LATTICES IN NUMBER THEORY

Transcription

1 LATTICES IN NUMBER THEORY Ha Tran ICTP CIMPA summer school 2016 HCM University of Science Saigon University 1 / 32

2 Lattices in Number Theory We will study ideal lattices. They are lattices with structure of ideals. They are used in coding theory and cryptography. They have nice algebraic structure (so have nice and cheap representation, allows fast arithmetic,...) 2 / 32

3 We will Lattices in Number Theory 1 Review about number fields, ideal lattices and... Study Arakelov divisors. 2 Study the Arakelov class group. 3 Discuss about reduced Arakelov divisors and their properties. Discuss the reduction algorithm. 3 / 32

4 We will Lattices in Number Theory 1 Review about number fields, ideal lattices and... Study Arakelov divisors. 2 Study the Arakelov class group. 3 Discuss about reduced Arakelov divisors and their properties. Discuss the reduction algorithm. PS: Questions are very welcome :) 3 / 32

5 ICTP CIMPA summer school 2016 Lecture 1. NUMBER FIELDS AND... HCM University of Science Saigon University 4 / 32

6 Content 1 Review Number fields and the ring of integers Fractional ideals The class group F R The Φ map The L map 2 Ideal lattices 5 / 32

7 Let F be a number field of degree n. 1 F = Q, 2 F = Q( 2), 3 F = Q(i) (the Gaussian field), 4 F = Q( 5), Review 5 F is the splitting field of x 3 + mx 2 (m + 3)x + 1 (m 1, m 3 mod 9) (simplest cubic field), 6 F = Q( 4 2)? 7 F = Q(ζ m ) (cyclotomic field)? 6 / 32

8 Review Let F be a number field of degree n. σ 1,..., σ r1 are r 1 real infinite primes σ r1 +1,..., σ r1 +r 2 are r 2 complex infinite primes. 1 F = Q, 2 F = Q( 2), 3 F = Q(i) (the Gaussian field), 4 F = Q( 5), 5 F is the splitting field of x 3 + mx 2 (m + 3)x + 1 (m 1, m 3 mod 9) (simplest cubic field), 6 F = Q( 4 2)? 7 F = Q(ζ m ) (cyclotomic field)? 6 / 32

9 Review Let F be a number field of degree n. σ 1,..., σ r1 are r 1 real infinite primes σ r1 +1,..., σ r1 +r 2 are r 2 complex infinite primes. O F : The ring of integers of F (n = r 1 + 2r 2 ). 1 F = Q, 2 F = Q( 2), 3 F = Q(i) (the Gaussian field), 4 F = Q( 5), 5 F is the splitting field of x 3 + mx 2 (m + 3)x + 1 (m 1, m 3 mod 9) (simplest cubic field), 6 F = Q( 4 2)? 7 F = Q(ζ m ) (cyclotomic field)? 6 / 32

10 Fractional ideals I is called a fractional ideal of F if I is an additive subgroup of F, and there exists α F st αi is an ideal of O F. 7 / 32

11 Fractional ideals I is called a fractional ideal of F if I is an additive subgroup of F, and there exists α F st αi is an ideal of O F. Ex: F = Q( 5). I i is a factional ideal of F? 1 I 1 = {m 1 + 2m 2 : m 1, m 2 Z}? 7 / 32

12 Fractional ideals I is called a fractional ideal of F if I is an additive subgroup of F, and there exists α F st αi is an ideal of O F. Ex: F = Q( 5). I i is a factional ideal of F? 2 I 2 = {m m 2 : m 1, m 2 Z}? 7 / 32

13 Fractional ideals I is called a fractional ideal of F if I is an additive subgroup of F, and there exists α F st αi is an ideal of O F. Ex: F = Q( 5). I i is a factional ideal of F? 2 I 2 = {m m 2 : m 1, m 2 Z}? 3 I 3 = {m m 2 : m 1, m 2 Z}? 7 / 32

