Pure Anderson Motives and Abelian \tau-Sheaves

Pure t-motives were introduced by G. Anderson as higher dimensional generalizations of Drinfeld modules, and as the appropriate analogs of abelian varieties in the arithmetic of function fields. In order to construct moduli spaces for pure t-motives the second author has previously introduced the concept of abelian \tau-sheaf. In this article we clarify the relation between pure t-motives and abelian \tau-sheaves. We obtain an equivalence of the respective quasi-isogeny categories. Furthermore, we develop the elementary theory of both structures regarding morphisms, isogenies, Tate modules, and local shtukas. The later are the analogs of p-divisible groups.Comment: final version as it appears in Mathematische Zeitschrif

Extention of Finite Solvable Torsors over a Curve

Let $R$ be a discrete valuation ring with fraction field $K$ and with algebraically closed residue field of positive characteristic $p$. Let $X$ be a smooth fibered surface over $R$ with geometrically connected fibers endowed with a section $x\in X(R)$. Let $G$ be a finite solvable $K$-group scheme and assume that either $|G|=p^n$ or $G$ has a normal series of length 2. We prove that every quotient pointed $G$-torsor over the generic fiber $X_{\eta}$ of $X$ can be extended to a torsor over $X$ after eventually extending scalars and after eventually blowing up $X$ at a closed subscheme of its special fiber $X_s$.Comment: 16 page

Computing Hilbert Class Polynomials

We present and analyze two algorithms for computing the Hilbert class polynomial $H_D$ . The first is a p-adic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm which uses the class group action on CM-curves over finite fields. Our run time analysis gives tighter bounds for the complexity of all known algorithms for computing $H_D$, and we show that all methods have comparable run times

Cubic Curves, Finite Geometry and Cryptography

Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational points are also surveyed. A possible strengthening of the security of elliptic curve cryptography is proposed using a `shared secret' related to the group law. Cubic curves are also used in a new way to construct sets of points having various combinatorial and geometric properties that are of particular interest in finite Desarguesian planes.Comment: This is a version of our article to appear in Acta Applicandae Mathematicae. In this version, we have corrected a sentence in the third paragraph. The final publication is available at springerlink.com at http://www.springerlink.com/content/xh85647871215644

L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields

The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to high order at the center point of their functional equation. The main application is that for every prime p and every integer g>0 there are absolutely simple abelian varieties of dimension g over Fp(t) for which the BSD conjecture holds and which have arbitrarily large rank.Comment: To appear in Inventiones Mathematica

Abelian Varieties with Prescribed Embedding Degree

We present an algorithm that, on input of a CM-field $K$, an integer $k\ge1$, and a prime $r \equiv 1 \bmod k$, constructs a $q$-Weil number \pi \in \O_K corresponding to an ordinary, simple abelian variety $A$ over the field \F of $q$ elements that has an \F-rational point of order $r$ and embedding degree $k$ with respect to $r$. We then discuss how CM-methods over $K$ can be used to explicitly construct $A$.Comment: to appear in ANTS-VII

Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms

We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.Comment: 39 page

The genome of the stable fly, Stomoxys calcitrans, reveals potential mechanisms underlying reproduction, host interactions, and novel targets for pest control.

The stable fly, Stomoxys calcitrans, is a major blood-feeding pest of livestock that has near worldwide distribution, causing an annual cost of over $2 billion for control and product loss in the USA alone. Control of these flies has been limited to increased sanitary management practices and insecticide application for suppressing larval stages. Few genetic and molecular resources are available to help in developing novel methods for controlling stable flies. This study examines stable fly biology by utilizing a combination of high-quality genome sequencing and RNA-Seq analyses targeting multiple developmental stages and tissues. In conjunction, 1600 genes were manually curated to characterize genetic features related to stable fly reproduction, vector host interactions, host-microbe dynamics, and putative targets for control. Most notable was characterization of genes associated with reproduction and identification of expanded gene families with functional associations to vision, chemosensation, immunity, and metabolic detoxification pathways. The combined sequencing, assembly, and curation of the male stable fly genome followed by RNA-Seq and downstream analyses provide insights necessary to understand the biology of this important pest. These resources and new data will provide the groundwork for expanding the tools available to control stable fly infestations. The close relationship of Stomoxys to other blood-feeding (horn flies and Glossina) and non-blood-feeding flies (house flies, medflies, Drosophila) will facilitate understanding of the evolutionary processes associated with development of blood feeding among the Cyclorrhapha

Genomic analysis of two phlebotomine sand fly vectors of Leishmania from the New and Old World.

Phlebotomine sand flies are of global significance as important vectors of human disease, transmitting bacterial, viral, and protozoan pathogens, including the kinetoplastid parasites of the genus Leishmania, the causative agents of devastating diseases collectively termed leishmaniasis. More than 40 pathogenic Leishmania species are transmitted to humans by approximately 35 sand fly species in 98 countries with hundreds of millions of people at risk around the world. No approved efficacious vaccine exists for leishmaniasis and available therapeutic drugs are either toxic and/or expensive, or the parasites are becoming resistant to the more recently developed drugs. Therefore, sand fly and/or reservoir control are currently the most effective strategies to break transmission. To better understand the biology of sand flies, including the mechanisms involved in their vectorial capacity, insecticide resistance, and population structures we sequenced the genomes of two geographically widespread and important sand fly vector species: Phlebotomus papatasi, a vector of Leishmania parasites that cause cutaneous leishmaniasis, (distributed in Europe, the Middle East and North Africa) and Lutzomyia longipalpis, a vector of Leishmania parasites that cause visceral leishmaniasis (distributed across Central and South America). We categorized and curated genes involved in processes important to their roles as disease vectors, including chemosensation, blood feeding, circadian rhythm, immunity, and detoxification, as well as mobile genetic elements. We also defined gene orthology and observed micro-synteny among the genomes. Finally, we present the genetic diversity and population structure of these species in their respective geographical areas. These genomes will be a foundation on which to base future efforts to prevent vector-borne transmission of Leishmania parasites