The Growth of CM Periods over False Tate Extensions

Abstract

We prove weak forms of Kato's K1-congruences for elliptic curves with complex multiplication, subject to two technical hypotheses. We next use Magma to calculate the µ-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 µ-invariant using arithmetic of the Z 2 p -extension.

Non-commutative Iwasawa theory for modular forms

“…In addition, for the two forms of weight 4 and conductors 7 and 121, we prove that (6) holds for all integers m > 1, and happily, our numerical results show that the sharper congruence (7) holds for these two forms and the prime p = 3 for a good range of cube free integers m > 1. When f is a complex mutliplication form, some cases of the congruence (4) have already been established theoretically by Delbourgo and Ward [3] and Kim [15]. However, when f is not a complex multiplication form, our numerical data seems to provide the first hard evidence in support of the mysterious non-abelian congruence (4) between abelian p-adic L-functions.…”

“…The work of Manin [16] proves that there exists a unique power series H(σ, T ) in the ring R = Z p [[T ]] such that (2) H(σ, ψ(γ)u r − 1) = L can p (f, σψ, k/2 + r), for all ψ in Ξ, and all integers r with −k/2 + 1 ≤ r ≤ k/2 − 1. On the other hand, the conjectural existence of a good p-adic L-function for f over the field F ∞ would imply, in particular, the existence of a power series H(ρ, T ) in the ring R such that (3) H(ρ, ψ(γ)u r − 1) = L can p (f, ρψ, k/2 + r), for all ψ in Ξ, and all integers r with −k/2 + 1 ≤ r ≤ k/2 − 1. Then Kato's work [14] implies the following conjectural congruence between formal power series (4) H(ρ, T ) ≡ H(σ, T ) mod pR.…”

p-adic L-functions over the false Tate curve extensions

“…So far, these non-abelian p-adic L-functions have only been proven to exist when J is the Tate motive, M ∞ is totally real and contains the cyclotomic Z p -extension of Q, and the finally the relevant Iwasawa μ = 0 conjecture is valid. However, several authors [1,2,5,6,7] have obtained partial results about the existence of these p-adic L-functions, when J is the motive attached to a primitive modular form and…”

Non-Abelian Congruences Between Special Values of L-Functions of Elliptic Curves: The Cm Case

“…Actually the only known results in this direction are mainly restricted to the Tate motive over particular p-adic Lie extensions as for example in [25,21,20,15]. We should also mention here that for elliptic curves there are some evidences for the existence of such non-abelian p-adic L-functions offered in [4,10] and also some computational evidences offered in [13,11].…”