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Ideal arithmetic and infrastructure in purely cubic function fields

Renate Scheidler (2001)

Journal de théorie des nombres de Bordeaux

This paper investigates the arithmetic of fractional ideals of a purely cubic function field and the infrastructure of the principal ideal class when the field has unit rank one. First, we describe how irreducible polynomials decompose into prime ideals in the maximal order of the field. We go on to compute so-called canonical bases of ideals; such bases are very suitable for computation. We state algorithms for ideal multiplication and, in the case of unit rank one and characteristic at least five,...

Improvement of Grauert-Riemenschneider's theorem for a normal surface

Jean Giraud (1982)

Annales de l'institut Fourier

Let X ˜ be a desingularization of a normal surface X . The group Pic ( X ˜ ) is provided with an order relation L ≫ _ 0 , defined by L . V ≤ 0 for any effective exceptional divisor V . Comparing to the usual order relation we define the ceiling of L which is an exceptional divisor. This notion allows us to improve the usual vanishing theorem and we deduce from it a numerical criterion for rationality and a genus formula for a curve on a normal surface; the difficulty lies in the case of a Weil divisor which is not a Cartier...

Indecomposable parabolic bundles

William Crawley-Boevey (2004)

Publications Mathématiques de l'IHÉS

We study the possible dimension vectors of indecomposable parabolic bundles on the projective line, and use our answer to solve the problem of characterizing those collections of conjugacy classes of n×n matrices for which one can find matrices in their closures whose product is equal to the identity matrix. Both answers depend on the root system of a Kac-Moody Lie algebra. Our proofs use Ringel’s theory of tubular algebras, work of Mihai on the existence of logarithmic connections, the Riemann-Hilbert...

Infinite rank of elliptic curves over ℚ a b

Bo-Hae Im, Michael Larsen (2013)

Acta Arithmetica

If E is an elliptic curve defined over a quadratic field K, and the j-invariant of E is not 0 or 1728, then E ( ℚ a b ) has infinite rank. If E is an elliptic curve in Legendre form, y² = x(x-1)(x-λ), where ℚ(λ) is a cubic field, then E ( K ℚ a b ) has infinite rank. If λ ∈ K has a minimal polynomial P(x) of degree 4 and v² = P(u) is an elliptic curve of positive rank over ℚ, we prove that y² = x(x-1)(x-λ) has infinite rank over K ℚ a b .

Injective endomorphisms of algebraic and analytic sets

Sławomir Cynk, Kamil Rusek (1991)

Annales Polonici Mathematici

We prove that every injective endomorphism of an affine algebraic variety over an algebraically closed field of characteristic zero is an automorphism. We also construct an analytic curve in ℂ⁶ and its holomorphic bijection which is not a biholomorphism.

Integrable systems and moduli spaces of rank two vector bundles on a non-hyperelliptic genus 3 curve

Pol Vanhaecke (2005)

Annales de l’institut Fourier

We use the methods that were developed by Adler and van Moerbeke to determine explicit equations for a certain moduli space, that was studied by Narasimhan and Ramanan. Stated briefly it is, for a fixed non-hyperelliptic Riemann surface Γ of genus 3 , the moduli space of semi-stable rank two bundles with trivial determinant on Γ . They showed that it can be realized as a projective variety, more precisely as a quartic hypersurface of ℙ 7 , whose singular locus is the Kummer variety of Γ . We first construct...

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