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. 
View on Springer
arxiv.org
13 Citations
Highly Influential Citations
1
Background Citations
5
Methods Citations
3
Results Citations
1

13 Citations

Citation Type

Citation Type

The fundamental group and extensions of motives of Jacobians of curves
In this paper, we construct extensions of mixed Hodge structure coming from the mixed Hodge structure on the graded quotients of the group ring of the fundamental group of a smooth projective pointed…
On the harmonic volume of Fermat curves
We prove that B. Harris’ harmonic volume of the Fermat curve of degree n n is of infinite order if n n has a prime divisor greater than 7. The statement is equivalent to the…
Extensions of Motives and the Fundamental Group
In this paper we construct extensions of the Mixed Hodge structure on the fundamental group of a pointed algebraic curve. These extensions correspond to the regulator of certain explicit motivic…
Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves
We obtain the trace map image of the values of certain harmonic volumes for some quotients of Fermat curves. This provides the algorithm that the algebraic cycles called by the k-th Ceresa cycles are…
Albanese kernels and Griffiths groups
We describe the Griffiths group of the product of a curve $C$ and a surface $S$ as a quotient of the Albanese kernel of $S$ over the function field of $C$. When $C$ is a hyperplane section of $S$…
Pointed harmonic volume and its relation to the extended Johnson homomorphism
  • Y. Tadokoro
  • Mathematics
    Journal of Topology and Analysis
  • 2018
The period for a compact Riemann surface, defined by the integral of differential 1-forms, is a classical complex analytic invariant, strongly related to the complex structure of the surface. In this…
Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type
Given a smooth projective variety, a Chow-K\"unneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial…
Distinguished cycles on varieties with motive of abelian type and the Section Property
A remarkable result of Peter O’Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a…
DISTINGUISHED CYCLES ON VARIETIES WITH MOTIVE OF ABELIAN TYPE AND THE SECTION PROPERTY
A remarkable result of Peter O’Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a…
Some new Fano varieties with a multiplicative Chow-K\"unneth decomposition
Let Y be a smooth dimensionally transverse intersection of the GrassmannianGr(2, n) with 3 Plücker hyperplanes. We show that Y admits a multiplicative Chow–Künneth decomposition, in the sense of…
...
...

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…
...
...