A Carlitz module analogue of a conjecture of Erdos and Pomerance

Abstract. Let A = Fq[T] be the ring of polynomials over the finite field Fq and 0 = a ∈ A. Let C be the A-Carlitz module. For a monic polynomial m ∈ A, let C(A/mA) and ¯a be the reductions of C and a modulo mA respectively. Let fa(m) be the monic generator of the ideal {f ∈ A, Cf (¯a) = ¯0} on C(A/mA). We denote by ω(fa(m)) the number of distinct monic irreducible factors of fa(m). If q = 2 or q = 2 and a = 1, T, or (1 + T), we prove that there exists a normal distribution for the quantity ω(fa(m)) − 1 2 (log deg m)2 √1 3 (log deg m)3/2 . This result is analogous to an open conjecture of Erd˝os and Pomerance concerning the distribution of the number of distinct prime divisors of the multiplicative order of b modulo n, where b is an integer with |b| > 1, and n a positive integer

The Erdős–Kac theorem and its generalizations

Abstract. We give a survey of the Erd}o-Kac theorem and its various generalizations. In particular, we discuss an open conjecture of Erd}os and Pomerance about the distribution of the number of distinct prime divisors of the order of a xed integer in the multiplicative groups (Z=nZ) . We also formulate a Carlitz module analogue of this conjecture and provide a sketch of its proof

Gaussian Laws on Drinfeld Modules

Let A = q[T] be the polynomial ring over the finite field q, k = q(T) the rational function field, and K a finite extension of k. Let ϕ be a Drinfeld A-module over K of rank r. For a place of K of good reduction, write , where is the valuation ring of and its maximal ideal. Let P, ϕ(X) be the characteristic polynomial of the Frobenius automorphism of acting on a Tate module of ϕ. Let χϕ() = P, ϕ(1), and let ν(χϕ()) be the number of distinct primes dividing χϕ(). If ϕ is of rank 2 with , we prove that there exists a normal distribution for the quantity For r ≥ 3, we show that the same result holds under the open image conjecture for Drinfeld modules. We also study the number of distinct prime divisors of the trace of the Frobenius automorphism of acting on a Tate module of ϕ and obtain similar results

Cyclicity of finite Drinfeld modules

This is the peer reviewed version of the following article: Kuo, W., & Liu, Y.-R. (2009b). Cyclicity of finite Drinfeld modules. Journal of the London Mathematical Society, 80(3), 567–584, which has been published in final form at https://doi.org/10.1112/jlms/jdp043. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.Le tA=Fq[T] be the polynomial ring over the finite field Fq,letk=Fq(T) be the rational function field, and let K be a finite extension of k. For a prime P of K, we denote by OP the valuation ring of P, by MP the maximal ideal of OP, and by FP the residue field OP/MP. Let φ be a DrinfeldA-module over K of rankr. If φ has good reduction at P, let φ ⊗ FP denote the reduction of φ at P and letφ(FP) denote the A-module (φ⊗FP)(FP). Ifφis of rank 2 with End ̄K(φ)=A, then we obtain an asymptotic formula for the number of primes P of K of degree x for which φ (FP) is cyclic. This result can be viewed as a Drinfeld module analogue of Serre’s cyclicity result on elliptic curves. We also show that whenφis of rankr 3 a similar result follows.This research was supported by an NSERC Discovery Grant

Multidimensional Vinogradov-type Estimates in Function Fields

This article has been published in a revised form in the Canadian Journal of Mathematics https://doi.org/10.4153/CJM-2013-014-9. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. Copyright © Canadian Mathematical Society 2014.Let Fq[t] denote the polynomial ring over the finite field Fq. We employ Wooley’s new efficient congruencing method to prove certain multidimensional Vinogradov-type estimates in Fq[t]. These results allow us to apply a variant of the circle method to obtain asymptotic formulas for a system connected to the problem about linear spaces lying on hypersurfaces defined over Fq[t].Research supported by NSERC Discovery Grants || NSFC, 11126191 || NSFC, 11201163

The asymptotic estimates and Hasse principle for multidimensional Waring's problem

This article is made available through Elsevier's Open Archive https://doi.org/10.1016/j.aim.2019.06.028. © 2019 Elsevier Inc. All rights reserved.Motivated by the asymptotic estimates and Hasse principle for multidimensional Waring's problem via the circle method, we prove for the first time that the corresponding singular series is bounded below by an absolute positive constant without any nonsingular local solubility assumption. The number of variables we need is near-optimal. By proving a more general uniform density result over certain complete discrete valuation rings with finite residue fields, we also establish uniform lower bounds for both singular series and singular integral in Fq[t]. We thus obtain asymptotic formulas and the Hasse principle for multidimensional Waring's problem in Fq[t] via a variant of the circle method.NSERC Discovery Grant, No. RGPIN-2015-03709 || NSERC Discovery Grant, RGPIN-2016-03720 || National Natural Science Foundation of China, Grant No. 11201163

The Shifted Turan Sieve Method on Tournaments

This article has been published in a revised form in the Canadian Mathematical Bulletin http://dx.doi.org/10.4153/S000843951900016X. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. Copyright © Canadian Mathematical Society 2019.Abstract. We construct a shi ed version of the Turán sieve method developed by R. Murty and the second author and apply it to counting problems on tournaments. More precisely, we obtain upper bounds for the number of tournaments which contain a fixed number of restricted r-cycles. These are the first concrete results which count the number of cycles over “all tournaments”.Research partially supported by NSERC Discovery Grants || CAPES and CSF/CNPQ, Brazil