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Structure of cubic mapping graphs for the ring of Gaussian integers modulo n

Yangjiang Wei, Jizhu Nan, Gaohua Tang (2012)

Czechoslovak Mathematical Journal

Let ℤ n [ i ] be the ring of Gaussian integers modulo n . We construct for ℤ n [ i ] a cubic mapping graph Γ ( n ) whose vertex set is all the elements of ℤ n [ i ] and for which there is a directed edge from a ∈ ℤ n [ i ] to b ∈ ℤ n [ i ] if b = a 3 . This article investigates in detail the structure of Γ ( n ) . We give suffcient and necessary conditions for the existence of cycles with length t . The number of t -cycles in Γ 1 ( n ) is obtained and we also examine when a vertex lies on a t -cycle of Γ 2 ( n ) , where Γ 1 ( n ) is induced by all the units of ℤ n [ i ] while Γ 2 ( n ) is induced by all the...

Structure of three interval exchange transformations I: an arithmetic study

Sébastien Ferenczi, Charles Holton, Luca Q. Zamboni (2001)

Annales de l’institut Fourier

In this paper we describe a 2 -dimensional generalization of the Euclidean algorithm which stems from the dynamics of 3 -interval exchange transformations. We investigate various diophantine properties of the algorithm including the quality of simultaneous approximations. We show it verifies the following Lagrange type theorem: the algorithm is eventually periodic if and only if the parameters lie in the same quadratic extension of ℚ .

Sturm type theorem for Siegel modular forms of genus 2 modulo p

Dohoon Choi, YoungJu Choie, Toshiyuki Kikuta (2013)

Acta Arithmetica

Suppose that f is an elliptic modular form with integral coefficients. Sturm obtained bounds for a nonnegative integer n such that every Fourier coefficient of f vanishes modulo a prime p if the first n Fourier coefficients of f are zero modulo p. In the present note, we study analogues of Sturm's bounds for Siegel modular forms of genus 2. As an application, we study congruences involving an analogue of Atkin's U(p)-operator for the Fourier coefficients of Siegel modular forms of genus 2.

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