# Genus and Spinor genus of a lattice

Hi, I'm looking for a motivation for the names genus and spinor genus of a lattice (and spinor norm of an isometry).

Is there any relation between the genus of a lattice and the genus of an algebraic curve ?

Is the spinor norm used in number theory related to physics ? I mean, if I have an isometry in a quadratic or skew-hermitian space and I calculate its spinor norm... Is there a physical interpretation for that ?

Thanks a lot !!

• The term spinor just means an element of a certain representation of an orthogonal group (namely a spin representation) (or maybe they have the spin group itself lying around). The group is usually considered as the underlying symmetry of the physics that's going on. Similarly, when a physicist says the word vector, they usually implicitly mean an element of the standard representation of O(3) (or a bigger O(n)) that transforms under the quotient SO(3). This is why they have the term pseudovector: these flip under reflections. – Rob Harron Oct 13 '11 at 14:46
• The genus of a lattice developed from the genus of quadratic forms. See Frei, On the development of the genus of quadratic forms, Ann. Sci. Math. Quebec 3, 5-62 (1979). – Franz Lemmermeyer Oct 14 '11 at 18:19

Given a quadratic space $(V, q)$, and an orthogonal transformation $\sigma \in O(V,q)$, the spinor norm of $\sigma$ is nonzero if and only if $\sigma$ is not in the image of the canonical map $Pin(V,q) \to O(V,q)$. If $\sigma \in SO(V,q)$, we can say the same using the map $Spin(V,q) \to SO(V,q)$. It is a refined way to measure the failure of an element to come from the spin group, and you can view it as a boundary map in Galois cohomology (see, e.g., Wikipedia).
I don't know of a deep connection to physics, but it does come up in the following sense: if $(V,q)$ is an indefinite real space of dimension at least 3, like $\mathbb{R}^{3,1}$, then the pin group only has 2 connected components, while the orthogonal group has 4 - we can reflect in vectors of either positive or negative norm to get P or T type discrete symmetries. The pin group only maps to the subgroup of $O(3,1)$ generated by reflections in positive norm vectors, as these are precisely the transformations with positive spinor norm. The spin group is connected, and maps to the connected group $SO_0(3,1)$, while $SO(3,1)$ contains PT symmetries that reverse orientation of both space and time.
Adding to the previous answer: Actually in its original form due to Eichler and Kneser one uses not $Pin$ but the spin group, which is the two fold simply connected cover of the special orthogonal group and has its name from its use in quantum theory.