| 1. |
Complete intersection of two quadrics and Galois cohomology : a research announcement (Algebraic Number Theory and Related Topics 2012)
In this paper, we give an overview of a method of explicit 2-descent for hyperelliptic Jacobian varieties ([1]), and relate it to the set of complete intersections of two quadrics. This is an announcement of the results contained in the paper [4].. |
| 2. |
Yasuhiro Ishitsuka, Tetsushi Ito, On the symmetric determinantal representations of the Fermat curves of prime degree, INTERNATIONAL JOURNAL OF NUMBER THEORY, 10.1142/S1793042116500597, 12, 4, 955-967, 2016.06, We prove that the defining equations of the Fermat curves of prime degree cannot be written as the determinant of symmetric matrices with entries in linear forms in three variables with rational coefficients. In the proof, we use a relation between symmetric matrices with entries in linear forms and non-effective theta characteristics on smooth plane curves. We also use some results of Gross-Rohrlich on the rational torsion points on the Jacobian varieties of the Fermat curves of prime degree.. |
| 3. |
石塚 裕大, 伊藤 哲史, The local–global principle for symmetric determinantal representations of smooth plane curves, The Ramanujan Journal, 10.1007/s11139-016-9775-3, 43, 1, 141-162, 2017.05. |
| 4. |
石塚 裕大, 伊藤 哲史, The local–global principle for symmetric determinantal representations of smooth plane curves in characteristic two, Journal of Pure and Applied Algebra, 10.1016/j.jpaa.2016.09.013, 221, 6, 1316-1321, 2017.06. |
| 5. |
石塚 裕大, A positive proportion of cubic curves over Q admit linear determinantal representations, Journal of the Ramanujan Mathematical Society, 32, 3, 231-257, 2017.09. |
| 6. |
Yasuhiro Ishitsuka, Tetsushi Ito, Tatsuya Ohshita, On algorithms to obtain linear determinantal representations of smooth plane curves of higher degree, JSIAM Letters, 10.14495/jsiaml.11.9, 11, 0, 9-12, 2019.03, We give two algorithms to compute linear determinantal representations of
smooth plane curves of any degree over any field. As particular examples, we
explicitly give representatives of all equivalence classes of linear
determinantal representations of two special quartics over the field
$mathbb{Q}$ of rational numbers, the Klein quartic and the Fermat quartic.
This paper is a summary of third author's talk at the JSIAM JANT workshop on
algorithmic number theory in March 2018. Details will appear elsewhere.. |
| 7. |
Yasuhiro Ishitsuka, Tetsushi Ito, Tatsuya Ohshita, Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic, International Journal of Number Theory, 10.1142/s1793042120500451, 16, 04, 881-905, 2020.05, We use explicit methods to study the [Formula: see text]-torsion points on the Jacobian variety of the Fermat quartic. With the aid of computer algebra systems, we explicitly give a basis of the group of [Formula: see text]-torsion points. We calculate the Galois action, and show that the image of the mod [Formula: see text] Galois representation is isomorphic to the dihedral group of order [Formula: see text]. As applications, we calculate the Mordell–Weil group of the Jacobian variety of the Fermat quartic over each subfield of the [Formula: see text]th cyclotomic field. We determine all of the points on the Fermat quartic defined over quadratic extensions of the [Formula: see text]th cyclotomic field. Thus, we complete Faddeev’s work in 1960.. |
| 8. |
Yasuhiro Ishitsuka, Takeo Uramoto, On the integrality of algebraic Witt vectors over imaginary quadratic fields, 9th International Symposium on Symbolic Computation in Software Science (SCSS 2021), short and work-in-progress papers, RISC Report Series No. 21-16, https://doi.org/10.35011/risc.21-16, 28-33, 2022.06, The aim of this short note is to report some progress on numerical computations of algebraic Witt
vectors over imaginary quadratic fields K based on a previous work of the second author. In partic-
ular, when K is of class number one, we describe an effective procedure to decide whether a given
algebraic Witt vector ξ∈EK =K ⊗WaOK (O ̄K)is an integral Witt vector ξ∈WaOK (O ̄K).. |
| 9. |
Yasuhiro Ishitsuka
Tetsushi Ito, The Hasse principle for finite Galois modules allowing exceptional sets of positive density, International Journal of Number Theory, 10.1142/S179304212250097X, 18, 9, 1895-1919, 2022.10, We study a variant of the Hasse principle for finite Galois modules, allowing exceptional sets of positive density. For a Galois module whose underlying abelian group is isomorphic to F_p^r (r≤2), we show that the product of the restriction maps for places in a set of places S is injective if the Dirichlet density of 𝑆 is strictly larger than 1-p^{-r}. We give applications to the local–global divisibility problem for elliptic curves and the Hasse principle for flexes on plane cubic curves.. |
