RWTH Aachen University - phpLDAPadmin (1.2.1.1) › ~haensch › index_files › Kirschmer.pdf1 Abilinearspace(V,) is a ﬁnite dimensional vector space V over K equipped with some

Algorithms for lattices

Markus Kirschmer

RWTH Aachen University

June 2016

Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 1 / 27

Bilinear spaces

Let K be an algebraic number field with ring of integers ZK . Let ⌦ be theset of places of K and P denotes the prime ideals of ZK .

Definition1 A bilinear space (V,�) is a finite dimensional vector space V over K

equipped with some regular bilinear form � : V ⇥ V ! K.2 Two bilinear spaces (V,�) and (V 0,�0) are said to be isometric, if

there exists some isomorphism ' 2 HomK(V, V 0) such that

�

0('(x),'(y)) = �(x, y) for all x, y 2 V .

Then ' is called an isometry.

3 O(V,�) = {' : V ! V | ' an isometry} the orthogonal group of(V,�).

Given a place v 2 ⌦, let Kv and Vv := V ⌦K Kv be the completions of Kand V at v.

Then � extends to some bilinear form � : Vv ⇥ Vv ! Kv.Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 2 / 27

Theorem (Local-Global Principle, Hasse-Kneser-Landherr-Springer)

The spaces (V,�) and (V

0,�

0) are isometric if and only if the completions

(Vv,�) and (V0v ,�

0) are isometric for every place v of K.

This yields classifications of global bilinear spaces via the followinginvariants:

1 The rank m of V .2 The determinant.3 The signature of (Vv,�) at the real places v of K.4 The finite set {p 2 P | Hasse(Vp,�) = �1}.

Definition

A bilinear space (V,�) over K is said to be (totally positive) definite if

1K is totally real

2�(x, x) is totally positive for all nonzero x 2 V .

We will concentrate on definite spaces for most of the talk.Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 3 / 27

Deciding if two bilinear spaces are isometric is trivial (compare invariants!).

Constructing one with given invariants is more di�cult:1 The case m = 1 is trival (determinant).2 If m � 3 the local-global principle yields some a 2 K⇤ represented by

�. So(V,�) = hai ? (V 0,�0)

for some space (V 0,�0) for which we know its invariants.3 So this leaves the case m = 2:

1 Let S = {v 2 ⌦ | Hasse(Vv,�) = �1}.2 Let a 2 K⇤ such that a /2 (K⇤v )2 for all v 2 S.3 There exists some b 2 K⇤ supported at (S \ P) [ {q} with q “small”

such that{v 2 ⌦ | (a, b)v = �1} = S

and ha, bi has the correct signature at the real places of K.One can find such a b using linear algebra over F2 provided that oneknown tha class group Cl(K) and generator for the unit group Z⇤K .

4 Then (V,�) ⇠=

ha, bi.This is the same as “Finding a quaternion algebra with givenramification”.


Definition

Let L be a lattice in a bilinear space (V,�), i.e. a finitely generatedZK-submodule of V such that KL = V .

1 The dual L# = {x 2 V | �(x, L) ✓ ZK} is also a lattice.2

L is called unimodular if L = L#.

3 More generally, if L = aL# for some fractional ideal a of ZK , then Lis called a-modular.

4 Lattices L,L0 in bilinear spaces (V,�) and (V 0,�0) are isometric ifthere exists some isometry ' : (V,�) ! (V 0,�) such that '(L) = L0.

5 The automorphism group of L is

Aut(L) = {' : L ! L | ' an isometry } .

Similarly, one can define isometries between Lp and L0p whereLp = L⌦ZK ZKp is the completion of L at a prime ideal p of ZK .


Pseudo bases

A lattice L in V is represented e�ciently on a computer using apseudo-basis

L =

mM

i=1

aixi where xi 2 V and fractional ideals ai of K.

