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Representations of $\mathrm{GL}_2$ over $\mathbb{Z}/p^n\mathbb{Z}$ and supercongruences for hypergeometric polynomials

Atsushi Ichino, Kartik Prasanna

Abstract

For an odd prime $p$, we realize the trivial representation of $\mathrm{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$ on the free $\mathbb{Z}/p^n \mathbb{Z}$-module of rank one as a subquotient of a direct sum of symmetric power representations (twisted by appropriate powers of the determinant) of rank strictly greater than one. The proof eventually reduces to establishing some novel supercongruences for hypergeometric polynomials.

Representations of $\mathrm{GL}_2$ over $\mathbb{Z}/p^n\mathbb{Z}$ and supercongruences for hypergeometric polynomials

Abstract

For an odd prime , we realize the trivial representation of on the free -module of rank one as a subquotient of a direct sum of symmetric power representations (twisted by appropriate powers of the determinant) of rank strictly greater than one. The proof eventually reduces to establishing some novel supercongruences for hypergeometric polynomials.

Paper Structure

This paper contains 14 sections, 15 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Motivation and main result
  3. Sketch of the proof
  4. Representations of $\mathrm{GL}_2$ over $\mathbb{Z}/p^n\mathbb{Z}$
  5. Symmetric and divided powers
  6. Another realization
  7. Congruences for hypergeometric polynomials
  8. Reduction to congruences for hypergeometric polynomials
  9. Reduction to congruences for binomial coefficients
  10. Congruences for binomial coefficients
  11. Generalized congruences
  12. A result of Granville
  13. Further generalizations
  14. Proof of Theorem \ref{['t:gen2']}

Key Result

Theorem 1

Assume that $p \ne 2$. For $n=2$ (so that $R = \mathbb{Z}/p^2 \mathbb{Z}$), the trivial representation of $\mathrm{GL}_2(R)$ on the free $R$-module of rank one appears in as a subquotient, where each direct summand is twisted by an appropriate power of the determinant.

Theorems & Definitions (23)

  • Theorem
  • Remark
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • ...and 13 more