Part 1 Algebraic methods: finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratic forms with discriminant +-1. Part 2 Analytic methods: the theorem on arithmetic progressions modular forms.

Some aspects of light-like compactifications of superstring theory and their implications for the matrix model of M-theory are discussed

background

A Course in Arithmetic

Worldsheet of the discrete light front string

The first symmetry by base substitutions of degeneracy in the genetic code was described by Rumer (1966) and the other symmetries were identified later by Jestin (2006) and Jestin and Soule (2007). Here, a rationale accounting for these symmetries is reported. The number of non-synonymous substitutions over the replicated coding sequence is written as a function of the substitution matrix, whose elements are the number of substitutions from any codon to any other codon. The p-adic distance used as a similarity measure and applied to this matrix is shown to be biologically relevant. The rationale indicates that symmetries by base substitutions of degeneracy in the genetic code are symmetries of the measures of the number of non-synonymous substitutions for sets of synonymous codons.

A rationale for the symmetries by base substitutions of degeneracy in the genetic code

It was recently shown by Aka, Einsiedler and Shapira that if . It also follows that equidistribution takes place with a polynomial error term with respect to the length of the primitive points.

Distribution of shapes of orthogonal lattices

We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of $GL(2,\mathbb{Q}_p)$ for automorphic forms belonging to the principal series. We interpret this conclusion in terms of the equidistribution of eigenfunctions on covers of a fixed regular graph or along nested sequences of regular graphs. Our results are the first of their kind on any p-adic arithmetic quotient. They may be understood as analogues of Lindenstrauss's theorem on the equidistribution of Maass forms on a compact arithmetic surface. The new ingredients here include the introduction of a representation-theoretic notion of "p-adic microlocal lifts" with favorable properties, such as diagonal invariance of limit measures, the proof of positive entropy of limit measures in a p-adic aspect, following the method of Bourgain--Lindenstrauss, and some analysis of local Rankin--Selberg integrals involving the microlocal lifts introduced here as well as classical newvectors. An important input is a measure-classification result of Einsiedler--Lindenstrauss.

A Course in Arithmetic

Citations

Worldsheet of the discrete light front string

A rationale for the symmetries by base substitutions of degeneracy in the genetic code

Distribution of shapes of orthogonal lattices

Microlocal lifts and quantum unique ergodicity on $GL(2,\mathbb{Q}_p)$

Chapter 1 Elliptic Functions

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