Using pseudo-bases one can perform basic operations like comparison,addition, intersection,...But is also good for taking completions at p 2 P, since Lp has thebasis

(⇡

ordp(a1)x1, . . . ,⇡

ordp(am)xm)

where ⇡ 2 K denotes a uniformizer of p.Pseudo bases are also useful for local “manipulations”. For example,to compute maximal sublattices X1, . . . , Xr of L that contain pL, dothis:

1 Let M be the lattice with the basis above basis.2 Since M is free and M/pM ⇠

=

(ZK/p)m, one can write down themaximal sublattices Y1, . . . , Yr of M that contain pM .

3 Set Xi = (Yi + pL) \ L.Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 6 / 27

Jordan decomposition

A variation of the Gram-Schmidt orthogonalization shows that

Lp⇠=

L0 ? L1 ? . . . ? Lswhere Li is pi-modular.

If p - 2, then the isometry class of Lp is uniquely determined by

rank(L0), . . . , rank(Ls) and det(L0), . . . , det(Ls) .

The description of the isometry classes in the case p | 2 is much moreinvolved and is due to O’Meara.


Example

Suppose K = Q. Let L and L0 be lattices with Gram matrices✓

1 1/2

1/2 8

◆and

✓2 1/2

1/2 4

◆.

The envelopping quadratic spaces of L and L0 are isometric and Lp ⇠= L0pfor all primes p but L 6⇠

=

L

0 since L represents 1 while L0 does not.

So the local-global principle does not hold for lattices. This leads to thefollowing definition.

Definition

The class and the genus of a lattice L in a bilinear space (V,�) are

cls(L) = {L0 ⇢ V | L0 a lattice such that L ⇠=

L

0}gen(L) = {L0 ⇢ V | L0 a lattice such that Lp ⇠= L0p for all p 2 P} .


Theorem (Borel)

The genus decomposes into finitely many isometry classes

gen(L) =h]

i=1

cls(Li)

and h(L) = h(gen(L)) = h is called the class number of L or of gen(L).

The local-global principle holds for L if and only if h = 1. Otherwise hmeasures “by how much” it fails.

Problems1 Find some lattice in a given genus.

2 Decide if L0 2 cls(L).3 Find representatives L1, . . . , Lh of the isometry classes in gen(L).


A lattice M in (V,�) is called maximal, if �(x, x) 2 ZK for all x 2 M andM is maximal with this property. The maximal lattices always form asingle genus.

Finding a lattice L in a genus G given by invariants1 Let M be a maximal lattice in (V,�).

2 Let P = {p 2 P | Lp 6⇠= Mp for L 2 G}.3 For p 2 P , find a sublattice X(p) of M such that

X

(p)p

⇠=

Lp for L 2 G and X(p)q = Mq for q 6= p .4

L :=

Tp2P X

(p) 2 G does the trick.

Step (3) is easy if p - 2 (manipulate a Jordan decomposition of M).For p | 2, one can do it like that: Let Y be a lattice with Yp ⇠= Lp.Construct a minimal chain of overlattices

Y = Y

(0) ( Y (1) ( . . . ( Y (r)

such that Y (r)p is maximal and Y(i) ✓ p�1Y (i�1).

Find a lattice X(r�1) between pM and M such that X(r�1)p ⇠= Y (r�1)p etc.Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 10 / 27

Computing isometries of definite lattices I

Suppose first K = Q and let L be a lattice in a definite space (V,�).Let (b1, . . . , bm) be a basis of L and B > 0.

First: Enumerate LB := {x 2 L | �(x, x) B}The Finke-Pohst method is based on the Cholesky decomposition: Thereare qi,j 2 Q such that

�(x, x) =

mX

i=1

qi,i

0

@xi +

mX

j=i+1

qijxj

1

A2

for all x =P

i xibi 2 L.

Then �(x, x) B implies x2mqm,m B. Hence there are only finitelymany possibilities for xm.

