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Displaying similar documents to “Jacobians in isogeny classes of abelian surfaces over finite fields”

Torsion and Tamagawa numbers

Dino Lorenzini (2011)

Annales de l’institut Fourier

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Let K be a number field, and let A / K be an abelian variety. Let c denote the product of the Tamagawa numbers of A / K , and let A ( K ) tors denote the finite torsion subgroup of A ( K ) . The quotient c / | A ( K ) tors | is a factor appearing in the leading term of the L -function of A / K in the conjecture of Birch and Swinnerton-Dyer. We investigate in this article possible cancellations in this ratio. Precise results are obtained for elliptic curves over ℚ or quadratic extensions K / ℚ , and for abelian surfaces A / ℚ . The smallest possible...

Class Invariants for Quartic CM Fields

Eyal Z. Goren, Kristin E. Lauter (2007)

Annales de l’institut Fourier

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One can define class invariants for a quartic primitive CM field K as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to K . Such constructions were given by de Shalit-Goren and Lauter. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct S -units in certain abelian extensions of a reflex field of K , where S is effectively determined by K , and to bound the...

Jacobian Nullwerte, periods and symmetric equations for hyperelliptic curves

Jordi Guàrdia (2007)

Annales de l’institut Fourier

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We propose a solution to the hyperelliptic Schottky problem, based on the use of Jacobian Nullwerte and symmetric models for hyperelliptic curves. Both ingredients are interesting on its own, since the first provide period matrices which can be geometrically described, and the second have remarkable arithmetic properties.

Selmer groups for elliptic curves in ℤ l d -extensions of function fields of characteristic p

Andrea Bandini, Ignazio Longhi (2009)

Annales de l’institut Fourier

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Let F be a function field of characteristic p > 0 , ℱ / F a ℤ l d -extension (for some prime l ≠ p ) and E / F a non-isotrivial elliptic curve. We study the behaviour of the r -parts of the Selmer groups ( r any prime) in the subextensions of ℱ via appropriate versions of Mazur’s Control Theorem. As a consequence we prove that the limit of the Selmer groups is a cofinitely generated (in some cases cotorsion) module over the Iwasawa algebra of ℱ / F .