scholarly journals Traces, high powers and one level density for families of curves over finite fields

2017 ◽  
Vol 165 (2) ◽  
pp. 225-248 ◽  
Author(s):  
ALINA BUCUR ◽  
EDGAR COSTA ◽  
CHANTAL DAVID ◽  
JOÃO GUERREIRO ◽  
DAVID LOWRY–DUDA
Keyword(s):  
Finite Field ◽  
Moduli Spaces ◽  
Function Field ◽  
Level Density ◽  
Zeta Functions ◽  
Hyperelliptic Curves ◽  
Unitary Matrix ◽  
Moduli Spaces Of Curves ◽  
A New Technique ◽  
The One

AbstractThe zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix ΘC. We develop and present a new technique to compute the expected value of tr(ΘCn) for various moduli spaces of curves of genus g over a fixed finite field in the limit as g is large, generalising and extending the work of Rudnick [Rud10] and Chinis [Chi16]. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given in [BDF+16] and [Zha]. We extend [BDF+16] by describing explicit dependence on the place and give an explicit proof of the Lindelöf bound for function field Dirichlet L-functions L(1/2 + it, χ). As applications, we compute the one-level density for hyperelliptic curves, cyclic ℓ-covers, and cubic non-Galois covers.

scholarly journals Picard Groups of Moduli Spaces of Curves with Symmetry

2018 ◽  
Vol 2020 (23) ◽  
pp. 9293-9335
Author(s):  
Kevin Kordek
Keyword(s):  
Moduli Spaces ◽  
Topological Type ◽  
Hyperelliptic Curves ◽  
Mapping Class ◽  
Finitely Generated ◽  
Smooth Curves ◽  
Group Of Automorphisms ◽  
Picard Groups ◽  
Moduli Spaces Of Curves ◽  
Special Cases

Abstract We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the 1st part of the paper, we show that, under mild restrictions, the moduli spaces of smooth curves with an abelian group of automorphisms of a fixed topological type have finitely generated Picard groups. In certain special cases, we are able to compute them exactly. In the 2nd part of the paper, we show that finite abelian level covers of the hyperelliptic locus in the moduli space of smooth curves have finitely generated Picard groups. We also compute the Picard groups of the moduli spaces of hyperelliptic curves of compact type.

scholarly journals Computing the Characteristic Polynomials of a Class of Hyperelliptic Curves for Cryptographic Applications

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
Lin You ◽  
Guangguo Han ◽  
Jiwen Zeng ◽  
Yongxuan Sang
Keyword(s):  
Finite Field ◽  
Characteristic Polynomial ◽  
Hyperelliptic Curve ◽  
Zeta Functions ◽  
Hyperelliptic Curves ◽  
Common Method ◽  
Characteristic Polynomials ◽  
Computational Data ◽  
Prime Factors

Hyperelliptic curves have been widely studied for cryptographic applications, and some special hyperelliptic curves are often considered to be used in practical cryptosystems. Computing Jacobian group orders is an important operation in constructing hyperelliptic curve cryptosystems, and the most common method used for the computation of Jacobian group orders is by computing the zeta functions or the characteristic polynomials of the related hyperelliptic curves. For the hyperelliptic curveCq:v2=up+au+bover the fieldFqwithqbeing a power of an odd primep, Duursma and Sakurai obtained its characteristic polynomial forq=p,a=−1,andb∈Fp. In this paper, we determine the characteristic polynomials ofCqover the finite fieldFpnforn=1, 2 anda,b∈Fpn. We also give some computational data which show that many of those curves have large prime factors in their Jacobian group orders, which are both practical and vital for the constructions of efficient and secure hyperelliptic curve cryptosystems.

scholarly journals A heuristic for the distribution of point counts for random curves over a finite field

2015 ◽  
Vol 373 (2040) ◽  
pp. 20140310 ◽  
Author(s):  
Jeffrey D. Achter ◽  
Daniel Erman ◽  
Kiran S. Kedlaya ◽  
Melanie Matchett Wood ◽  
David Zureick-Brown
Keyword(s):  
Finite Field ◽  
Poisson Distribution ◽  
Moduli Spaces ◽  
Algebraic Curve ◽  
Rational Points ◽  
Random Curves ◽  
Point Counts ◽  
Moduli Spaces Of Curves ◽  
Large Genus

How many rational points are there on a random algebraic curve of large genus g over a given finite field ? We propose a heuristic for this question motivated by a (now proven) conjecture of Mumford on the cohomology of moduli spaces of curves; this heuristic suggests a Poisson distribution with mean q +1+1/( q −1). We prove a weaker version of this statement in which g and q tend to infinity, with q much larger than g .

scholarly journals Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field

2009 ◽  
Vol 146 (1) ◽  
pp. 81-101 ◽  
Author(s):  
Dmitry Faifman ◽  
Zeév Rudnick
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Finite Field ◽  
Central Limit ◽  
Riemann Hypothesis ◽  
Zeta Functions ◽  
Hyperelliptic Curves ◽  
Distribution Of Zeros ◽  
Expected Number ◽  
Large Genus

AbstractWe study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval ℐ will contain asymptotically 2g∣ℐ∣ angles as the genus grows. We show that for the variance of number of angles in ℐ is asymptotically (2/π2)log (2g∣ℐ∣) and prove a central limit theorem: the normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles 2g∣ℐ∣ tends to infinity.

