An official website of the United States government
We consider the identity component of the Sato–Tate group of the Jacobian of curves of the form C1 : y2 = x2g+2 + c, C2 : y2 = x2g+1 + cx, C3 : y2 = x2g+1 + c, where g is the genus of the curve and c in Q* is constant. We approach this problem in three ways. First we use a theorem of Kani-Rosen to determine the splitting of Jacobians for C1 curves of genus 4 and 5 and prove what the identity component of the Sato–Tate group is in each case. We then determine the splitting of Jacobians of higher genus C1 curves by finding maps to lower genus curves and then computing pullbacks of differential 1-forms. In using this method, we are able to relate the Jacobians of curves of the form C1, C2 and C3. Finally, we develop a new method for computing the identity component of the Sato–Tate groups of the Jacobians of the three families of curves. We use this method to compute many explicit examples, and find surprising patterns in the shapes of the identity components ST^0(C) for these families of curves.
more »
« less
Sato-Tate distributions of y^2 = x^p − 1 and y^2 = x^{2p} − 1
We determine the Sato-Tate groups and prove the generalized Sato-Tate conjecture for the Jacobians of curves of the form y^2 = x^p−1 and y2 = x^{2p}−1, where p is an odd prime. Our results rely on the fact the Jacobians of these curves are nondegenerate, a fact that we prove in the paper. Furthermore, we compute moment statistics associated to the Sato-Tate groups. These moment statistics can be used to verify the equidistribution statement of the generalized Sato-Tate conjecture by comparing them to moment statistics obtained for the traces in the normalized L-polynomials of the curves.
more »
« less
- Award ID(s):
- 2002085
- PAR ID:
- 10544954
- Publisher / Repository:
- Science Direct
- Date Published:
- Journal Name:
- Journal of Algebra
- Volume:
- 597
- Issue:
- C
- ISSN:
- 0021-8693
- Page Range / eLocation ID:
- 241 to 265
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
The term degenerate is used to describe abelian varieties whose Hodge rings contain exceptional cycles -- Hodge cycles that are not generated by divisor classes. We can see the effect of the exceptional cycles on the structure of an abelian variety through its Mumford-Tate group, Hodge group, and Sato-Tate group. In this article we examine degeneracy through these different but related lenses. We specialize to a family of abelian varieties of Fermat type, namely Jacobians of hyperelliptic curves of the form $y^2=x^m-1$. We prove that the Jacobian of the curve is degenerate whenever $$m$$ is an odd, composite integer. We explore the various forms of degeneracy for several examples, each illustrating different phenomena that can occur.more » « less
-
Given an abelian variety over a number field, its Sato–Tate group is a compact Lie group which conjecturally controls the distribution of Euler factors of the -function of the abelian variety. It was previously shown by Fité, Kedlaya, Rotger, and Sutherland that there are 52 groups (up to conjugation) that occur as Sato–Tate groups of abelian surfaces over number fields; we show here that for abelian threefolds, there are 410 possible Sato–Tate groups, of which 33 are maximal with respect to inclusions of finite index. We enumerate candidate groups using the Hodge-theoretic construction of Sato–Tate groups, the classification of degree-3 finite linear groups by Blichfeldt, Dickson, and Miller, and a careful analysis of Shimura’s theory of CM types that rules out 23 candidate groups; we cross-check this using extensive computations inGAP,SageMath, andMagma. To show that these 410 groups all occur, we exhibit explicit examples of abelian threefolds realizing each of the 33 maximal groups; we also compute moments of the corresponding distributions and numerically confirm that they are consistent with the statistics of the associated -functions.more » « less
-
Abstract We prove the integral Hodge conjecture for all 3-folds $$X$$ of Kodaira dimension zero with $$H^{0}(X,K_{X})$$ not zero. This generalizes earlier results of Voisin and Grabowski. The assumption is sharp, in view of counterexamples by Benoist and Ottem. We also prove similar results on the integral Tate conjecture. For example, the integral Tate conjecture holds for abelian 3-folds in any characteristic.more » « less
-
We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi–Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use this to deduce cases of the usual integral Hodge conjecture for varieties. Along the way, we prove a version of the variational integral Hodge conjecture for families of two-dimensional Calabi–Yau categories, as well as a general smoothness result for relative moduli spaces of objects in such families. Our machinery also has applications to the structure of intermediate Jacobians, such as a criterion in terms of derived categories for when they split as a sum of Jacobians of curves.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.jalgebra.2022.01.002
