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Special values of Hilbert modular functions.

Martin L. Karel (1986)

Revista Matemática Iberoamericana

Recently, Baily has established new foundation for complex multiplication in the context of Hilbert modular functions; see [1]-[4]. However, in his treatment there is a restriction on the class of CM-points treated. Namely, the order of complex multiplications associated to the point must be the maximal order in its quotient field. The purpose of this paper is two-fold: (1) to remove the restriction just mentioned; (2) to recover a result of Tate on the conjugates of CM-points by arbitrary Galois...

Specializations of one-parameter families of polynomials

Farshid Hajir, Siman Wong (2006)

Annales de l’institut Fourier

Let K be a number field, and suppose λ ( x , t ) ∈ K [ x , t ] is irreducible over K ( t ) . Using algebraic geometry and group theory, we describe conditions under which the K -exceptional set of λ , i.e. the set of α ∈ K for which the specialized polynomial λ ( x , α ) is K -reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed n ≥ 10 , all but finitely many K -specializations of the degree n generalized Laguerre polynomial L n ( t ) ( x ) are K -irreducible and have Galois group S n . Second, we study specializations...

The p -rank stratification of Artin-Schreier curves

Rachel Pries, Hui June Zhu (2012)

Annales de l’institut Fourier

We study a moduli space 𝒜𝒮 g for Artin-Schreier curves of genus g over an algebraically closed field k of characteristic p . We study the stratification of 𝒜𝒮 g by p -rank into strata 𝒜𝒮 g . s of Artin-Schreier curves of genus g with p -rank exactly s . We enumerate the irreducible components of 𝒜𝒮 g , s and find their dimensions. As an application, when p = 2 , we prove that every irreducible component of the moduli space of hyperelliptic k -curves with genus g and 2 -rank s has dimension g - 1 + s . We also determine all pairs ( p , g ) for...

Currently displaying 81 – 100 of 113