Given a finite field F we can define one type of "discrete torus" as F^n, the vector space of dimension n over F. If |F| = p^k, then |F^n| = p^(kn). But there are many different kinds of tori. Equating R^2 with the complex plane C, there are e.g. both T_4 = C / Z[i] and T_6 = C / Z[w] where w = exp(2πi/3), to list my two favorite 2-dimensional ones. Here Z[i] is the Gaussian integers and Z[w] is the Eisenstein integers. (The notation comes from obtaining T_n from identifying the opposite edges of a regular n-gon.) These two 2-tori are not distinct topologically but metrically, since they each inherit a metric from the Euclidean plane R^2 that is the same locally — a surface of Gaussian curvature K = 0 — but which are distinct globally: There is no isometry carrying one to the other, and there is not even any global "similarity" mapping that multiplies all distances by the same factor. So I'm wondering what kinds of *discrete* tori there are, and what is the appropriate structure on them (like a metric) that may be globally distinct. One way to get other kinds of discrete tori is to imitate how T_4 and T_6 are obtained from R^2 (or from C = C^1): Find a discrete subgroup of the vector space and factor out by it. Since F(p^k) is a vector space over the prime field F(p), its additive group is just the direct sum Z_p + ... + Z_p of k copies of Z_p (where Z_p denotes Z/pZ). So we define an d-dimensional torus T over *the base field* F_p to be T = F^n / A where A is any abelian subgroup of the additive group (F,+) of the field. Just as for the real field R, T is an abelian group as the quotient of abelian groups. We'll say the "dimension" of T is d when d = log_p |T|: dim_(F_p)(T) = log_p(p^(kn) / |A|) = kn - log_p(|A|). I'm not sure what "isometric" or a "similarity" might mean *geometrically* for F^n in the absence of an obvious metric on the vector space F^n. But it makes sense to define the corresponding group of matrices over the field F: Definitions: ------------ An "orthogonal" matrix P over F is any invertible n x n matrix with entries in F such that P^t = P^(-1) (where P^t denotes the transpose). A "similarity" matrix S over F is any constant c ≠ 0 in F times an orthogonal n x n matrix P over F: S = c P. Now we can imitate the conditions under which two tori based on the field R are isometric or similar: Say T' = T (isometric) exactly when T' = P T P^(-1) for some orthogonal matrix P. Say T' ~ T (similar) exactly when T' = S T S^(-1) for some similarity matrix S. —Dan