When the branch character has root number $- 1$, the corresponding anticyclotomic Katz $p$-adic $L$-function vanishes identically. For this case, we determine the $\mu $-invariant of the cyclotomic derivative of the Katz $p$-adic $L$-function. The result proves, as an application, the non-vanishing of the anticyclotomic regulator of a self-dual CM modular form with root number $- 1$. The result also plays a crucial role in the recent work of Hsieh on the Eisenstein ideal approach to a one-sided divisibility of the CM main conjecture.

We study the action of the formal affine Hecke algebra on the formal group algebra, and show that the the formal affine Hecke algebra has a basis indexed by the Weyl group as a module over the formal group algebra. We also define a concept called the normal formal group law, which we use to simplify the relations of the generators of the formal affine Demazure algebra and the formal affine Hecke algebra.

Let $R\subset F$ be an extension of real closed fields, and let ${\mathcal{S}}(M,R)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset R^{n}$. We prove that every $R$-homomorphism ${\it\varphi}:{\mathcal{S}}(M,R)\rightarrow F$ is essentially the evaluation homomorphism at a certain point $p\in F^{n}$ adjacent to the extended semialgebraic set $M_{F}$. This type of result is commonly known in real algebra as a substitution lemma. In the case when $M$ is locally closed, the results are neat, while the non-locally closed case requires a more subtle approach and some constructions (weak continuous extension theorem, appropriate immersion of semialgebraic sets) that have interest of their own. We consider the same problem for the ring of bounded (continuous) semialgebraic functions, getting results of a different nature.