For modular elliptic curves over number fields of narrow class one, and with multiplicative reduction at a collection p-adic primes, we define new invariants. Inspired by Nekovář Scholl's plectic conjectures, believe these invariants control the Mordell–Weil group higher rank support our expectations numerical experiments.

Abstract When is an ideal of a ring radical or prime? By examining its generators, one may in many cases definably and uniformly test the ideal’s properties. We seek to establish such definable formulas rings p -adic power series, as $\mathbb Q_{p}\langle X\rangle $ , Z_{p}\langle related series over more general valuation their fraction fields. obtain definable, uniform for radicality, and, on...