Displaying similar documents to “Fields of definition of ℚ -curves”

Maximal unramified extensions of imaginary quadratic number fields of small conductors, II

Ken Yamamura (2001)

Journal de théorie des nombres de Bordeaux

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In the previous paper [15], we determined the structure of the Galois groups Gal ( K u r / K ) of the maximal unramified extensions K u r of imaginary quadratic number fields K of conductors ≦ 1000 under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors ≧ 723 ) and give a table of Gal ( K u r / K ) . We update the table (under GRH). For 19 exceptional fields K of them, we determine Gal ( K u r / K ) . In particular, for K = 𝐐 ( - 856 ) , we obtain Gal ( K u r / K ) ≅ S 4 ˜ × C 5 and K u r = K 4 , the fourth Hilbert class field of K . This is the first example of a number...

Maximal unramified extensions of imaginary quadratic number fields of small conductors

Ken Yamamura (1997)

Journal de théorie des nombres de Bordeaux

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We determine the structures of the Galois groups Gal ( K u r / K ) of the maximal unramified extensions K u r of imaginary quadratic number fields K of conductors ≦ 420 ( ≦ 719 under the Generalized Riemann Hypothesis). For all such K , K u r is K , the Hilbert class field of K , the second Hilbert class field of K , or the third Hilbert class field of K . The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We...

On the smallest degree of a surface containing a space curve

Margherita Roggero, Paolo Valabrega (1998)

Bollettino dell'Unione Matematica Italiana

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Sia C una curva dello spazio di grado D contenuta in una superficie di grado r e non in una di grado r - 1 . Se C è integra, allora r ≤ 6 ⁢ D - 2 - 2 ; questo limite superiore, raggiunto in alcuni casi (cfr. [5]), non vale però per curve arbitrarie (cfr. [?, 3 (iii)]). Ogni curva C dello spazio (anche non ridotta o riducibile) può essere ottenuta come schema degli zero di una sezione non nulla di un opportuno fascio riflessivo F di rango 2. Mediante i fasci riflessivi, siamo in grado di estendere alle curve...

On equations defining fake elliptic curves

Pilar Bayer, Jordi Guàrdia (2005)

Journal de Théorie des Nombres de Bordeaux

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Shimura curves associated to rational nonsplit quaternion algebras are coarse moduli spaces for principally polarized abelian surfaces endowed with quaternionic multiplication. These objects are also known as . We present a method for computing equations for genus 2 curves whose Jacobian is a fake elliptic curve with complex multiplication. The method is based on the explicit knowledge of the normalized period matrices and on the use of theta functions with characteristics. As in the...

Improved upper bounds for the number of points on curves over finite fields

Everett W. Howe, Kristin E. Lauter (2003)

Annales de l’institut Fourier

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We give new arguments that improve the known upper bounds on the maximal number N q ( g ) of rational points of a curve of genus g over a finite field 𝔽 q , for a number of pairs ( q , g ) . Given a pair ( q , g ) and an integer N , we determine the possible zeta functions of genus- g curves over 𝔽 q with N points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus- g curve over 𝔽 q with N points must have a low-degree map to another curve over 𝔽 q , and often this...