Displaying similar documents to “Local height functions and the Mordell-Weil theorem for Drinfeld modules”

Torsion Z-module and Torsion-free Z-module

Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, Yasunari Shidama (2014)

Formalized Mathematics

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In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].

Divisible ℤ-modules

Yuichi Futa, Yasunari Shidama (2016)

Formalized Mathematics

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In this article, we formalize the definition of divisible ℤ-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible ℤ-modules are not finitely-generated.We introduce a divisible ℤ-module, equivalent to a vector space of a torsion-free ℤ-module with a coefficient ring ℚ. ℤ-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [15], cryptographic systems with lattices [16] and coding theory [8]. ...

Rank of Submodule, Linear Transformations and Linearly Independent Subsets of Z-module

Kazuhisa Nakasho, Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2014)

Formalized Mathematics

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In this article, we formalize some basic facts of Z-module. In the first section, we discuss the rank of submodule of Z-module and its properties. Especially, we formally prove that the rank of any Z-module is equal to or more than that of its submodules, and vice versa, and that there exists a submodule with any given rank that satisfies the above condition. In the next section, we mention basic facts of linear transformations between two Z-modules. In this section, we define homomorphism...