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Rademacher-Carlitz polynomials

Matthias Beck, Florian Kohl (2014)

Acta Arithmetica

We introduce and study the Rademacher-Carlitz polynomial R ( u , v , s , t , a , b ) : = ∑ k = ⌈ s ⌉ ⌈ s ⌉ + b - 1 u ⌊ ( k a + t ) ⌋ / b v k where a , b ∈ ℤ > 0 , s,t ∈ ℝ, and u and v are variables. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view R(u,v,s,t,a,b) as a polynomial analogue (in the sense of Carlitz) of the Dedekind-Rademacher sum r t ( a , b ) : = ∑ k = 0 b - 1 ( ( ( k a + t ) / b ) ) ( ( k / b ) ) , which appears in various number-theoretic, combinatorial, geometric, and computational contexts. Our results come in three flavors: we prove a reciprocity theorem for Rademacher-Carlitz...

Ramification and moduli spaces of finite flat models

Naoki Imai (2011)

Annales de l’institut Fourier

We determine the type of the zeta functions and the range of the dimensions of the moduli spaces of finite flat models of two-dimensional local Galois representations over finite fields. This gives a generalization of Raynaud’s theorem on the uniqueness of finite flat models in low ramifications.

Rankin–Cohen brackets and representations of conformal Lie groups

Michael Pevzner (2012)

Annales mathématiques Blaise Pascal

This is an extended version of a lecture given by the author at the summer school “Quasimodular forms and applications” held in Besse in June 2010.The main purpose of this work is to present Rankin-Cohen brackets through the theory of unitary representations of conformal Lie groups and explain recent results on their analogues for Lie groups of higher rank. Various identities verified by such covariant bi-differential operators will be explained by the associativity of a non-commutative product...

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