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 = r 1 + 2 r 2 distinct monomorphisms σ i : K − → C , where r 1 is the number of real monomorphisms and 2 r 2 is the number of complex monomorphisms. → R r 1 × R 2 r 2 given by The application σ : K − σ ( x ) = ( σ 1 ( x ) , . . . , σ r 1 + r 2 ( x )) ∈ R r 1 × C r 2 ≃ R r 1 × R 2 r 2 is called Minkowski Homomorphism . If J � = 0 is an ideal of the ring of integers O K of K , σ ( J ) is a lattice called algebraic lattice . The center density of σ ( J ) is t n / 2 J δ = 2 n � | D ( K ) | N ( J ) where t J = min { Tr K : 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 � O K = Z [ G ] η d = R K T . d ∈D ( n ) where G = Gal ( K : Q ) K d = Q ( ζ d ) ∩ K η d = Tr Q ( d ) : K d ( ζ d ) � T = η d R K = Z [ G ][ { ǫ d : d ∈ D ( n ) } ] ⊂ Q [ G ] d ∈D ( n ) D ( n ) = { d ∈ N : P n | d , d | n e d �≡ 2 ( mod 4) } in which P n is the product of the distinct primes divisors of n different of 2 and ǫ d 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 ( ζ 6 ) is the ideal that minimizes the 1 center density in dimension 2 (in this case, δ = 3 ); √ 2 The principal ideal ( − 1 − ζ 20 + ζ 2 20 + ζ 3 20 + ζ 4 20 ) O K in the ring of integers of Q ( ζ 20 ) 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 optimal 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¨ orpers , 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