Similarly, qm�1,m�1(xm�1 + qm�1,mxm)2 B � qm,mx2m. Thus for fixedxm there are only finitely many possibilities for xm�1, etc.

So LB is finite and can be enumerated by backtracking.Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 11 / 27

Computing isometries of definite lattices II

The following algorithm computes an isometry ' : L ! L0 between latticesL,L

0 in definite spaces (V,�) and (V 0,�0).

Plesken & Souvignier1 Let B > 0 such that LB := {x 2 L | �(x, x) B} generates L.2 Suppose {b1, . . . , bm} ✓ LB generates V , so ' is uniquely

determined by '(bi) 2 L0B.3 If '(b1), . . . ,'(bi�1) are already chosen, pick '(bi) 2 L0B such that

�(bi, bj) = �0('(bi),'(bj)) for all 1 j i .

If no such image '(bi) exists, backtrack and choose a di↵erent imagefor bi�1.

A modification can be used to compute generators of Aut(L).


There are several tricks that speed up this search

1 Every isometry ' must respect the fingerprint

#{y 2 L=D | �(x, y) = c}for D 2 {'(x, x) | x 2 LB} and c 2 {'(x, y) | x, y 2 LB}.

2 R. Bacher associates to any v 2 L with ` := �(v, v) a polynomialBv(T ) 2 Z[T ] as follows.For w 2 Wv := {x 2 L | �(x, x) = `,�(x, v) = `/2}. Let

nw = #{(x, y) 2 W 2v | �(x,w) = �(y, v) = �(x, y) = `/2}.Then Bv(T ) :=

Pw2Wv T

nw . Since Bv is defined by scalar products,we have Bv = B'(v) for each isometry '.

3 W. Unger uses J. Leon’s ideas on partition refinement to speed up thebacktrack search in recent versions of Magma.

4' induces isometries between certain canonical sub/overlattices of Land L0. E.g. between ⇢p(L) and ⇢p(L0) where ⇢p is Watson p-maps(more later).


Computing isometries of definite lattices III

Obvious changes to the above method only computes isometries L ! Lwhich preserve some additional bilinear forms.

Suppose now K 6= Q and let L be a ZK-lattice in a definite bilinear space(V,�). For a 2 K,

�a : V ⇥ V ! Q, (x, y) 7! TrK/Q(a�(x, y))

defines a bilinear form on the Q-vector space VQ.

Note that �1 is positive definite. Further, for any Z-linear map ' : L ! L,the following statements are equivalent:

' is an isometry in (V,�).

' is an isometry in (VQ,�1) which preserves �a where K = Q(a).The maps ' satisfying the latter property can be enumerated as seenbefore.


How to compute h(L)?

Problem:

The orthogonal group O(V,�) does not have the strong approximationproperty.

1 For an anisotropic vector v 2 V , the reflection

⌧v : V ! V, w 7! w � 2�(w, v)�(v, v)

v

is isometry on (V,�) and O(V,�) is generated by reflections.2 The spinor norm is the homomorphism defined by

✓ : O(V,�) ! K⇤/(K⇤)2, ⌧v 7! �(v, v) .The subgroup

S(V,�) = {' 2 O(V,�) | det(') = 1 and ✓(') = 1}does have the strong approximation property.


1 Two lattices L and M in (V,�) are said to be in the same spinorgenus if there exists some ' 2 O(V,�) such that

'(L)p = �p(Mp) with �p 2 S(Vp,�) for all p 2 P.

The spinor genus of L is denoted by sgen(L).

2 Every genus is a finite union of spinor genera.

3 Every spinor genus is a finite union of isometry classes.

Step (2) from above is completely understood and can be madeexplicit in terms of a ray class groups of K and the spinor normgroups of Lp. The latter can be computed by work of Kneser and Beli.

If (V,�) is indefinite, then cls(L) = sgen(L).

So we have to discuss step (3) for definite spaces.


Representatives of the isometry classes in sgen(L).

