For a topological manifold $M$, one can speak of the cohomology ring structure $H^*(M, k)$ where $k$ is a ring. If one replace $M$ by an arithmetic schemes $X$ over a base ring $S$, and replace $k$ by a torsion sheaf $\mu_n$, then one can define the "cup product" $H^i(X, \mu_n)\times H^j(X, \mu_n)\to H^{i+j}(X, \mu_n^2)$. However, if $S$ contains the n-th root of unity, then one can "untwist" the torsions by the canonical isomorphism $\mathbb{Z}/n\cong \mu_n^r$. (E.g., One can view the isomorphism of etale sheaves $\mu_n\cong \mathbb{Z}/n$ as induced by the isomprhism of the corresponding group schemes that represent them. ) Then one indeed has a cohomological "ring" structure for $H^*(X, \mathbb{Z}/n)$. My question was, people don't seem to be using this information a lot to talk about properties of arithmetic schemes. Maybe I am wrong. To my knowledge, even when $X$ is an elliptic curve over a local field $\mathbb{Q}_p$, with coefficients $\mathbb{Z}/2$, not much is obvious to me. In particular, does the congruence condition on $p$ make a difference? (My real question was, how does "arithmetic" play with "geometry" in this sense?)
I computed this "ring" structure for elliptic curves, but I am also worried that this may be trivial in the eye of the experts. I am not sure if MO will be a good place to post this, anyway.
