Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On rank $2$ hypergeometric motives

Franco Golfieri Madriaga, Ariel Pacetti, Fernando Rodriguez Villegas

Abstract

Hypergeometric motives are family of motives associated to hypergeometric local systems. Their special features, in particular their rigidity, makes them more tractable than general motives. In the present article we prove most of the properties that they are expected to satisfy in the rank $2$ case.

On rank $2$ hypergeometric motives

Abstract

Hypergeometric motives are family of motives associated to hypergeometric local systems. Their special features, in particular their rigidity, makes them more tractable than general motives. In the present article we prove most of the properties that they are expected to satisfy in the rank case.
Paper Structure (27 sections, 58 theorems, 259 equations, 9 figures, 2 tables)

This paper contains 27 sections, 58 theorems, 259 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Rank one hypergeometric motives
  3. Hypergeometric Series and Monodromy Representation
  4. The monodromy representation of the differential equation
  5. Geometric representations
  6. Generalities on hypergeometric motives
  7. Hodge numbers and normalization
  8. Jacobi motives
  9. Finite Hypergeometric Sums
  10. Properties of $H_q(\pmb \alpha,\pmb \beta|z)$
  11. Relation with the $p$-adic Gamma function
  12. Hypergeometric character sums
  13. Zeta function of Superelliptic curves
  14. The new part
  15. Rank two Hypergeometric Motives
  16. ...and 12 more sections

Key Result

Theorem 1

Let ${\mathfrak{p}}$ be a prime ideal of ${\mathbb Q}(\zeta_N)$ satisfying prop-2. Then ${{{\mathcal{H}}({\pmb \alpha},{\pmb \beta}{\,|\,}{z})}}$ has good reduction at ${\mathfrak{p}}$ and the trace of the Frobenius automorphism $\mathop{\mathrm{Frob}}\nolimits_{{\mathfrak{p}}}$ acting on ${{{\mathc

Figures (9)

  • Figure 1: Fundamental group of $\pi_1(\mathbb P^1(\mathbb C)\setminus \{0,1,\infty\},z_0)$
  • Figure 2: The Hodge numbers of $(\frac{1}{8},\frac{7}{8}),(\frac{3}{8},\frac{5}{8})$ and $(\frac{3}{8},\frac{5}{8}),(\frac{1}{8},\frac{7}{8})$
  • Figure 3: The Hodge numbers of $(\frac{1}{2},\frac{1}{2}),(0,\frac{1}{4})$ and $(\frac{1}{2},\frac{1}{2}),(0,\frac{3}{4})$
  • Figure 4: Zigzag outcome when $a,b \in \mathbb Z$ or $c,d\in \mathbb Z$
  • Figure 5: The zigzag procedure in case $(1)$ for $a<c$ and $a>c$
  • ...and 4 more figures

Theorems & Definitions (182)

  • Theorem 1
  • Definition 1.1
  • Theorem 2
  • Theorem 3
  • Example 1
  • Example 2
  • Example 3
  • Remark
  • Definition 3.1
  • Remark 3.2
  • ...and 172 more