Let (V,�) be definite and let L be a lattice in (V,�) and let p 2 P.W.l.o.g. L ✓ L#, p - 2, Lp is unimodular and (Vp,�) is isotropic.Definition

A lattice L0 in V is called a p-neighbour of L, if L0p is unimodular and

L/(L \ L0) ⇠=

L

0/(L \ L0) ⇠

=

ZK/p as ZK-modules.

1 There exists an explicit finite subset X ⇢ L such that{{y 2 L | �(x, y) 2 p}+ p�1x | x 2 X}

is the set of all p-neighbours of L.

This allows us to compute the set of all p-neighbours quickly.2 Every p-neighbour of L lies in gen(L).3 Every isometry class in sgen(L) is represented by an iterated

p-neighbour of L.


Hermitian spaces

Let E be one of the following K-algebras with involution : E ! E:1

E = K and = idK .

2E/K a quadratic field extension with Gal(E/K) = {idE , }.

3E a quaternion algebra over K (i.e. a four-dimensional central simpleK-algebra) and the canonical involution on E.

Definition

A hermitian space is a finite dimensional (left) vector space V over Eequipped with a regular sesquilinear form � : V ⇥ V ! E such that

�(↵x+ x

0, y) = ↵�(x, y) + �(x

0, y) for all ↵ 2 E and x, x0, y 2 V ,

�(x, y) = �(y, x) for all x, y 2 V .


Definite lattices with class number BLet O be a maximal order in E and let L be a lattice in a hermitian space(V �) over E.

1 The definitions of isometries, genera, class numbers, etc. carry overto hermitian lattices.

2 The algorithms from before can be extended to lattices in hermitianspaces.

3 If (V,�) is indefinite, then the class number of L is again known apriori and depends only on some invariants of L and E (strongapproximation).

4 If (V,�) is definite, the class number of L is not known a priori, butit can be computed using Kneser’s method just as before.

GoalClassify all one-class genera of lattices in definite hermitian spaces.

Known: There are only finitely many such genera (up to similarity).Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 19 / 27

Watson’s transformations

Suppose E = K. For p 2 P define

⇢p(L) := L+ (p�1

L \ pL#)

If Lp = L0 ? . . . ? Ls is a Jordan decomposition, thenh(L) � h(⇢p(L)).⇢p(Lp) = (L0 ? p�1L2) ? (L1 ? p�1L3) ? p�1(L4 ? . . . ? Ls)⇢p(L) = L () Lp = L0 ? L1 if this is the case, then Lp is calledsquare-free.

Idea:It su�ces to enumerate the definite, square-free hermitian lattices withclass number 1.


One-class genera

Definition

If gen(L) =Uh

i=1 cls(Li), then Mass(L) :=Ph

i=11

#Aut(Li)is the mass of

L.

To avoid case distinctions, suppose that E/K is a CM-extension withn = [K : Q]. The other cases work along the same lines.

Let � be the non-trivial character of Gal(E/K). Set

LK(1, s) = ⇣K(s) the Dedekind zeta funkcion of K

LK(�, s) = ⇣E(s)/⇣K(s) the L-function attached to �


Siegels Maßformel

Theorem (Siegel, 1935)

Let L be a lattice in a definite hermitian space over E of rank m. Then

Mass(L) = 21�nm ·mY

i=1

|LK(�i, 1� i)| ·Y

p

�(Lp)

with some local factors �(Lp) 2 Q>0.

The local factors are known in almost all cases, e.g.

if 2 /2 p or p is unramified in E (Gan&Yu).if Lp is maximal (Shimura).

if 2 2 p is ramified in E and Kp/Q2 is unramified (Cho).if Lp is square-free (K.)

