Lattices that agree at every prime

How many lattices share a determinant counts the plane lattices whose metric has a given determinant, up to a change of basis by an integer matrix of determinant one. The answer is a class number and the computation is an enumeration of reduced forms.

There is a coarser equivalence above it, and the gap between the two is the subject here.

The question has a crystallographic form that is worth putting first, because the arithmetic below is otherwise a curiosity. Suppose two plane lattices give the same answer to every question of the form “does a vector of squared length n exist, modulo m”. Are they the same lattice? A crystallographer would reasonably expect so: congruence conditions are the whole content of systematic absences and centring rules, and if two lattices agree on all of them there seems to be nothing left to differ about. The answer is no, and the smallest counterexample fits in three lines.

Two ways for two lattices to be the same #

A plane lattice is a positive definite binary quadratic form ax² + bxy + cy², and the natural equivalence is a change of basis: two forms are the same lattice when an integer matrix of determinant one carries one to the other. Counting those classes is the class number.

The coarser one asks less. Two forms are in the same genus when they represent the same residues modulo the discriminant — equivalently, when they are equivalent over the p-adic integers for every prime and over the reals. Anything a congruence can detect is a genus invariant: a systematic absence, a parity rule, an argument modulo n for any n at all.

So the class is what a change of basis can see and the genus is what a congruence can see, and the second is weaker. The question is by how much.

The discriminant deserves a sentence, because it is the quantity everything is indexed by. For a form ax² + bxy + cy² it is b² − 4ac, and it is negative exactly when the form is positive definite — which is exactly when the form is the metric of a real lattice rather than of something indefinite. Up to a scale it is four times the determinant of the Gram matrix, so “lattices of a given discriminant” and “lattices of a given area” are the same family, and the reduction that enumerates them is the one the shortest basis performs.

The reason it takes only the values it does is worth a line too. A discriminant must be congruent to zero or one modulo four, because b² is, so the family of discriminants is half the integers rather than all of them. Feeding the machinery a number outside that set is refused rather than answered, which is one of the checks below.

One practical note about the enumeration, since it is the step everything rests on. A reduced form has |b| ≤ a ≤ c, with b ≥ 0 when a = |b| or a = c, and those conditions pick exactly one representative from each class — so counting reduced forms is counting classes, with no equivalence testing at all. The bound a ≤ √(|D|/3) follows from 4ac − b² = |D| and a ≤ c, and it is what makes the enumeration finite and quick. That is the same reduction theory two moves reach every basis describes as an algorithm; here it is used as a normal form rather than as a procedure.

The genus count is a theorem #

The genus count is not a measurement of the lattice so much as a fact about the discriminant’s factorisation.

That agreement is a check on both sides. The genus count comes from computing which residues each of the forms represents and grouping them; the prediction comes from factorising a number. The two share no step, and getting the even part of the factorisation wrong — which is the usual error, and was the first version’s — predicts twice or half the right answer on the discriminants divisible by four.

There is a structural reason the count is a power of two rather than an arbitrary number, and it is worth having because it explains why the genus is such a blunt instrument. The genus of a form is recorded by a list of signs — one per prime discriminant, each saying which of two residue classes the form’s values fall into modulo that prime. A list of t signs constrained to multiply to one has 2^(t−1) possibilities, and every possibility occurs. So the genus carries at most t − 1 bits, whatever the class number is, and a discriminant with a large class number and few prime factors necessarily has many classes per genus.

Minus seventy-one is the extreme case in the table: one prime factor, so one genus, and seven classes. Seven lattices, one bit of local information between them, and that bit is the same for all seven.

There is a second reading of the prediction which is the reason to bother with it. The genus count depends only on how the discriminant factorises, and the class number depends on much more — so the two are computed from different data, and the quotient is a quantity neither of them contains. A discriminant with one prime factor has one genus whatever its class number is, and a discriminant that is a product of many small primes has many genera and can still have a class number that outruns them. The quotient is not a function of either count alone.

