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Algorithms for lattices
Markus Kirschmer
RWTH Aachen University
June 2016
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 1 / 27
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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
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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
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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”.
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 4 / 27
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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 .
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 5 / 27
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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
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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.
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 7 / 27
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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} .
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 8 / 27
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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).
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 9 / 27
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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
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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
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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).
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 12 / 27
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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).
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 13 / 27
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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.
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 14 / 27
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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.
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 15 / 27
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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.
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 16 / 27
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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.
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 17 / 27
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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 .
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 18 / 27
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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
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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.
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 20 / 27
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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 �
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 21 / 27
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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
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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.
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 23 / 27
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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,�).
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 24 / 27
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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 .
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 25 / 27
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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.
Markus Kirschmer (RWTH Aachen University) Algorithms for lattices June 2016 26 / 27
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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