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Michel Mendès France's “Folding Lemma” for continued fraction expansions provides an unusual explanation for the well known symmetry in the expansion of a quadratic irrational integer.
We examine iteration graphs of the squaring function on the rings when , for a Fermat prime. We describe several invariants associated to these graphs and use them to prove that the graphs are not symmetric when and when and are symmetric when .
We provide a formula for the symplectic period of an Eisenstein series on and determine when it is not identically zero.
We define Picard cycles on each smooth three-sheeted Galois
cover C of the Riemann sphere. The moduli space of all these algebraic
curves is a nice Shimura surface, namely a symmetric quotient of the projective
plane uniformized by the complex two-dimensional unit ball. We show that
all Picard cycles on C form a simple orbit of the Picard modular group
of Eisenstein numbers. The proof uses a special surface classification in
connection with the uniformization of a classical Picard-Fuchs system....
On définit la notion de système d’Euler associé à une représentation -adique du groupe de Galois absolu de dans le cas cyclotomique. Cette notion a été introduite par Kolyvagin. L’existence d’un tel système a des conséquences très importantes sur l’étude des groupes de Selmer de que nous développons ici.
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