One more property of the residue sets is worth measuring rather than asserting, since it is what makes the genus computable at all. The set of residues a form represents modulo the discriminant is closed under multiplication: if a form represents u and v modulo the discriminant then it represents uv. So a genus is a coset of a subgroup of the multiplicative group modulo the discriminant, and comparing two forms means comparing two cosets rather than two arbitrary sets. That is why the computation is a grouping rather than a search, and why the number of genera divides the order of a group and is therefore forced to be a power of two.

Where they part #

Minus twenty-three is the smallest case and it is worth looking at in full, because the statement is stronger than the arithmetic makes it sound.

The forms are x² + xy + 6y², 2x² − xy + 3y² and 2x² + xy + 3y², and reading them as lattices makes the difference concrete: the first has a shortest vector of squared length one and the other two have shortest vectors of squared length two. So one of the three lattices has a vector the others do not, which is about as visible a difference as two lattices can have.

The consequence is worth stating plainly. Take any statement of the form “this lattice represents n modulo m”, for any m whatever. All three lattices give the same answer. Every systematic absence, every parity argument, every congruence condition — the whole apparatus by which crystallography usually separates one arrangement from another — is blind to the difference between them. And they are different: one contains a vector of length one and the others do not.

There is an appealing way to state what the genus does and does not know, and it is worth having because it is the reason the notion exists at all. Two forms in the same genus are equivalent over the rational numbers with denominators prime to the discriminant, and over the reals — which is to say, they become the same lattice as soon as any denominator is allowed. What distinguishes them is a fact about integers alone, and integrality is not a local condition. That is the whole content of a local-global failure, in one sentence, and here it has three explicit forms attached to it.

Counting what each equivalence sees makes the hierarchy concrete. The genus of a form at minus twenty-three is one bit of information and there is only one value it takes. The set of represented integers is more, and it separates the principal class from the other two. The multiset of represented integers with multiplicities — the theta series — is more still, and it separates all three. The class is the finest of all and here it coincides with the theta series; in four dimensions it does not, which is what makes the lengths do not name the lattice a separate result rather than a corollary of this one.

The two inverse classes are worth naming as lattices rather than as forms, since that is the reading this collection cares about. They are the same lattice reflected: one is a plane lattice and the other is its mirror image, and no rotation carries one onto the other. So the class number counts oriented lattices and the count of represented sets counts unoriented ones, and the difference is exactly the number of classes that are not their own inverse. At minus twenty-three that difference is one pair.

A third equivalence, coarser still #

There is one more way two lattices can look alike, and it catches the case above out.

That is a useful corrective, because “which integers does it represent” sounds like the strongest possible measurement and it is not. Three classes, two sets of represented integers, one genus. Each equivalence is strictly coarser than the last, and the class is finer than all of them.

It is also the reason this is not a statement about theta series. How many vectors of each length counts how many times each length occurs, not merely whether it occurs, and that finer measurement does separate a form from its inverse — the two have the same represented set and different multiplicities. The lengths do not name the lattice is where even that fails, and it fails in four dimensions rather than in two.

One more reading of the multiplicities is worth setting down. The forms 2x² − xy + 3y² and 2x² + xy + 3y² are carried onto each other by y → −y, which has determinant minus one — so they are equivalent under the full group of integer matrices and inequivalent under the subgroup of determinant one. The class group’s inverse operation is exactly that reflection, and a form equal to its own inverse is one an orientation-reversing change of basis fixes. So the difference between the class number and the number of represented sets is a question about orientation, which is the same question the law that hides handedness asks about a crystal and a diffraction pattern.

There is a third thing the failure at minus twenty-three does not mean, and it is worth ruling out because the temptation is real. It does not mean the three lattices are hard to tell apart in practice — they are trivially easy, since one has a shorter vector than the others. It means that a particular kind of argument cannot tell them apart, and the kind happens to be the one that a great deal of crystallographic reasoning is made of. The obstruction is to a method rather than to a measurement.