Fact: �(Lp) 2 12Z and if �(Lp) = 12 then m is odd and p ramifies in E.Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 22 / 27

The possible spaces

Suppose L is square-free, has rank m � 3 and h(L) B. Siegel’s Maßformula and the analytic class number formula show that

B � #µ(E) ·Mass(L)�

✓2⇡

�m

◆n· d(m2�1)/2K · NrK/Q(dE/K)m

0�1/2 ·Y

p

�(Lp)

| {z }�1

. (?)

where

µ(E) = roots of unity in E⇤.

m

0=

m(m�(�1)m)4 � 3.

�m =Qm

i=1(2⇡)i

(i�1)! .

dK = absolute value of the discriminant of K.

dE/K = relative discriminant of E/K.


The possible spaces

1 (?) yields an upper bound on the root discriminant d1/nK . For exampleB = 2 implies that m 16 and

m 3 4 5 6 7 8 9 ...

d

1/nK < 9.13 6.83 5.49 4.59 3.95 3.47 3.09

J. Voight lists all totally real number fields K with d1/nK 14, K.2 For fixed K, (?) yields all possibilities for dE/K . Class field theory E.

3 If det(Vp,�) /2 Nr(E⇤p ), then

p | dE/K or �(Lp) � 12

Nr(p)m�1 .

Hence (?) yields all possibilities for

{p | det(Vp,�) /2 Nr(E⇤p )} .All in all: Only finitely many combinations for K,E, (V,�).


The squarefree lattices in (V,�) with class number B

Let L be a squarefree lattice in a definite hermitian space (V,�) with classnumber B.

Let M be a maximal lattice in (V,�). W.l.o.g. L ✓ M .If Mp 6= Lp then

p | dE/K or �(Lp) � 12

Nr(p)m�1 .

Hence (?) yields all possibilities for

{p 2 P | Mp 6= Lp} .

This gives a bound a 2 N which does not depend on L such thataM ✓ L ✓ M .


Binary hermitian lattices

If m = dimE(V ) = 2, one gets all possible fields K just as before. But

B � #µ(E) ·Mass(L) � (2⇡)�2n · #Cl(O)Q ·#Cl(ZK) · d

3/2K · ⇣K(2)

where Q = [O⇤ : µ(E)Z⇤K ] 2 {1, 2} is Hasse’s unit undex.

For example K = Q(p5) and B = 2 yields #Cl(O) 120. If E/Q is

cyclic, Louboutin shows that

NrK/Q(dE/K) < 1.68 · 1011 .Without the assumptions on E/Q the bound becomes even worse. Som = 2 is out of reach with current methods.

If K = Q and B = 2 then #Cl(O) 48. All imaginary quadratic numberfields with class number 100 were enumerated by Watkins in his theis.So K = Q is feasible.


Some results

Similarly, if dimK(V ) = 2 this lead to relative class number problems(Gauß) which are currently out of reach.

In all other cases, a complete classification is possible. Below are thenumber of genera with class number one

case h(L) = 1 max rank(L) max [K : Q] #E #K #generaE = Q 10 1884 (Watson)

E = K 6= Q 6 5 29 4019 (Lorch)dimK(E) = 2 8 3 10 5 164

dimK(E) = 4 4 5 69 29 �and with class number two

case h(L) = 2 max rank(L) max [K : Q] #E #K #generaE = Q 16 7283

E = K 6= Q 8 5 75 17.064dimK(E) = 2 9 4 19 9 406

dimK(E) = 4 5 8 148 60 �Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 27 / 27

Bilinear spacesDefinitionsConstructing a bilinear space with given invariants

LatticesDefinitionsGeneraFinding a lattice in a given genusIsometries and automorphism groupsKneser's neighbour method

One-class generaHermitian spacesWatson's transformationsSiegel's mass formulaThe possible spacesThe squarefree latticesBinary hermitian latticesResults

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RWTH Aachen University - phpLDAPadmin (1.2.1.1) › ~haensch › index_files › Kirschmer.pdf1 Abilinearspace(V,) is a ﬁnite dimensional vector space V over K equipped with some