On the arithmetic of polynomials over a number field

TL;DR

This paper extends classical number theory results to polynomial rings over number fields, exploring quadratic forms, reciprocity laws, and properties of elliptic curves in this context.

Contribution

It introduces new analogues of classical theorems for polynomial rings over number fields, including quadratic reciprocity and genus theory, and applies invariant theory to elliptic curves.

Findings

Proves an analogue of Gauss's principal genus theorem for polynomial coefficients.

Analyzes failures of quadratic reciprocity in polynomial rings.

Counts cyclic subgroups in Jacobians of hyperelliptic curves.

Abstract

Counterparts of several classical results of number theory are proven for the ring of polynomials with coefficients in a number field. A theorem of Milnor that determines the Witt ring of a function field is applied to prove an analogue of Gauss's principal genus theorem for binary quadratic forms with polynomial coefficients. This is used to help understand when and why quadratic reciprocity fails in these polynomial rings. Another application is a count of the number of cyclic subgroups whose order is divisible by four in the primary decomposition of the torsion subgroup of the Jacobian of certain hyperelliptic curves. Invariant theory is applied to prove an analogue of a classical theorem of Fueter to give criteria for an elliptic curve with a polynomial discriminant and zero $j$-invariant to have no affine points over the associated function field.