The residue sets in that figure deserve one more look, because their size says something. Modulo twenty-three there are twenty-two residues coprime to it, and the forms represent eleven of them — exactly half, which are the quadratic residues. That is not an accident: a form’s values modulo its discriminant land in the subgroup of squares, and a genus is a coset of that subgroup. So the genus is not a fine-grained fingerprint being compared; it is a choice among a handful of cosets, and at minus twenty-three there is only one coset to choose.

There is one more caution about reading the table, and it concerns what “lattice” means here. A binary quadratic form is a lattice with a chosen orientation and scale, and two forms differing by a positive scale factor have different discriminants and are not compared. So the class number counts lattices of a fixed area rather than lattices of a fixed shape, and the moduli-space reading — the space every lattice lives in — is the continuous version of the same set with the area divided out. The discrete count here is a count of the integral points of that space at one level.

What this says about a crystal #

Two things, and the second is the one that generalises.

The first is a warning about local arguments. A great deal of crystallographic reasoning is congruence reasoning: an absence rule is a statement modulo two, a centring condition is a statement modulo the centring vector, and a superstructure argument is a statement modulo the supercell index. All of that is genus-level information, and the genus is provably coarser than the lattice. There are lattices no such argument can distinguish, and the smallest instance is small enough to draw.

The second is the shape of the failure. The genus is the “local” data — what every completion sees — and the class number is the “global” answer. That the two differ is the standard shape of a failure of a local-global principle, and it is the same shape as the two hundred and thirty against the two hundred and nineteen: two classifications that agree except where an orientation or a completion carries information the other cannot. The ratio h/g is the size of that gap and it is computable, which is more than the general theory usually offers.

The fourth is the one that keeps the result from being an artefact. Minus twenty also has two classes, and they are in two genera — so the machinery is capable of separating classes when the residues differ, and its failure to separate the three at minus twenty-three is a property of those forms rather than of the code.

A last observation about how the gap grows, since the table only reaches minus seventy-one. The class number of an imaginary quadratic discriminant grows roughly like the square root of its size, while the genus count grows like a power of two in the number of prime factors — which grows like the logarithm of the logarithm. So the quotient grows, and grows fast: for large discriminants almost all of the class number is invisible to congruences. The thirteen rows here are the small end of a phenomenon that becomes the normal case.

One more thing worth extracting from the whole exercise, because it is the shape rather than the content. Every count here was obtained by enumerating a finite set and grouping it, and every claim about the counts was then checked against a theorem that predicts them from somewhere else — the genus count against the factorisation, the class number against the reduction bound, the residue sets against the coset structure. Neither half is trusted on its own. That is the same discipline the rest of this collection applies to point groups and lattices, arriving here in a subject where the theorems are much older than the computation.

There is a small piece of history worth a sentence, since the objects are not modern. Gauss defined the genus and computed these tables by hand, two hundred years before anybody had a p-adic number to define it with — his criterion was exactly the residues, which is what is computed here. The local-global reading came later and is the one that generalises; the residue computation is the one that fits in a figure.

Where this stops #

Everything here is binary — plane lattices. In three dimensions the same two equivalences exist, the genus is still what the p-adic completions see, and the class number of a genus is still a count of lattices no congruence separates. The arithmetic is harder: genus symbols in three variables need the p-adic Jordan decomposition rather than a set of residues, and this collection does not carry it. What generalises without change is the statement, and it is the statement rather than the number that a crystallographer needs.

And the class number itself is not computed here as anything but an enumeration. There is a formula — the class number of an imaginary quadratic field is a sum over the discriminant’s residues, and it is one of the more beautiful things in the subject — and reproducing it would be a different essay about a different kind of argument. What is here is the counting and the distinction, which is what makes the counting mean something.