Let 𝔽_q be a finite field of characteristic p and π Y→ X be a finite 𝔽_q-morphism of separated 𝔽_q-schemes of finite type. Suppose π is generically Galois with group G of prime order r≠ p. We determine the mod-r reduction of the zeta function of Y in terms of the zeta function of X and the branch locus Z⊂ X of π. We give applications to curves and to numerators of hyperelliptic/superelliptic curves.

We first ask how hard it is to say of a countable well-ordered set A that it has order type at least some given ordinal α . We measure complexity in the Borel and effective Borel hierarchies. The class of well orderings is not Borel, so our results use Calvert’s notions of complexity and completeness within. We then turn to subsets of ordered Abelian groups. We ask how hard it is to say of well-ordered subsets A, B of such a group G that the set A+B = {a+b:a∈ A & b∈ B} has type at least α . The question is more interesting if we require that A and B have type strictly less than α . If α is multiplicatively indecomposable, so of the form ω ^ω ^γ , the answer is trivial—if tp(A),tp(B) < α , then tp(A+B) < α . Assuming that G is Archimedean, we have a precise result for α the n^th power of a multiplicatively indecomposable ordinal, for finite n≥ 2 . Finally, we ask how hard it is to say of a well-ordered set A of non-negative elements in an ordered Abelian group G that the set [A], consisting of finite sums of elements of A, has type at least α . Assuming that G is Archimedean, we have complete results.

We use knowledge of local fields to adapt Jonathan Lubin and Michael Rosen’s proof of Mazur’s Proposition 4.39. This changes the result about abelian varieties from only working over local fields with a finite residue field to working with local fields with an arbitrary perfect residue field of positive characteristic. We then briefly discuss the Local Class Field Theory implications of such information

We study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^{r-1}(x + 1)(x + t)$ over the function field $\mathbb{F}_p(t)$, when $p$ is prime and $r\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, we compute the $L$-function of $J$ over $\mathbb{F}_q(t^{1/d})$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\mathbb{F}_q(t^{1/d})$. When $d$ is divisible by $r$ and of the form $p^\nu +1$, and $K_d := \mathbb{F}_p(\mu_d,t^{1/d})$, we write down explicit points in $J(K_d)$, show that they generate a subgroup $V$ of rank $(r-1)(d-2)$ whose index in $J(K_d)$ is finite and a power of $p$, and show that the order of the Tate-Shafarevich group of $J$ over $K_d$ is $[J(K_d):V]^2$. When $r>2$, we prove that the new part of $J$ is isogenous over $\overline{\mathbb{F}_p(t)}$ to the square of a simple abelian variety of dimension $\phi(r)/2$ with endomorphism algebra $\mathbb{Z}[\mu_r]^+$. For a prime $\ell$ with $\ell \nmid pr$, we prove that $J[\ell](L)=\{0\}$ for any abelian extension $L$ of $\overline{\mathbb{F}}_p(t)$.

We compute the variances of sums in arithmetic progressions of generalized [Formula: see text]-divisor functions related to certain [Formula: see text]-functions in [Formula: see text], in the limit as [Formula: see text]. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when [Formula: see text], in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual [Formula: see text]-divisor function, when the [Formula: see text]-function in question has degree one. They illustrate the role played by the degree of the [Formula: see text]-functions; in particular, we find qualitatively new behavior when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over [Formula: see text], and we illustrate them by examining in some detail the generalized [Formula: see text]-divisor functions associated with the Legendre curve.

We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain L-functions of degree 2 and higher in F-q[t], in the limit as q -> infinity. This is achieved by establishing appropriate equidistribution results for the associated Frobenius conjugacy classes. The variances are thus related to matrix integrals, which may be evaluated. Our results differ significantly from those that hold in the case of degree-1 L-functions (i.e., situations considered previously using this approach). They correspond to expressions found recently in the number field setting assuming a generalization of the pair correlation conjecture. Our calculations apply, for example, to elliptic curves defined over F-q[t].

Let G be a finite connected graph, and let ρ be the spectral radius of its universal cover. For example, if G is k-regular then ρ=2√k−1. We show that for every r, there is an r-covering (a.k.a. an r-lift) of G where all the new eigenvalues are bounded from above by ρ. It follows that a bipartite Ramanujan graph has a Ramanujan r-covering for every r. This generalizes the r=2 case due to Marcus, Spielman and Srivastava (2013). Every r-covering of G corresponds to a labeling of the edges of G by elements of the symmetric group Sr. We generalize this notion to labeling the edges by elements of various groups and present a broader scenario where Ramanujan coverings are guaranteed to exist. In particular, this shows the existence of richer families of bipartite Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava, a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups. The core argument of this component is taken from Marcus-Spielman-Srivastava (2015). Another important ingredient of our proof is a new generalization of the matching polynomial of a graph. We define the r-th matching polynomial of G to be the average matching polynomial of all r-coverings of G. We show this polynomial shares many properties with the original matching polynomial. For example, it is real rooted with all its roots inside [−ρ,ρ].

The Hardy--Littlewood prime $k$-tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem. In this work, we extend the Maynard-Tao method to both number fields and the function field $\mathbb{F}_q(t)$.

We show that certain ramification invariants associated to a compatible system of $\ell$-adic sheaves on a curve are independent of $\ell$.

Let A be an abelian variety defined over a number field K. Let p be a prime of K of good reduction and A(p) the fiber of A over the residue field k(p). We call A(K)(p) the image of the Mordell-Weil group via reduction modulo p, which is a subgroup of A(p)(k)(p). We prove in particular that the size of A(K)(p), by varying p, encodes enough information to characterize the K-isogeny class of A, provided that the following necessary condition holds: the Mordell-Weil group A(K) is Zariski dense in A. This is an analogue a 1983 result of Faltings, considering instead the size of A(p)(k(p)).