14 Fractional ideals I is called a fractional ideal of F if I is an additive subgroup of F, and there exists α F st αi is an ideal of O F. Ex: F = Q( 5). I i is a factional ideal of F? 4 I 4 = {2m 1 + (1 5)m 2 : m 1, m 2 Z}? 7 / 32

15 Fractional ideals I is called a fractional ideal of F if I is an additive subgroup of F, and there exists α F st αi is an ideal of O F. Ex: F = Q( 5). I i is a factional ideal of F? 4 I 4 = {2m 1 + (1 5)m 2 : m 1, m 2 Z}? 5 I 5 = { 1 2 m m 2 : m 1, m 2 Z}? 7 / 32

16 Fractional ideals I is called a fractional ideal of F if I is an additive subgroup of F, and there exists α F st αi is an ideal of O F. Ex: F = Q( 5). I i is a factional ideal of F? 4 I 4 = {2m 1 + (1 5)m 2 : m 1, m 2 Z}? 5 I 5 = { 1 2 m m 2 : m 1, m 2 Z}? 6 I 6 = { 5 5 m m 2 : m 1, m 2 Z}? = 1 2 I / 32

17 The class group Id F : The group of all fractional ideals of F with multiplication. Let Princ F := the subgroup of principal ideals of F. The class group of F is Cl F = Id F /Princ F. 8 / 32

18 The class group Id F : The group of all fractional ideals of F with multiplication. Let Princ F := the subgroup of principal ideals of F. The class group of F is Cl F = Id F /Princ F. Cl F is finite and #Cl F = h F (the class number). 8 / 32

19 The class group Id F : The group of all fractional ideals of F with multiplication. Let Princ F := the subgroup of principal ideals of F. The class group of F is Cl F = Id F /Princ F. Cl F is finite and #Cl F = h F (the class number). Ex: Cl F =?, h F =? if 1 F = Q. 2 F = Q( 2). 3 F = Q(i). 4 F = Q( 10). 8 / 32

20 F R Let F R := F Q R σ real R σ complex C R r 1 Cr 2 an R-algebra. Ex: F R =? if 1 F = Q, 2 F = Q( 2), 3 F = Q(i), 4 F = Q( 5), 5 F is a simplest cubic field, 6 F = Q( 4 2), 7 F = Q(ζ m ) (cyclotomic field)? 9 / 32

21 F R Let F R := F Q R σ real R σ complex C R r 1 Cr 2 an R-algebra. 10 / 32

22 F R Let F R := F Q R σ real R σ complex C R r 1 Cr 2 an R-algebra. A scalar product on F R, for any u = (u σ ), v = (v σ ) F R, u, v := Tr(uv) = Re(u σ v σ ) + 2 Re(u σ v σ ). σ real σ complex u 2 = σ real u2 σ + 2 σ complex u σ / 32

23 F R Let F R := F Q R σ real R σ complex C R r 1 Cr 2 an R-algebra. A scalar product on F R, for any u = (u σ ), v = (v σ ) F R, u, v := Tr(uv) = Re(u σ v σ ) + 2 Re(u σ v σ ). σ real σ complex u 2 = σ real u2 σ + 2 σ complex u σ 2. Ex: F = Q, Q( 2), Q(i), Q( 5), simplest cubic fields, F = Q( 4 2)? 10 / 32

24 F R Let F R := F Q R σ real R σ complex C R r 1 Cr 2 an R-algebra. u 2 = uσ u σ 2. σ real σ complex 11 / 32

25 F R Let F R := F Q R σ real R σ complex C R r 1 Cr 2 an R-algebra. u 2 = uσ u σ 2. σ real σ complex N(u) := σ real u σ σ complex u σ / 32

26 F R Let F R := F Q R σ real R σ complex C R r 1 Cr 2 an R-algebra. u 2 = uσ u σ 2. σ real σ complex N(u) := σ real u σ σ complex u σ 2. Ex: F = Q, Q( 2), Q(i), Q( 5), simplest cubic fields, F = Q( 4 2)? 11 / 32

