational maps with double unramified fixed-point structure, and our second main result establishes a geometric Shafarevich-type non-Zariski-density result for the set of such maps with good reduct…
In an earlier work, the first author and Petsche used potential theoretic techniques to establish a lower bound for the height of algebraic numbers that satisfy splitting conditions, such as being totally real or p-adic, improving on earlier work of Bombieri and Zannier in the totally p-adic case. These bounds applied as the degree of the algebraic number over the rationals tended towards infinity. In this paper, we use discrete energy approximation techniques on the Berkovich projective line to make the dependence on the degree in these bounds explicit, and we establish lower bounds for algeb…
Let $k$ be an algebraic closed field of characteristic zero. Let $K$ be the rational function field $K=k(t)$. Let $ϕ$ be a non isotrivial rational function in $K(z)$. We prove a bound for the cardinality of the set of $K$--rational preperiodic points for $ϕ$ in terms of the number of places of bad reduction and the degree $d$ of $ϕ$.
Let $ϕ$ be a an endomorphism of degree $d\geq{2}$ of the projective line, defined over a number field $K$. Let $S$ be a finite set of places of $K$, including the archimedean places, such that $ϕ$ has good reduction outside of $S$. The article presents two main results: the first result is a bound on the number of $K$-rational preperiodic points of $ϕ$ in terms of the cardinality of the set $S$ and the degree $d$ of the endomorphism $ϕ$. This bound is quadratic in terms of $d$ which is a significant improvement to all previous bounds on the number of preperiodic points in terms of the degree $…
We give defining equations for function fields over finite fields with many rational places. They are obtained from composita of quadratic extensions of the rational function field.
If phi(z) is a rational function on P^1 of degree at least 2 with coefficients in a number field k, we compute the homogeneous transfinite diameter of the v-adic filled Julia sets of phi for all places v of k by introducing a new quantity called the homogeneous sectional capacity. In particular, we show that the product over all places of these homogeneous transfinite diameters is 1. We apply this product formula and some new potential-theoretic results concerning Green's functions on Riemann surfaces and Berkovich spaces to prove an adelic equidistribution theorem for dynamical systems on the…
Let $P\in\mathbb{P}_1(\mathbb{Q})$ be a periodic point for a monic polynomial with coefficients in $\mathbb{Z}$. With elementary techniques one sees that the minimal periodicity of $P$ is at most $2$. Recently we proved a generalization of this fact to the set of all rational functions defined over ${\mathbb{Q}}$ with good reduction everywhere (i.e. at any finite place of $\mathbb{Q}$). The set of monic polynomials with coefficients in $\mathbb{Z}$ can be characterized, up to conjugation by elements in PGL$_2({\mathbb{Z}})$, as the set of all rational functions defined over $\mathbb{Q}$ with a…
Let p be a rational prime and let F be a number field. Then, for each i>0, there is a short exact localization sequence for K_{2i}(F). If p is odd or F is nonexceptional, we find necessary and sufficient conditions for this exact sequence to split: these conditions involve coinvariants of twisted p-parts of the p-class groups of certain subfields of the fields F(μ_{p^n}) for n\in N. We also compare our conditions with the weaker condition WK^{et}_{2i}(F)=0 and give some example.
Let $K$ be a number field and let $ϕ$ in $K(z)$ be a rational function of degree $d\geq 2$. Let $S$ be the places of bad reduction for $ϕ$ (including the archimedan places). Let $Per(ϕ,K)$, $PrePer(ϕ, K)$, and $Tail(ϕ,K)$ be the set of $K$-rational periodic, preperiodic, and purely preperiodic points of $ϕ$, respectively. The present paper presents two main results. The first result gives a bound for $|PrePer(ϕ,K)|$ in terms of the number of places of bad reduction $|S|$ and the degree $d$ of the rational function $ϕ$. This bound significantly improves a previous bound given by J. Canci and L.…
Let $ϕ$ be an endomorphism of the projective line defined over a global field $K$. We prove a bound for the cardinality of the set of $K$-rational preperiodic points for $ϕ$ in terms of the number of places of bad reduction. The result is completely new in the function fields case and it is an improvement of the number fields case. An important tool is an $S$-unit equation theorem in 2 variables.
In this work, we use the notion of ``symmetry'' of functions for an extension $K/L$ of finite fields to produce extensions of a function field $F/K$ in which almost all places of degree one split completely. Then we introduce the notion of ``quasi-symmetry'' of functions for $K/L$, and demonstrate its use in producing extensions of $F/K$ in which all places of degree one split completely. Using these techniques, we are able to restrict the ramification either to one chosen rational place, or entirely to non-rational places. We then apply these methods to the related problem of building asympto…
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…
