The Growth of CM Periods over False Tate Extensions

Abstract

We prove weak forms of Kato's ${\rm K}_1$-congruences for elliptic curves with complex multiplication, subject to two technical hypotheses. We next use "Magma" to calculate the $\mu$-invariant measuring the discrepancy between the "motivic'' and "automorphic'' {$p$-adic} $L$-functions. Via the two-variable main conjecture, one can then estimate growth in this $\mu$-invariant using arithmetic of the $\Z_p^2$-extension.