An application of the effective Sato-Tate conjecture

Based on the Lagarias-Odlyzko effectivization of the Chebotarev density theorem, Kumar Murty gave an effective version of the Sato-Tate conjecture for an elliptic curve conditional on analytic continuation and Riemann hypothesis for the symmetric power $L$-functions. We use Murty's analysis to give a similar conditional effectivization of the generalized Sato-Tate conjecture for an arbitrary motive. As an application, we give a conditional upper bound of the form $O((\log N)^2 (\log \log 2N)^2)$ for the smallest prime at which two given rational elliptic curves with conductor at most $N$ have Frobenius traces of opposite sign.Comment: 12 pages; v2: refereed versio

The probability that a complete intersection is smooth

Given a smooth subscheme of a projective space over a finite field, we compute the probability that its intersection with a fixed number of hypersurface sections of large degree is smooth of the expected dimension. This generalizes the case of a single hypersurface, due to Poonen. We use this result to give a probabilistic model for the number of rational points of such a complete intersection. A somewhat surprising corollary is that the number of rational points on a random smooth intersection of two surfaces in projective 3-space is strictly less than the number of points on the projective line.Comment: 14 pages; v3: final journal versio

Organizational Wellness Modeling

The aim of the present paper is to establish certain mathematical models for organizational wellness as well as to create some wellness optimization problems applicable to any type of organization (including universities) that might be mathematically solved resorting to aspects of operational research of mathematical analysis. The results obtained associated with a mathematical apparatus enable one to perform analyses, comparisons, interpretations, predictions. All of us have, consciously or not, a genuine curiosity in creating and shaping organizational wellness. This concept represents a highly topical issue since professional activity, irrespective of its field, holds a very significant place for each of us. The hard-to-dinstinguish „border” between personal and professional life urges all companies, smaller or larger, to attempt to find solution at an individual and organizational level in order to support and improve the concept of organizational wellness, in its current and future understanding. Regardless of their thoughts, feelings or actions, all individuals belong to that organization.organizational wellness; mathematical models; social processes modelling.

The fluctuations in the number of points of smooth plane curves over finite fields

In this note, we study the fluctuations in the number of points of smooth projective plane curves over finite fields $\mathbb{F}_q$ as $q$ is fixed and the genus varies. More precisely, we show that these fluctuations are predicted by a natural probabilistic model, in which the points of the projective plane impose independent conditions on the curve. The main tool we use is a geometric sieving process introduced by Poonen.Comment: 12 page

Effective Sato-Tate conjecture for abelian varieties and applications

From the generalized Riemann hypothesis for motivic L-functions, we derive an effective version of the Sato-Tate conjecture for an abelian variety A defined over a number field k with connected Sato-Tate group. By effective we mean that we give an upper bound on the error term in the count predicted by the Sato-Tate measure that only depends on certain invariants of A. We discuss three applications of this conditional result. First, for an abelian variety defined over k, we consider a variant of Linnik's problem for abelian varieties that asks for an upper bound on the least norm of a prime whose normalized Frobenius trace lies in a given interval. Second, for an elliptic curve defined over k with complex multiplication, we determine (up to multiplication by a nonzero constant) the asymptotic number of primes whose Frobenius trace attain the integral part of the Hasse-Weil bound. Third, for a pair of abelian varieties defined over k with no common factors up to k-isogeny, we find an upper bound on the least norm of a prime at which the respective Frobenius traces have opposite sign.Comment: 28 page

Statistics for traces of cyclic trigonal curves over finite fields

We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over a field of q elements as the curve varies in an irreducible component of the moduli space. We show that for q fixed and g increasing, the limiting distribution of the trace of the Frobenius equals the sum of q+1 independent random variables taking the value 0 with probability 2/(q+2) and 1, e^{(2pi i)/3}, e^{(4pi i)/3} each with probability q/(3(q+2)). This extends the work of Kurlberg and Rudnick who considered the same limit for hyperelliptic curves. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution and how to generalize these results to p-fold covers of the projective line.Comment: 30 pages, added statement and sketch of proof in Section 7 for generalization of results to p-fold covers of the projective line, the final version of this article will be published in International Mathematics Research Notice

Galois representations and Galois groups over Q

In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve, let J(C) be the associated Jacobian variety and let ¯ρℓ : GQ → GSp(J(C)[ℓ]) be the Galois representation attached to the ℓ-torsion of J(C). Assume that there exists a prime p such that J(C) has semistable reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes ℓ (if they exist) such that ¯ρℓ is surjective. In particular we realize GSp6 (Fℓ) as a Galois group over Q for all primes ℓ ∈ [11, 500000]