Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mixed motives and linear forms in the Catalan constant

Payman Eskandari, Kumar Murty, Yusuke Nemoto

Abstract

We first give a geometric construction of a 2-dimensional mixed motive over $\mathbb{Q}$ with the Catalan constant $\mathbf{G}=1-1/3^2+1/5^2-1/7^2+\cdots$ as a period. We then use this motive to obtain a supply of linear forms in 1 and $\mathbf{G}$. We also explicitly compute the coefficients of 1 and $\mathbf{G}$ in these linear forms.

Mixed motives and linear forms in the Catalan constant

Abstract

We first give a geometric construction of a 2-dimensional mixed motive over with the Catalan constant as a period. We then use this motive to obtain a supply of linear forms in 1 and . We also explicitly compute the coefficients of 1 and in these linear forms.
Paper Structure (27 sections, 31 theorems, 178 equations, 2 figures)

This paper contains 27 sections, 31 theorems, 178 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation and context
  3. Contents of the paper
  4. The motivic setting and notation
  5. Construction of the motive $M$
  6. The weight filtration of $M$
  7. The de Rham and Betti realizations
  8. The Betti realization
  9. The de Rham realization
  10. The period matrix of $M$
  11. The motive of the Catalan constant
  12. Linear forms in 1 and ${\mathbf{G}}$ - I: The simple pole case
  13. Overview
  14. Symmetric differential forms with logarithmic singularity
  15. Further comments on symmetric differential forms with logarithmic singularity
  16. ...and 12 more sections

Key Result

Theorem 1.1.1

Let $t$ be an integer $\geq 0$. Suppose $F=F(x,y)\in {\mathbb Q}[x,y]$ satisfies conditions (i) and (ii) below: Then the integral of ${\frac{F dxdy}{(1-x^2-y^2)^{t+1}}}$ over the simplex converges to a ${\mathbb Q}$-linear combination of $1$ and ${\mathbf{G}}$.

Figures (2)

  • Figure 1:
  • Figure 2: Boundary of $\partial{\widetilde{\Delta}}$

Theorems & Definitions (57)

  • Theorem 1.1.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Lemma 4.1
  • proof
  • ...and 47 more