Book contents
- Frontmatter
- Contents
- Preface
- 1 Construction of Drinfeld modular varieties
- 2 Drinfeld A-modules with finite characteristic
- 3 The Lefschetz numbers of Hecke operators
- 4 The fundamental lemma
- 5 Very cuspidal Euler–Poincaré functions
- 6 The Lefschetz numbers as sums of global elliptic orbital integrals
- 7 Unramified principal series representations
- 8 Euler–Poincaré functions as pseudocoefficients of the Steinberg representation
- Appendices
- A Central simple algebras
- B Dieudonné's theory: some proofs
- C Combinatorial formulas
- D Representations of unimodular, locally compact, totally discontinuous, separated, topological groups
- References
- Index
C - Combinatorial formulas
from Appendices
Published online by Cambridge University Press: 03 May 2010
- Frontmatter
- Contents
- Preface
- 1 Construction of Drinfeld modular varieties
- 2 Drinfeld A-modules with finite characteristic
- 3 The Lefschetz numbers of Hecke operators
- 4 The fundamental lemma
- 5 Very cuspidal Euler–Poincaré functions
- 6 The Lefschetz numbers as sums of global elliptic orbital integrals
- 7 Unramified principal series representations
- 8 Euler–Poincaré functions as pseudocoefficients of the Steinberg representation
- Appendices
- A Central simple algebras
- B Dieudonné's theory: some proofs
- C Combinatorial formulas
- D Representations of unimodular, locally compact, totally discontinuous, separated, topological groups
- References
- Index
Summary
Introduction
In the proof of (4.6.1), we have made use of combinatorial formulas which involve the q-binomial coefficients. They are well known (see [MM] (Ch. V)) but for the convenience of the reader we will recall their proofs.
q-binomial coefficients
Let x and q be two independent indeterminates over ℤ. For any integersd, r, with d ≥ 0, the q-binomial coefficient is defined by
If d ≥ 1, we have
and
and it follows by induction on d that
for any integers d ≥ 0 and r.
LEMMA (C.1.1). — For any positive integer d, we have
in ℤ[x, q].
Proof : The formula is trivial if d = 1. Let us prove it in general by induction on d. If d ≥ 2, we have
by the induction hypothesis and the lemma follows.
Let us set
for any non-negative integer d and let us denote by the coefficient of xr in the expansion of Fd(x, q) as a polynomial in x with coefficients in ℤ[q] (r ∈ ℤ, r ≥ 0; if r ∈ ℤ, r < 0, we set.
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- Information
- Cohomology of Drinfeld Modular Varieties , pp. 281 - 283Publisher: Cambridge University PressPrint publication year: 1995