27 The Φ map Let Φ : F F R with Φ(f ) = (σ 1 (f ),..., σ r1 +r 2 (f )) for f F. Ex: Φ =? if 1 F = Q, 2 F = Q( 2), 3 F = Q(i), 4 F = Q( 5), 5 F is a simplest cubic field, 6 F = Q( 4 2)? 12 / 32

28 Definition (Lattices) A lattice is a pair (L, q) where Lattices L is a free Z-module of finite rank, and q : L L R is a non-degenerate symmetric bilinear form. 13 / 32

29 Definition (Lattices) A lattice is a pair (L, q) where Lattices L is a free Z-module of finite rank, and q : L L R is a non-degenerate symmetric bilinear form. Ex: L = Z n with the standard metric inherited from R n is a lattice. 13 / 32

30 Definition (Lattices) A lattice is a pair (L, q) where Lattices L is a free Z-module of finite rank, and q : L L R is a non-degenerate symmetric bilinear form. Ex: L = Z n with the standard metric inherited from R n is a lattice. Ex: Let F = Q( 5). Then Φ(O F ) is a lattice in F R R / 32

31 The Φ map Let Φ : F F R with Φ(f ) = (σ 1 (f ),..., σ r1 +r 2 (f )) for f F. 14 / 32

32 The Φ map Let Φ : F F R with Φ(f ) = (σ 1 (f ),..., σ r1 +r 2 (f )) for f F. Ex: Let F = Q( 5). What Φ(O F ) looks like? 14 / 32

33 The Φ map Let Φ : F F R with Φ(f ) = (σ 1 (f ),..., σ r1 +r 2 (f )) for f F. Ex: Let F = Q( 5). What Φ(O F ) looks like? 14 / 32

34 The Φ map Let Φ : F F R with Φ(f ) = (σ 1 (f ),..., σ r1 +r 2 (f )) for f F. Ex: Let F = Q( 5). What Φ(O F ) looks like? Φ(O F ) is a lattice in F R. 14 / 32

35 The Φ map and the images of fractional ideals Let Φ : F F R with Φ(f ) = (σ 1 (f ),..., σ r1 +r 2 (f )) for f F. Ex: Let F = Q( 5) and I 5 = { 1 2 m m 2 : m 1, m 2 Z}: a fractional ideal of F. What Φ(I ) looks like? 15 / 32

36 The Φ map and the images of fractional ideals Let Φ : F F R with Φ(f ) = (σ 1 (f ),..., σ r1 +r 2 (f )) for f F. Ex: Let F = Q( 5) and I 5 = { 1 2 m m 2 : m 1, m 2 Z}: a fractional ideal of F. What Φ(I ) looks like? Proposition Let I be a factional ideal of F. Then Φ(I ) is a lattice in F R. 15 / 32

37 The Φ map and the images of fractional ideals Let Φ : F F R with Φ(f ) = (σ 1 (f ),..., σ r1 +r 2 (f )) for f F. Ex: Let F = Q( 5) and I 5 = { 1 2 m m 2 : m 1, m 2 Z}: a fractional ideal of F. What Φ(I ) looks like? Proposition Let I be a factional ideal of F. Then Φ(I ) is a lattice in F R. Ex: Draw the lattice Φ(I 5 ). 15 / 32

38 The L map Consider to the map L : F F R where L(f ) = (log( σ(f ) )) σ. Ex: F is the splitting field of x 3 x 2 2x + 1 (simplest cubic field). The roots: α 1 = α, α 2 = 1 1 α, α 3 = 1 1 α. Then What L(O F ) looks like? O F =< ±1, α 1, α 2 >. 16 / 32

39 The L map Ex: F is the splitting field of x 3 x 2 2x + 1. The roots: α 1 = α, α 2 = 1 1 α, α 3 = 1 1 α. Then O F =< ±1, α 1, α 2 >. 17 / 32