Let E be an elliptic curve defined over a number field K and let S be a density-one set of primes of K of good reduction for E. Faltings proved in 1983 that the K-isogeny class of E is characterized by the function p bar right arrow #E(k(p)), which maps a prime p is an element of S to the order of the group of points of E over the corresponding field k(p). We show that, in this statement, the integer #E(k(p)) can be replaced by its radical. (c) 2013 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

The IEEE 802.16 (or) Worldwide Interoperability for Microwave Access (WiMAX) [7] is a standards-based technology enabling the delivery of last mile wireless broadband access with quality of service (QoS) guarantees, security, and mobility. For security in WiMAX the privacy sub layer of the MAC layer has the main objective to protect service providers against theft of service but not securing network users [12]. It is obvious that the privacy sub layer only secures data at the data link layer, but it does not ensure complete encryption of user data. To secure the networks, the IPSec protocol may be the most effective and suitable protocol to secure end-to-end network layer communication [2]. In the current scenario, existing method do not provide security level for end to end communications. The security algorithms namely TWOFISH [5] and BLOWFISH [26] are to encrypt the packets with best security levels for end to end communications. Analyses are done between these two algorithms with the existing encryption algorithms, from that TWOFISH is the best security algorithm and BLOWFISH is the fastest algorithm. General Terms Security, Algorithms, Throughput, Processing Time.

This paper presents empirical evidence supporting Goldfelds conjecture on the average analytic rank of a family of quadratic twists of a fixed elliptic curve in the function field setting. In particular, we consider representatives of the four classes of nonisogenous elliptic curves over with (q, 6)=1 possessing two places of multiplicative reduction and one place of additive reduction. The case of q=5 provides the largest data set as well as the most convincing evidence that the average analytic rank converges to 1/2, which we also show is a lower bound following an argument of Kowalski. The data were generated via explicit computation of the L-function of these elliptic curves, and we present the key results necessary to implement an algorithm to efficiently compute the L-function of nonisotrivial elliptic curves over by realizing such a curve as a quadratic twist of a pullback of a versal elliptic curve. We also provide a reference for our open-source library ELLFF, which provides all the necessary functionality to compute such L-functions, and additional data on analytic rank distributions as they pertain to the density conjecture.

We show that families of coverings of an algebraic curve where the associated Cayley-Schreier graphs form an expander family exhibit strong forms of geometric growth. We then give many arithmetic applications of this general result, obtained by combining it with finiteness statements for rational points of curves with large gonality. In particular we derive a number of results concerning the variation of Galois representations in one-parameter families of abelian varieties.

Let K be a number field and A/K be a polarized abelian variety with absolutely trivial endomorphism ring. We show that if the Neron model of A/K has at least one fiber with potential toric dimension 1, then, for almost all rational primes l, the Galois group of the splitting field of the l-torsion of A is GSp(2g)(Z/l).

Let K be a number field, and let E be an elliptic curve over K. A famous result by Faltings of 1983 can be reformulated for elliptic curves as follows: if S is a set of primes of good reduction for E having density one, then the K-isogeny class of E is determined by the function which maps a prime in S to the size of the group of points over the residue field. In this paper, we prove that it suffices to look at the radical of the size.

We solve the Hurwitz monodromy problem for degree 4 covers. That is, the Hurwitz space H4,g of all simply branched covers of P1 of degree 4 and genus g is an unramified cover of the space P2g+6 of (2g+6)-tuples of distinct points in P1. We determine the monodromy of π1(P2g+6) on the points of the fiber. This turns out to be the same problem as the action of π1(P2g+6) on a certain local system of Z/2-vector spaces. We generalize our result by treating the analogous local system with Z/N coefficients, 3∤N, in place of Z/2. This in turn allows us to answer a question of Ellenberg concerning families of Galois covers of P1 with deck group (Z/N)2:S3.

A social epidemiologic perspective considers factors at multiple levels of influence (e.g., social networks, neighbourhoods, states) that may individually or jointly affect health and health behaviour. This provides a useful lens through which to understand the production of health behaviours in general, and drug use in particular. However, the analytic models that are commonly applied in population health sciences limit the inference we are able to draw about the determination of health behaviour by factors, likely interrelated, across levels of influence. Complex system dynamic modelling techniques may be useful in enabling the adoption of a social epidemiologic approach in health behaviour and drug use research. We provide an example of a model that aims to incorporate factors at multiple levels of influence in understanding drug dependence. We conclude with suggestions about future directions in the field and how such models may serve as virtual laboratories for policy experiments aimed at improving health behaviour.

We consider, in the special case of certain one-parameter families of Jacobians of curves defined over a number field, the problem of how the property that the generic fiber of such a family is absolutely simple 'spreads' to other fibers. We show that this question can be approached using arithmetic geometry or with more analytic methods based on sieve theory. In the first setting, non-trivial group-theoretic information is needed, while the version of the sieve we use is also of independent interest.

Let k be a field not of characteristic two and L be a set of almost all rational primes invertible in k. Suppose we have a variety X/k and strictly compatible system {M_ell -> X : ell in L} of constructible F_ell-sheaves. If the system is orthogonally or symplectically self-dual, then the geometric monodromy group of M_ell is a subgroup of a corresponding isometry group G_ell over F_ell, and we say it has big monodromy if it contains the derived subgroup DG_ell=[G_ell,G_ell]. We prove a theorem which gives sufficient conditions for M_ell to have big monodromy. We apply the theorem to explicit systems arising from the middle cohomology of families of hyperelliptic curves and elliptic surfaces to show that the monodromy is uniformly big as we vary ell and the system.