Reciprocity Laws
نویسنده
چکیده
Informal notes for a talk at UNCG on reciprocity laws (such as quadratic reciprocity and Shimura-Taniyama). This material is from Fearless Symmetry by Ash and Gross [1] and Galois Representations and Modular Forms by Ribet [2].
منابع مشابه
Rational Quartic Reciprocity
In 1985, K. S. Williams, K. Hardy and C. Friesen [11] published a reciprocity formula that comprised all known rational quartic reciprocity laws. Their proof consisted in a long and complicated manipulation of Jacobi symbols and was subsequently simplified (and generalized) by R. Evans [3]. In this note we give a proof of their reciprocity law which is not only considerably shorter but also she...
متن کاملDerived Categories and the Analytic Approach to General Reciprocity Laws. Part I
We reformulate Hecke’s open problem of 1923, regarding the Fourier-analytic proof of higher reciprocity laws, as a theorem about morphisms involving stratified topological spaces. We achieve this by placing Kubota’s formulations of n-Hilbert reciprocity in a new topological context, suited to the introduction of derived categories of sheaf complexes. Subsequently, we begin to investigate condit...
متن کاملThe Elliptic Apostol-dedekind Sums Generate Odd Dedekind Symbols with Laurent Polynomial Reciprocity Laws
Abstract. Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a new elliptic analogue of the Apostol-Dedekind sums. Then we will show that the newly defined sums generate all odd Dedekind symbols with Laurent poly...
متن کاملGeneralized Reciprocity Laws for Sums of Harmonic Numbers
We present summation identities for generalized harmonic numbers, which generalize reciprocity laws discovered when studying the algorithm quickselect. Furthermore, we demonstrate how the computer algebra system Sigma can be used in order to find/prove such identities. We also discuss alternating harmonic sums, as well as limiting relations.
متن کاملA Categorical Proof of the Parshin Reciprocity Laws on Algebraic Surfaces
We define and study the 2-category of torsors over a Picard groupoid, a central extension of a group by a Picard groupoid, and commutator maps in this central extension. Using this in the context of two-dimensional local fields and two-dimensional adèle theory we obtain the two-dimensional tame symbol and a new proof of Parshin reciprocity laws on an algebraic surface.
متن کامل