40 The L map Ex: F is the splitting field of x 3 x 2 2x + 1. The roots: α 1 = α, α 2 = 1 1 α, α 3 = 1 1 α. Then O F =< ±1, α 1, α 2 >. L(O F ) is the hexagonal lattice. 17 / 32

41 The L map Λ := L(O F ) = {L(x) : x O F } H. 18 / 32

42 The L map Λ := L(O F ) = {L(x) : x O F } H. 18 / 32

43 The L map Λ := L(O F ) = {L(x) : x O F } H. H = (x σ) σ R : x σ + 2 σ real σ complex x σ = / 32

44 The L map Λ := L(O F ) = {L(x) : x O F } H. H = (x σ) σ R : x σ + 2 σ real σ complex x σ = 0. Λ is a lattice contained in the vector space H. 18 / 32

45 The L map Λ := L(O F ) = {L(x) : x O F } H. H = (x σ) σ R : x σ + 2 σ real σ complex x σ = 0. Λ is a lattice contained in the vector space H. Let T 0 = H/Λ. Then T 0 is a compact real torus of dimension r 1 + r 2 1 (Dirichlet). 18 / 32

46 T 0 and Λ = L(O F ) Ex: Let F = Q( 2). Then r 1 = 0, r 2 = 1 and T 0 =? dim(t 0 ) =? 19 / 32

47 T 0 and Λ = L(O F ) Ex: Let F = Q( 2). Then r 1 = 2, r 2 = 0 and H = {(x, y) R 2 : x + y = 0} R. T 0 =? dim(t 0 ) =? 20 / 32

48 T 0 and Λ = L(O F ) Ex: Let F = Q( 2). Then r 1 = 2, r 2 = 0 and H = {(x, y) R 2 : x + y = 0} R. 21 / 32

49 T 0 and Λ = L(O F ) Ex: Let F be a totally real cubic field. Then r 1 = 3, r 2 = 0 and H = {(x, y, z) R 3 : x + y + z = 0} R 2. T 0 =? dim(t 0 ) =? 22 / 32

50 T 0 and Λ = L(O F ) Ex: Let F be a totally real cubic field. Then r 1 = 3, r 2 = 0 and H = {(x, y, z) R 3 : x + y + z = 0} R 2. T 0 =? dim(t 0 ) =? Ex: F = Q( 4 2)? 22 / 32

51 Ideal lattices Definition (Ideal lattices) An ideal lattice is a lattice (I, q), where I is a (fractional) O F -ideal and q : I I R is st q(λx, y) = q(x, λy) (Hermitian property) for all x, y I and for all λ O F. 23 / 32

52 Ideal lattices Definition (Ideal lattices) An ideal lattice is a lattice (I, q), where I is a (fractional) O F -ideal and q : I I R is st q(λx, y) = q(x, λy) (Hermitian property) for all x, y I and for all λ O F. Ex 1: Let I be a factional ideal of F. q(x, y) := x, y for any x, y I. (the scalar product defined on F R ) 23 / 32

53 Ideal lattices Definition (Ideal lattices) An ideal lattice is a lattice (I, q), where I is a (fractional) O F -ideal and q : I I R is st q(λx, y) = q(x, λy) (Hermitian property) for all x, y I and for all λ O F. Ex 1: Let I be a factional ideal of F. q(x, y) := x, y for any x, y I. (the scalar product defined on F R ) Then (I, q) is an ideal lattice. 23 / 32

54 Ideal lattices Ex 2: Let I be a factional ideal of F and u = (u σ ) (R >0 ) r 1+r 2. z I, uz := (u σ σ(z)) σ F R. We define q u (x, y) := ux, uy for any x, y I. (the scalar product defined on F R ) 24 / 32

55 Ideal lattices Ex 2: Let I be a factional ideal of F and u = (u σ ) (R >0 ) r 1+r 2. z I, uz := (u σ σ(z)) σ F R. We define q u (x, y) := ux, uy for any x, y I. (the scalar product defined on F R ) Proposition (I, q u ) is an ideal lattice. 24 / 32

