ZpL : a p-adic precision package Xavier Caruso, David Roe, Tristan - PowerPoint PPT Presentation

ZpL : a p-adic precision package ZpL : a p-adic precision package Xavier Caruso, David Roe, Tristan Vaccon Univ. Rennes 1 → Univ. Bordeaux; MIT; Université de Limoges ISSAC 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction What are p -adic numbers? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction What are p -adic numbers? p refers to a prime number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction What are p -adic numbers? p refers to a prime number p -adic numbers are numbers written in p -basis of the shape: a = . . . a i . . . a 2 a 1 a 0 , a − 1 a − 2 . . . a − n with 0 ≤ a i < p for all i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction What are p -adic numbers? p refers to a prime number p -adic numbers are numbers written in p -basis of the shape: a = . . . a i . . . a 2 a 1 a 0 , a − 1 a − 2 . . . a − n with 0 ≤ a i < p for all i . Addition and multiplication on these numbers are defined by applying SchoolBook algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction What are p -adic numbers? p refers to a prime number p -adic numbers are numbers written in p -basis of the shape: a = . . . a i . . . a 2 a 1 a 0 , a − 1 a − 2 . . . a − n with 0 ≤ a i < p for all i . Addition and multiplication on these numbers are defined by applying SchoolBook algorithms. The valuation v p ( a ) of a is the smallest v such that a v ̸ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction What are p -adic numbers? p refers to a prime number p -adic numbers are numbers written in p -basis of the shape: a = . . . a i . . . a 2 a 1 a 0 , a − 1 a − 2 . . . a − n with 0 ≤ a i < p for all i . Addition and multiplication on these numbers are defined by applying SchoolBook algorithms. The valuation v p ( a ) of a is the smallest v such that a v ̸ = 0. The p -adic numbers form the field Q p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction What are p -adic numbers? p refers to a prime number p -adic numbers are numbers written in p -basis of the shape: a = . . . a i . . . a 2 a 1 a 0 , a − 1 a − 2 . . . a − n with 0 ≤ a i < p for all i . Addition and multiplication on these numbers are defined by applying SchoolBook algorithms. The valuation v p ( a ) of a is the smallest v such that a v ̸ = 0. The p -adic numbers form the field Q p . A p -adic number with no digit after the comma is a p -adic integer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction What are p -adic numbers? p refers to a prime number p -adic numbers are numbers written in p -basis of the shape: a = . . . a i . . . a 2 a 1 a 0 , a − 1 a − 2 . . . a − n with 0 ≤ a i < p for all i . Addition and multiplication on these numbers are defined by applying SchoolBook algorithms. The valuation v p ( a ) of a is the smallest v such that a v ̸ = 0. The p -adic numbers form the field Q p . A p -adic number with no digit after the comma is a p -adic integer. The p -adic integers form a subring Z p of Q p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction Summary on p-adics Proposition Z p / p Z p = Z / p Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction Summary on p-adics Proposition Z p / p Z p = Z / p Z . ∀ k ∈ N , Z p / p k Z p = Z / p k Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction Summary on p-adics Proposition Z p / p Z p = Z / p Z . ∀ k ∈ N , Z p / p k Z p = Z / p k Z . A first idea Q p is an extension of Q where one can perform calculus , as simply as over R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction Summary on p-adics Proposition Z p / p Z p = Z / p Z . ∀ k ∈ N , Z p / p k Z p = Z / p k Z . A first idea Q p is an extension of Q where one can perform calculus , as simply as over R . We are closer to arithmetic : we can reduce modulo p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction Summary on p-adics Proposition Z p / p Z p = Z / p Z . ∀ k ∈ N , Z p / p k Z p = Z / p k Z . A first idea Q p is an extension of Q where one can perform calculus , as simply as over R . We are closer to arithmetic : we can reduce modulo p . Remark Q p R Z p Z p Q Z / p Z Z / p Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction Why should one work with p -adic numbers? p -adic methods Working in Q p instead of Q , one can handle more efficiently the coefficients growth; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction Why should one work with p -adic numbers? p -adic methods Working in Q p instead of Q , one can handle more efficiently the coefficients growth; e.g. Dixon’s method (used in F4), polynomial factorization via Hensel’s lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction Why should one work with p -adic numbers? p -adic methods Working in Q p instead of Q , one can handle more efficiently the coefficients growth; e.g. Dixon’s method (used in F4), polynomial factorization via Hensel’s lemma. p -adic algorithms Going from Z / p Z to Z p and then back to Z / p Z enables more computation, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction Why should one work with p -adic numbers? p -adic methods Working in Q p instead of Q , one can handle more efficiently the coefficients growth; e.g. Dixon’s method (used in F4), polynomial factorization via Hensel’s lemma. p -adic algorithms Going from Z / p Z to Z p and then back to Z / p Z enables more computation, e.g. solving differential equations over finite fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction Why should one work with p -adic numbers? p -adic methods Working in Q p instead of Q , one can handle more efficiently the coefficients growth; e.g. Dixon’s method (used in F4), polynomial factorization via Hensel’s lemma. p -adic algorithms Going from Z / p Z to Z p and then back to Z / p Z enables more computation, e.g. solving differential equations over finite fields. Kedlaya’s and Lauder’s counting-point algorithms via p -adic cohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction Why should one work with p -adic numbers? p -adic methods Working in Q p instead of Q , one can handle more efficiently the coefficients growth; e.g. Dixon’s method (used in F4), polynomial factorization via Hensel’s lemma. p -adic algorithms Going from Z / p Z to Z p and then back to Z / p Z enables more computation, e.g. solving differential equations over finite fields. Kedlaya’s and Lauder’s counting-point algorithms via p -adic cohomology. My personal (long-term) motivation Computing (some) moduli spaces of p -adic Galois representations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction Motivations and goal Today’s goal We present the ZpL package for Sage. It features: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ZpL : a p-adic precision package Introduction Motivations and goal Today’s goal We present the ZpL package for Sage. It features: Tracking of the optimal behaviour of p -adic precision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .