On the Abel–Jacobi maps of Fermat Jacobians

@article{Otsubo2010OnTA,
  title={On the Abel–Jacobi maps of Fermat Jacobians},
  author={Noriyuki Otsubo},
  journal={Mathematische Zeitschrift},
  year={2010},
  volume={270},
  pages={423-444}
}
We study the Abel–Jacobi image of the Ceresa cycle $${W_k-W_k^-}$$, where Wk is the image of the kth symmetric product of a curve X on its Jacobian variety. For the Fermat curve of degree N, we express it in terms of special values of generalized hypergeometric functions and give a criterion for the non-vanishing of $${W_k-W_k^-}$$ modulo algebraic equivalence, which is verified numerically for some N and k. 
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References

SHOWING 1-10 OF 39 REFERENCES
Harmonic volume, symmetric products, and the Abel-Jacobi map
The author generalizes B. Harris' definition of harmonic volume to the algebraic cycle W k − W k − for k > 1 in the Jacobian of a nonsingular algebraic curve X. We define harmonic volume, determine…
A nontrivial algebraic cycle in the Jacobian variety of the Klein quartic
We prove some value of the harmonic volume for the Klein quartic C is nonzero modulo $$\frac{1}{2}{\mathbb{Z}}$$ , using special values of the generalized hypergeometric function 3F2. This result…
On the periods of abelian integrals and a formula of Chowla and Selberg
Given an imaginary quadratic field k of discriminant d , let E be an elliptic curve defined over Q, the algebraic closure of Q in C, which admits complex multiplication by some order in k. Let ~ be a…
The Conjectures of Birch and Swinnerton-Dyer, and of Tate
In the last few years, it has become increasingly evident that the study of the zeta-function of an algebraic variety can yield valuable information about that variety, some of which cannot easily be…
Note on curves in a Jacobian
For a curve C, viewed as a cycle in its Jacobian, we study its image n_*C under multiplication by n on JC. We prove that the subgroup generated by these cycles, in the Chow group modulo algebraic…
On the regulator of Fermat motives and generalized hypergeometric functions
Abstract We calculate the Beilinson regulator of motives associated to Fermat curves and express them by special values of generalized hypergeometric functions. As a result, we obtain surjectivity…
Homological versus algebraic equivalence in a jacobian.
  • B. Harris
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1983
TLDR
A formula for nu is given for nu for the case of V the jacobian of an algebraic curve C and Z=C - C' (C' = "inverse" of C') in terms of iterated integrals of holomorphic 1-forms on C.
Higher Dimensional Harmonic Volume Can be Computed as an Iterated Integral
Abstract In this paper it is shown that the computation of higher dimensional harmonic volume, defined in [1], can be reduced to Harris' computation in the onedimensional case (See [3]), so that…
Extensions of mixed Hodge structures
According to Deligne, the cohomology groups of a complex algebraic variety carry a generalized Hodge structure, or, in precise terms, a mixed Hodge structure [2]. The purpose of this paper is to…
A nontrivial algebraic cycle in the Jacobian variety of the Fermat sextic
We compute some value of the harmonic volume for the Fermat sextic. Using this computation, we prove that some special algebraic cycle in the Jacobian variety of the Fermat sextic is not…
...
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3
4
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