Concept

Modular region — where it appears

The modular region is the part of the upper half-plane holding one point for each shape of plane lattice, up to size and rotation. It is bounded below by an arc and on the sides by two vertical lines, and it reaches upward without limit toward ever longer, thinner cells.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A lattice placed in the region, and its distance to each special shape. The modular region, with the two special points marked — the square lattice at the top of the arc and the hexagonal one at its corner — and a third lattice placed by reducing its form. The distances are hyperbolic rather than Euclidean, and the choice is forced rather than aesthetic: a distance between lattice shapes has to be unchanged by every change of basis, and the hyperbolic metric is the one defined by being invariant under exactly that group. Writing the same lattice down on three other bases and measuring again gives the same two numbers to the last digit.

How far one lattice is from another

A crystal that is nearly hexagonal twins where an exactly hexagonal one would not, and 'nearly' does real work in that sentence. Giving it a number needs a distance that no change of basis can move — which forces the geometry to be hyperbolic rather than flat.

lattices · Moduli

Discriminant −23 at the prime 2: the crater and three levels down. The lattices of discriminant minus 23 and of 4, 16 and 64 times that, joined by index-two sublattices. Three lattices form the crater, a cycle in which each has two neighbours of its own discriminant. Each crater lattice has one sublattice going down, and every lattice below has one edge up and two down. The levels hold 3, 3, 6, 12 lattices.

One way up and p ways down

Every plane lattice has exactly p + 1 sublattices of prime index p. Restrict an integer form to each and the discriminant can do only three things, so the lattices become the vertices of a graph with a definite shape: a ring of lattices at the top, and below it a tree in which every lattice has one way up and p ways down. Whether the ring exists at all is decided by how p factors, which is why the square lattice has two copies of itself at index five and none at index three.

lattices · Moduli

Named alongside it

The objects these essays reach for when they reach for this one.

Moduli space Binary quadratic form Class number Discriminant Hyperbolic distance Lattice shape Pseudo-merohedry Quadratic form Reduction Similarity Sublattice

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