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The fluctuations in the number of points on a family of curves over a finite field

Maosheng Xiong (2010)

Journal de Théorie des Nombres de Bordeaux

Let l 2 be a positive integer, 𝔽 q a finite field of cardinality q with q 1 ( mod l ) . In this paper, inspired by [6, 3, 4] and using a slightly different method, we study the fluctuations in the number of 𝔽 q -points on the curve F given by the affine model F : Y l = F ( X ) , where F is drawn at random uniformly from the set of all monic l -th power-free polynomials F 𝔽 q [ X ] of degree d as d . The method also enables us to study the fluctuations in the number of 𝔽 q -points on the same family of curves arising from the set of monic irreducible...

The formal completion of the Néron model of J0(p).

Enric Nart (1991)

Publicacions Matemàtiques

For any prime number p > 3 we compute the formal completion of the Néron model of J0(p) in terms of the action of the Hecke algebra on the Z-module of all cusp forms (of weight 2 with respect to Γ0(p)) with integral Fourier development at infinity.

The fractional dimensional theory in Lüroth expansion

Luming Shen, Kui Fang (2011)

Czechoslovak Mathematical Journal

It is well known that every x ( 0 , 1 ] can be expanded to an infinite Lüroth series in the form of x = 1 d 1 ( x ) + + 1 d 1 ( x ) ( d 1 ( x ) - 1 ) d n - 1 ( x ) ( d n - 1 ( x ) - 1 ) d n ( x ) + , where d n ( x ) 2 for all n 1 . In this paper, sets of points with some restrictions on the digits in Lüroth series expansions are considered. Mainly, the Hausdorff dimensions of the Cantor sets F φ = { x ( 0 , 1 ] : d n ( x ) φ ( n ) , n 1 } are completely determined, where φ is an integer-valued function defined on , and φ ( n ) as n .

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