sublattices of index 3 contained in a lattice $L $ in $\mathbb{R}^2$

Asked Nov 05 '24 at 06:14

Active Nov 05 '24 at 13:18

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How many sublattices of index 3 are contained in a lattice $L $ in $\mathbb{R}^2 $?

This is a problem in the book Algebra by Michael Artin.

I guess there are only two， but have no clue to find a proof.

A sublattice of index 3 can be generated by an integral matrix

$$\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$$

whose determinant is 3. The problem is that many matrices generate the same lattice.

asked Nov 05 '24 at 06:14

poisson

5

Hint: Every sublattice of $\mathbb Z^2$ of index $3$ contains the lattice $(3\mathbb Z)^2$. – Carl Schildkraut Nov 05 '24 at 06:43
1

See my answer to a more general question: https://math.stackexchange.com/questions/2368026/number-of-sublattices-of-bbb-z2/2694288#2694288 – colt_browning Nov 07 '24 at 13:04

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