Seminar

The residue rings of integers modulo powers of a prime $p$ have been around since 300 BC, while their projective limit $\mathbb Z_p$, the ring of $p$-adic integers, was introduced in 1897. The metamathematics of the class of the residue rings has been little studied, whereas the metamathematics of the rings $\mathbb Z_p$ includes some of the main successes of model-theoretic algebra. The decidability of the class of all the residue rings modulo powers of $p$ is readily obtained from the decidability of $\mathbb Z_p$, but the interpretation involved gives almost no information about axioms or definability in the residue rings themselves. Moreover, the interpretation gives almost no explicit algorithmic information for the class of residue rings. Reflection on why this is so will remind one that our algorithmic information about $\mathbb Z_p$ is very scanty compared with what we know for the real field. In the talk, we will use heavier methods relating both to Denef's rationality results for $p$-adic Poincar\'e series (including uniformities in $p$), together with some basic theory of recurrence relations, to give fairly explicit axioms for the class of residue rings. There remain some algorithmic issues, which will be discussed.

The work, joint with Paola D'Aquino, is part of a bigger project on residue rings in models of Peano arithmetic.

Our work owes a great deal to work of Ax, Denef, and Skolem. Related work with Derakhshan solves a 60 year-old problem of Ax on residue rings $\mathbb Z/n\mathbb Z$, posed in his Annals paper. This is done using definability theory in the adeles over $\mathbb Q$.

A Closer Look At The Model Theory Of The Rings $\Mathbb Z /P^N\Mathbb Z$