56 Isometric ideal lattices Definition Two ideal lattices (I, q) and (I, q ) are called isometric if there exists f F such that I = fi and q (fx, fy) = q(x, y) for all x, y I. 25 / 32

57 Lattices from fractional ideals A full rank lattice in F R is also a full rank lattice in R n via F R R r 1 C r 2 R n x 1 x 1 x 2 x 2 x r1 x r1 Re(x r1 +1). x r1 +1 Im(x r1 +1) x r1 +r 2 Re(x r1 +r 2 ) Im(x r1 +r 2 ) EX: F = Q( 4 2)? 26 / 32

58 Lattices from fractional ideals Z 2? 27 / 32

59 Lattices from fractional ideals Z 2? The hexagonal lattice? 27 / 32

60 Lattices from fractional ideals Z 2? The hexagonal lattice? Let p is an odd prime, p = (1 ζ p )O F : principal ideal of F = Q(ζ p ). Then (p, q 1/p ) is an ideal lattice? 27 / 32

61 Lattices from fractional ideals Z 2? The hexagonal lattice? Let p is an odd prime, p = (1 ζ p )O F : principal ideal of F = Q(ζ p ). Then (p, q 1/p ) is an ideal lattice? Let I = 1 (1 ζ 9 ) 4 O F : principal ideal of F = Q(ζ 9 ). Then (I, q 1 ) is an ideal lattice? 27 / 32

62 Lattices from fractional ideals Let I = 1 (1 ζ 9 ) 4 O F : principal ideal of F = Q(ζ 9 ). Then (I, q 1 ) is an ideal lattice? Figure: The Gram matrix of (I, q 1 ). 28 / 32

63 Lattices from fractional ideals Let F = Q(ζ 15 ) and I = βo F for some β F. Then (O F, q β ) is an ideal lattice? 29 / 32

64 Lattices from fractional ideals Let F = Q(ζ 15 ) and I = βo F for some β F. Then (O F, q β ) is an ideal lattice? Let F = Q(ζ 35 ) and α = (ζ ζ3 35 )(ζ ζ6 35 )(ζ ζ9 35 )(ζ ζ35 12)/ψ (ζ 35 + ζ35 1 ). Then (O F, q α ) is an ideal lattice? 29 / 32

65 Recap Number field, the ring of integers. Fractional ideals: 1/αJ for some α O F and J O F is an ideal. The class group Cl F = Id F /Princ F and class number h F = #Cl F. The Φ map: Φ = (σ 1,, σ r1, σ r1 +1,, σ r1 +r 2 ). The L map: L(x) = (log σ(f ) ) σ, x F. Ideal lattices: (I, q), where I is a (fractional) O F -ideal and q : L L R is a non-degenerate symmetric bilinear form with Hermitian property. I : factional ideal; u = (R >0 ) r 1+r 2. Then (I, q u ) is an ideal lattice. 30 / 32

66 Oh, no : ( : ( : ( Exercise: Let F = Q( 4 2). Prove that O F = Z[ 4 2]. 31 / 32

67 Oh, no : ( : ( : ( Exercise: Let F = Q( 4 2). Prove that O F = Z[ 4 2]. There will be a gift for this :). 31 / 32

68 References Eva Bayer-Fluckiger. Lattices and number fields. In Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), volume 241 of Contemp. Math., pages Amer. Math. Soc., Providence, RI, Hendrik W. Lenstra, Jr. Lattices. In Algorithmic number theory: lattices, number fields, curves and cryptography, volume 44 of Math. Sci. Res. Inst. Publ., pages Cambridge Univ. Press, Cambridge, René Schoof. Computing Arakelov class groups. In Algorithmic number theory: lattices, number fields, curves and cryptography, volume 44 of Math. Sci. Res. Inst. Publ., pages Cambridge Univ. Press, Cambridge, / 32