1 results
On the
$\mu$-invariant of two-variable 
$2$-adic 
$\boldsymbol{L}$-functions
- Part of
-
- Journal:
- Glasgow Mathematical Journal , First View
- Published online by Cambridge University Press:
- 24 February 2025, pp. 1-31
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- Article
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Let
$K={\mathbb {Q}}(\sqrt {-7})$ and 
$\mathcal {O}$ the ring of integers in 
$K$. The prime 
$2$ splits in 
$K$, say 
$2{\mathcal {O}}={\mathfrak {p}}\cdot {\mathfrak {p}}^*$. Let 
$A$ be an elliptic curve defined over 
$K$ with complex multiplication by 
$\mathcal {O}$. Assume that 
$A$ has good ordinary reduction at both 
$\mathfrak {p}$ and 
${\mathfrak {p}}^*$. Write 
$K_\infty$ for the field generated by the 
$2^\infty$–division points of 
$A$ over 
$K$ and let 
${\mathcal {G}}={\mathrm {Gal}}(K_\infty /K)$. In this paper, by adopting a congruence formula of Yager and De Shalit, we construct the two-variable 
$2$-adic 
$L$-function on 
$\mathcal {G}$. Then by generalizing De Shalit’s local structure theorem to the two-variable setting, we prove a two-variable elliptic analogue of Iwasawa’s theorem on cyclotomic fields. As an application, we prove that every branch of the two-variable measure has Iwasawa 
$\mu$ invariant zero.
