### 作ったプログラム Computer programs

Bases of the space of algebraic modular forms

We present an algorithm to compute a basis of the space of algebraic modular forms over the maximal order of the definite quaternion algebra of discriminant 2. In what follows, we present numerical examples for bases of $H_{l,\pm}^{\Gamma}$ with l up to 100. This is joint work with Hiroyuki Ochiai (Kyushu University) and Satoshi Wakatsuki (Kanazawa University).

Main program is HERE

Database: 3-10 / 11-20 / 21-30 / 31-40 / 41-50 / 51-60 / 61-70 / 71-80 / 81-90 / 91-100

Execution times: Single threaded times in seconds taken on a 3.2GHz Apple M1 Max processor (firestorm) with 64GB unified memory.

Distribution of toric periods of modular forms on definite quaternion algebras

We study distribution of the toric periods of algebraic modular forms of some level. Especially, we focus of 2 problems: non-vanishing and sign changes. Using our database, we provide evidence for non-vanishing conjectures for central values of twisted automorphic L-functions, and the sequence of toric periods having infinitely many sign changes. This is joint work with Miyu Suzuki (Kyoto University) and Satoshi Wakatsuki (Kanazawa University).

Some graphs for distribution of toric periods are available at HERE.

C++ library for optimal ate pairing on BLS48

BLS48 curve is one of the pairing friendly curves, and recommended parameters for realizing 256-bit security are known (by Kiyomura et al. SCIS2017). We implemented C++ library for Optimal Ate Pairing on BLS48 by using libsnark library. This is joint work with Keishi Mabuchi (Kyushu University) and Tsunekazu Saito (NTT Secure Platform Laboratories).

Constructing all abelian extensions over a p-adic field

We provide 3 algorithms on the computation of abelian extensions over a p-adic field:

(1) Identification of p-adic algebra

(2) Constructing all abelian extensions over a p-adic field

(3) The lower and upper numbering ramification breaks

This is joint work with Manabu Yoshida (National Institute of Technology, Toyama College) and all programs and databases are available at his webpage.

Computing resultant matrix and its determinant using Magma

We produce an efficient program to compute resultant matrix and its determinant for a given pair of multivariate polynomials on Magma. This program works much more faster than the Magma's built-in function "Resultant" for multivariate polynomials. It is the author's pleasure to thank Professor Kinji Kimura (Fukui University) for his valuable discussions, suggestions, and assistance for this trial. The author also would like to thank Professor Yukihiro Uchida (Tokyo Metropolitan University) for giving him (=the author) some comments.

Type SR (seki_res + built-in determinant): HERE

Type BR (seki_res + det_berkowitz): HERE

Benchmark Problem (general formula of the discriminant of degree 9): SR / BR

Sample code that only can verify the computation of the discriminant up to degree 12 using Cayley's method: HERE