Inversion of Abelian integrals

“…(ii) All our results apply equally well to the linearized inversion of abelian integrals problem [5]. It is only necessary to supply the correct technical means for translating the section Do + ~b0-1 ( ) into a section of the linearized bundle.…”

mentioning

confidence: 53%

Inversion of abelian integrals on small genus curves

Coombes

Fisher

1986

Math. Ann.

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Let M be a smooth projective curve of genus g > 0. Let J be the Jacobian of M and let M ") be the i th symmetric product. Fix a base point PeM and define a map ~bi: M ") ~ J by Oi(D) = DiP. Mattuck [6, 7] has shown that, if i > 2g-1, then 0i is a Ipi-g-bundle. The most interesting case occurs when i = 2g-1, since in [1 ] the bundles for larger i are determined by this one, The inversion of abelian integrals problem asks: What is an explicit description of the transition functions of the bundle 0~ ? For an excellent introduction to the problem, see Kempf's article [5]. Until now, the only complete answer was given in genus 1 by the Abel Inversion Theorem, the Riemann-Roch Theorem, and Riemann's approach through theta functions. Even without the transition functions, Gunning [1, 2] and Kempf [3,4] were able to extract a great deal of information about M from the bundles 0,. In this paper, we present a solution to the inversion of abelian integrals problem in the cases (i) curves of genus 2 (ii) non-hyperelliptic curves of genus 3. The techniques we use are very geometric but essentially elementary. We strongly emphasize the role played by effective divisors and by the Riemann-Roch Theorem. To build sections, we use the commutative diagram M(g) +Oo ~ M(2g-1) j +no-(g-l)e ~ j where D O is a fixed effective divisor of degree g-1. The Jacobi Inversion Theorem says that 4~ is a birational surjection; hence, it has an inverse on an open set. It is then necessary to choose enough different D0's to trivialize the bundle. The geometry first enters when we must describe the open set over which a given collection of Do, s determines a trivialization. We then use explicit knowledge of the

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“…Hence we have a surjective map of vector bundles p* -> xg+] ■ This is equivalent to having an injective map of the dual bundles xg+l -* M ■ Let Q be the quotient bundle. So we have an exact sequence (12) o-^-zi-iß-O. Now, Q is a bundle of rank 2g -(g + 1) = g -1.…”

Section: This Completes the Proof Of Part (1)mentioning

confidence: 98%

“…However, there is of yet no explicit description of these Picard bundles. They were initially investigated by Kempf [12], and Mattuck [16,17]. Gunning has shown [6] that the rank g + 1 Picard bundle sits inside the rank 2s Clifford bundle, representing the well-known transformation properties of second order theta function under half periods.…”

Section: Introductionmentioning

confidence: 99%

Second order theta functions and vector bundles over Jacobi varieties

Yuen¹

1990

Trans. Amer. Math. Soc.

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Abstract.We consider the Picard vector bundles defined over Jacobi varieties. The rank g + 1 Picard bundle imbeds in the rank 2g Clifford bundle, so the second order theta functions, viewed appropriately, span the dual of the Picard bundle over each fiber. We prove a result on the minimum number of such second order theta functions required to span the whole bundle at each point. We give an application of using these functions to describe subvarieties of the Jacobian. There follow comments on which functions we could use, and generalizations to higher order theta functions. IntroductionOver the Jacobi variety of a compact Riemann surface, there are the naturally occurring Picard vector bundles. These bundles are derived by considering sufficiently high symmetric products of the curve, which map to the Jacobi variety. They reflect many essential properties of the Jacobi variety and of the curve. However, there is of yet no explicit description of these Picard bundles. They were initially investigated by Kempf [12], and Mattuck [16,17]. Gunning has shown [6] that the rank g + 1 Picard bundle sits inside the rank 2s Clifford bundle, representing the well-known transformation properties of second order theta function under half periods. One aim of this paper is to find in some sense a smaller explicitly describable bundle in which this Picard bundle sits. We prove a result that there exist 2g+ 1 second order theta functions that span the dual of this Picard bundle, and that this is the minimum number required. This implies that the pullback of this Picard bundle under an isogeny of the Jacobi variety sits inside a rank 2g + 1 bundle which is the rank 2g + 1 trivial bundle tensored with a line bundle. This result generalizes to the the rank (d -l)g + l Picard bundle. There follow some comments on whether we can say which 2g + 1 second order theta functions we can use.

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Inversion of Abelian integrals

Cited by 8 publications

References 16 publications

The action of the Frobenius map on rank 2 vector bundles in characteristic 2

The action of the Frobenius map on rank 2 vector bundles in characteristic 2

Inversion of abelian integrals on small genus curves

Second order theta functions and vector bundles over Jacobi varieties

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