scholarly journals A BOUND FOR THE INDEX OF A QUADRATIC FORM AFTER SCALAR EXTENSION TO THE FUNCTION FIELD OF A QUADRIC

2018 ◽  
Vol 19 (2) ◽  
pp. 421-450 ◽  
Author(s):  
Stephen Scully
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Function Field ◽  
The Other ◽  
Direct Generalization ◽  
Other Hand ◽  
General Field ◽  
The One ◽  
Anisotropic Quadratic Form

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

scholarly journals Moduli spaces of curves with homology chains and c = 1 matrix models

1994 ◽  
Vol 327 (3-4) ◽  
pp. 221-225 ◽  
Author(s):  
A.S. Cattaneo ◽  
A. Gamba ◽  
M. Martellini
Keyword(s):  
Matrix Models ◽  
Moduli Spaces ◽  
Moduli Spaces Of Curves

Random Dieudonné modules, random -divisible groups, and random curves over finite fields

2013 ◽  
Vol 12 (3) ◽  
pp. 651-676 ◽  
Author(s):  
Bryden Cais ◽  
Jordan S. Ellenberg ◽  
David Zureick-Brown
Keyword(s):  
Probability Distribution ◽  
Finite Field ◽  
Random Matrix ◽  
Hyperelliptic Curve ◽  
Hyperelliptic Curves ◽  
Plane Curves ◽  
Random Curves ◽  
Isomorphism Classes ◽  
Curves Over Finite Fields ◽  
Divisible Groups

AbstractWe describe a probability distribution on isomorphism classes of principally quasi-polarized $p$-divisible groups over a finite field $k$ of characteristic $p$ which can reasonably be thought of as a ‘uniform distribution’, and we compute the distribution of various statistics ($p$-corank, $a$-number, etc.) of $p$-divisible groups drawn from this distribution. It is then natural to ask to what extent the $p$-divisible groups attached to a randomly chosen hyperelliptic curve (respectively, curve; respectively, abelian variety) over $k$ are uniformly distributed in this sense. This heuristic is analogous to conjectures of Cohen–Lenstra type for $\text{char~} k\not = p$, in which case the random $p$-divisible group is defined by a random matrix recording the action of Frobenius. Extensive numerical investigation reveals some cases of agreement with the heuristic and some interesting discrepancies. For example, plane curves over ${\mathbf{F} }_{3} $ appear substantially less likely to be ordinary than hyperelliptic curves over ${\mathbf{F} }_{3} $.

scholarly journals One-Time Cortical Lamina: A New Technique for Horizontal Ridge Augmentation. A Case Series

2021 ◽  
Vol 8 (6) ◽  
pp. 22-30
Author(s):  
Vincenzo Foti ◽  
Davide Savio ◽  
Roberto Rossi
Keyword(s):  
Fibrin Sealant ◽  
Case Series ◽  
X Rays ◽  
Marginal Bone Loss ◽  
Ridge Augmentation ◽  
Keratinized Tissue ◽  
A New Technique ◽  
The One ◽  
Hard Consistency

The aim of this case series is to introduce the One-Time Cortical Lamina Technique, a simplification of the F.I.R.S.T. (Fibrinogen-Induced Regeneration Sealing Technique) in cases where only horizontal augmentation is needed. The indications for this technique are ASA2 and ASA1 anxious patients. Pre-requisites for this surgical technique are: a good amount of keratinized tissue, sufficient alveolar ridge width for placement of implants, thickness of vestibular bone at CBCT planning less than 1 mm with risk of threads exposure. Five patients with horizontal deficiencies were selected to test the efficacy of this approach. The defects were augmented using a porcine cortical bone lamina in combination with collagenated porcine bone mixed with fibrin sealant. The cortical lamina was placed only buccal to the implants and stabilized with fibrin sealant, without pins or screws. Upon completion of the implant surgery, healing abutments were connected to the implants and the soft tissue sutured around them. The healing was uneventful in all cases. Six months after surgery impressions for final restorations were taken and screwed crowns delivered. The new volume had hard consistency and the follow-up CBCT measured an average of 4.17 mm of horizontal bone augmentation. One to three years of follow up demonstrated the maintenance of vestibular volume, hard consistency and clinical stability. Intraoral X-rays showed no marginal bone loss. An advantage of this technique could be the one stage surgery that creates a stable environment for regeneration from day one.

scholarly journals Dynamical behavior of a stochastic predator-prey model with general functional response and nonlinear jump-diffusion

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xinhong Zhang ◽  
Qing Yang
Keyword(s):  
Functional Response ◽  
Dynamical Behavior ◽  
Uniform Boundedness ◽  
Jump Diffusion ◽  
Necessary Condition ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
A New Technique ◽  
The One

<p style='text-indent:20px;'>In this paper, we consider a stochastic predator-prey model with general functional response, which is perturbed by nonlinear Lévy jumps. Firstly, We show that this model has a unique global positive solution with uniform boundedness of <inline-formula><tex-math id="M1">\begin{document}$ \theta\in(0,1] $\end{document}</tex-math></inline-formula>-th moment. Secondly, we obtain the threshold for extinction and exponential ergodicity of the one-dimensional Logistic system with nonlinear perturbations. Then based on the results of Logistic system, we introduce a new technique to study the ergodic stationary distribution for the stochastic predator-prey model with general functional response and nonlinear jump-diffusion, and derive the sufficient and almost necessary condition for extinction and ergodicity.</p>

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