Abstract and Explicit Constructions of Jacobian Varieties

Abelian varieties, in particular Jacobian varieties, have long attracted interest in mathematics. Their influence pervades arithmetic geometry and number theory, and understanding their construction was a primary motivator for Weil in his work on developing new foundations for algebraic geometry in the 1930s and 1940s. Today, these exotic mathematical objects find applications in cryptography and computer science, where they can be used to secure confidential communications and factor integers in subexponential time. Although in many respects well-studied, working in concrete, explicit ways with abelian varieties continues to be difficult. The issue is that, aside from the case of elliptic curves, it is often difficult to find ways of modelling and understanding these objects in ways amenable to computation. Often, the approach taken is to work ``indirectly'' with abelian varieties, in particular with Jacobians, by working instead with divisors on their associated curves to simplify computations. However, properly understanding the mathematics underlying the direct approach --- why, for instance, one can view the degree zero divisor classes on a curve as being points of a variety --- requires sophisticated mathematics beyond what is usually understood by algorithms designers and even experts in computational number theory. A direct approach, where explicit polynomial and rational functions are given that define both the abelian variety and its group law, cannot be found in the literature for dimensions greater than two. In this thesis, we make two principal contributions. In the first, we survey the mathematics necessary to understand the construction of the Jacobian of a smooth algebraic curve as a group variety. In the second, we present original work with gives the first instance of explicit rational functions defining the group law of an abelian variety of dimension greater than two. In particular, we derive explicit formulas for the group addition on the Jacobians of hyperelliptic curves of every genus g, and so give examples of explicit rational formulas for the group law in every positive dimension

Deutsche Grammatik : Regeln, Normen, Sprachgebrauch : Bericht von der 44. Jahrestagung des Instituts für Deutsche Sprache

Sie ist schon ein erstaunliches Phänomen, die Sprache, bedenkt man, dass es auch ohne Einfluss einer steuernden Instanz so etwas wie Standarddeutsch gibt und die deutsche Sprache nicht in unzählige Variationen und Varietäten auseinanderdriftet. Die Verwunderung über den Zusammenhalt der Sprache ließ sich auch im Laufe der diesjährigen Jahrestagung des IDS immer wieder vernehmen, die unter dem Motto „Deutsche Grammatik. Regeln, Normen, Sprachgebrauch“ vom 11. bis 13. März 2008 im neugestalteten Rosengarten in Mannheim stattfand. Da man auf einer wissenschaftlichen Tagung beim Wundern nicht stehen bleibt, versuchten die versammelten Linguistinnen und Linguisten, der Natur von sprachlichen Regeln und Normen erklärend auf die Spur zu kommen. Wie entstehen sprachliche Normen? Welche Faktoren entscheiden, dass manche der neuen grammatischen Formen sich durchsetzen und zur Norm werden und andere nicht? Welche Bedeutung hat Sprachnormierung in verschiedenen gesellschaftlichen Bereichen wie Schule, Wirtschaft oder Recht? Und nicht zuletzt: Wie kann das grammatische Regelsystem erfasst werden

Absolute Hodge and ℓ\ell-adic Monodromy

Let $\mathbb{V}$ be a motivic variation of Hodge structure on a $K$-variety $S$, let $\mathcal{H}$ be the associated $K$-algebraic Hodge bundle, and let $\sigma \in \textrm{Aut}(\mathbb{C}/K)$ be an automorphism. The absolute Hodge conjecture predicts that given a Hodge vector $v \in \mathcal{H}_{\mathbb{C}, s}$ above $s \in S(\mathbb{C})$ which lies inside $\mathbb{V}_{s}$, the conjugate vector $v_{\sigma} \in \mathcal{H}_{\mathbb{C}, s_{\sigma}}$ is Hodge and lies inside $\mathbb{V}_{s_{\sigma}}$. We study this problem in the situation where we have an algebraic subvariety $Z \subset S_{\mathbb{C}}$ containing $s$ whose algebraic monodromy group $\mathbf{H}_Z$ fixes $v$. Using relationships between $\mathbf{H}_Z$ and $\mathbf{H}_{Z_{\sigma}}$ coming from the theories of complex and $\ell$-adic local systems, we establish a criterion that implies the absolute Hodge conjecture for $v$ subject to a group-theoretic condition on $\mathbf{H}_{Z}$. We then use our criterion to establish new cases of the absolute Hodge conjecture.Comment: Comments welcome

Weathering the Storm: Effects of the National Recession and Statewide Property Tax Caps in Northwest Indiana

This paper outlines the economic health of Northwest Indiana communities following the latest national recession and the passage of statewide property tax reforms in 2008. This paper identifies the communities with the highest concentrations of economically distressed residents as measured by poverty, unemployment, and participation in the free- and reduced-lunch program during the time period from 2008 to 2012. These communities historically have had the highest property tax rates in the region. In the past, these high tax rates may have served as a disincentive for residential and business investment, but now, with the passage of statewide tax restructuring, the high rates have resulted in a new type of disparity in the form of significant funding losses for local government. For purposes of this paper, Northwest Indiana is defined as consisting of Lake, Porter, and LaPorte counties