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Register a new userA General Reciprocity Law on arbitrary Vector Spaces
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Authors
Fernando Pablos Romo
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The paper contained a preliminary version of a general theory of reciprocity laws on vector spaces.
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The aim of this article is to establish the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain R that is module-finite over $O[[x_1, ..., x_d]]$, where $O$ is the ring of integers of a finite extension of the field of p-adic integers $Q_p$. The specialization method is a technique that recovers the information on the characteristic ideal $char_R(M)$ from $char_{R/I}(M/IM)$, where I varies in a certain family of nonzero principal ideals of R. As applications, we prove Euler system bound over Cohen-Macaulay normal domains by combining the main results in an earlier article of the first named author and then we prove one of divisibilities of the Iwasawa main conjecture for two-variable Hida deformations generalizing the main theorem obtained in an article of the first named author.
Let $p$ be a prime. Let $R$ be a regular local ring of dimension $dge 2$ whose completion is isomorphic to $C(k)[[x_1,ldots,x_d]]/(h)$, with $C(k)$ a Cohen ring with the same residue field $k$ as $R$ and with $hin C(k)[[x_1,ldots,x_d]]$ such that its reduction modulo $p$ does not belong to the ideal $(x_1^p,ldots,x_d^p)+(x_1,ldots,x_d)^{2p-2}$ of $k[[x_1,ldots,x_d]]$. We extend a result of Vasiu-Zink (for $d=2$) to show that each Barsotti-Tate group over $text{Frac}(R)$ which extends to every local ring of $text{Spec}(R)$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $R$. This result corrects in many cases several errors in the literature. As an application, we get that if $Y$ is a regular integral scheme such that the completion of each local ring of $Y$ of residue characteristic $p$ is a formal power series ring over some complete discrete valuation ring of absolute ramification index $ele p-1$, then each Barsotti-Tate group over the generic point of $Y$ which extends to every local ring of $Y$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $Y$.
We study conditional independence (CI) models in statistical theory, in the case of discrete random variables, from the point of view of algebraic geometry and matroid theory. Any CI model with hidden random variables corresponds to a variety defined by certain determinantal conditions on a matrix whose entries are probabilities of events involving the observed random variables. We show that any CI variety, and more generally any hypergraph variety, admits a matroid stratification. Our main motivation for studying decompositions of CI varieties is the realizability problem: given a collection of CI relations, the goal is to determine the existence of random variables that satisfy these constraints and violates the rest. We show that the realization spaces of CI models and the matroid varieties in their decompositions are closely related. We use ideas from incidence geometry, in particular point and line configurations, to find minimal decompositions of general hypergraph varieties in terms of matroid varieties, which are not necessarily irreducible by Mnev--Sturmfels universality theorem, and may have arbitrary singularities. We focus on various families of hypergraph varieties for which we explicitly compute an irredundant irreducible decomposition. Our main findings in this direction are threefold: (1) we describe minimal matroids of such hypergraphs; (2) we prove that the varieties of these matroids are irreducible and their union is the hypergraph variety; and (3) we show that every such matroid is realizable over real numbers. Our decomposition strategy gives immediate matroid interpretations of the irreducible components of many families of CI varieties in algebraic statistics, and unravels the symmetric structures in CI varieties which hugely simplifies the computations.
This is an English translation of Felix Kleins classical paper Uber die Auflosung der allgemeinen Gleichungen funften und sechsten Grades (Auszug aus einem Schreiben an Herrn K. Hensel) from 1905 and is put in modern notation. The original work first appeared in the Journal for Pure and Applied Mathematics (Volume 129) and then was reprinted in Mathematische Annalen (Volume 61, Issue 1). Kleins work (including his Lectures on the Icosahedron and the Solution of Equations of Fifth Degree) lies at the heart of the 19th and 20th work on solving generic polynomials. In this paper, Klein summarizes his approach to solving the generic quintic and sextic polynomials. He also lays the foundation for the modern framework of resolvent degree.
We prove that the central values of additive twists of a cuspidal $L$-function define a quantum modular form in the sense of Zagier, generalizing recent results of Bettin and Drappeau. From this we deduce a reciprocity law for the twisted first moment of multiplicative twists of cuspidal $L$-functions, similar to reciprocity laws discovered by Conrey for the twisted second moment of Dirichlet $L$-functions. Furthermore we give an interpretation of quantum modularity at infinity for additive twists of $L$-functions of weight 2 cusp forms in terms of the corresponding functional equations.
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