Ring of Integers of Abelian Number Fields and Algebraic Lattices - - PowerPoint PPT Presentation

Ring of Integers of Abelian Number Fields and Algebraic Lattices

Robson Ricardo de Araujo dearaujorobsonricardo@gmail.com

• Prof. Dr. Antonio Aparecido de Andrade (orientador)

andrade@ibilce.unesp.br SP Coding and Information School January 19th to 30th 2015 UNICAMP - Campinas, Brazil

Araujo, R.R., Andrade, A.A. (IBILCE/UNESP) Ring of Integers of Abelian Number Fields and Algebraic Lattices SPCodingSchool 1 / 6

Let K be a number field of degree n (over Q). It exists n = r1 + 2r2 distinct monomorphisms σi : K − → C, where r1 is the number of real monomorphisms and 2r2 is the number of complex monomorphisms. The application σ : K − → Rr1 × R2r2 given by σ(x) = (σ1(x), . . . , σr1+r2(x)) ∈ Rr1 × Cr2 ≃ Rr1 × R2r2 is called Minkowski Homomorphism. If J = 0 is an ideal of the ring of integers OK of K, σ(J) is a lattice called algebraic lattice. The center density of σ(J) is δ = tn/2

J

2n |D(K)|N(J) where tJ = min{TrK:Q(xx) : x ∈ J, x = 0} and D(K) is the discriminant of the field K.

Araujo, R.R., Andrade, A.A. (IBILCE/UNESP) Ring of Integers of Abelian Number Fields and Algebraic Lattices SPCodingSchool 2 / 6

Leopoldt-Lettl Theorem

Let K be an abelian number field of conductor n and G = Gal(K : Q). The ring of integers of K is OK =

• d∈D(n)

Z[G]ηd = RKT. where G = Gal(K : Q) Kd = Q(ζd) ∩ K ηd = TrQ(d):Kd(ζd) T =

• d∈D(n)

ηd RK = Z[G][{ǫd : d ∈ D(n)}] ⊂ Q[G] D(n) = {d ∈ N : Pn | d, d | n e d ≡ 2 (mod 4)} in which Pn is the product of the distinct primes divisors of n different of 2 and ǫd are idempotent orthogonal elements of Q[G].

Araujo, R.R., Andrade, A.A. (IBILCE/UNESP) Ring of Integers of Abelian Number Fields and Algebraic Lattices SPCodingSchool 3 / 6

Open Problem

It is known algebraic lattices of optimal density center in dimensions 2, 4, 6 and 8. For example: The ring of integers of K = Q(ζ6) is the ideal that minimizes the center density in dimension 2 (in this case, δ =

1 2 √ 3);

The principal ideal (−1 − ζ20 + ζ2

20 + ζ3 20 + ζ4 20)OK in the ring of

integers of Q(ζ20) minimizes the center density in dimension 8 (in this case, δ = 1/16). Thinking... However, we don’t know yet an example of algebraic lattice that has

• ptimal center density in the odd dimensions (in dimension 3, the best

density center for lattices is 1/4 √ 2; in dimension 5, it is 1/8 √ 2; in the dimension 7, it is 1/16). This is an open problem. The Leopoldt-Lettl Theorem has been used in our attempt to solve this problem.

Araujo, R.R., Andrade, A.A. (IBILCE/UNESP) Ring of Integers of Abelian Number Fields and Algebraic Lattices SPCodingSchool 4 / 6

Bibliografia Lettl, G¨

• unter. The ring of integers of an abelian number field, J. reine
• angew. Math. 404 (1990), 162-170.

Leopoldt, H.-W., ¨ Uber die Hauptordnung der ganzen Elemente eines abelschen Zahlk¨

• rpers, J. reine angew. Math. 201 (1959), 119-149.

Thanks! dearaujorobsonricardo@gmail.com andrade@ibilce.unesp.br

Araujo, R.R., Andrade, A.A. (IBILCE/UNESP) Ring of Integers of Abelian Number Fields and Algebraic Lattices SPCodingSchool 5 / 6