Return to Question

In page 16 of these notes on $p$-adic $L$-functions, it makes the following claim:

Let $\alpha$ be a $p$-adic measure on $\mathbf{Z}_p$ which corresponds to a power series $F_{\alpha}(T) \in \mathbf{Z}_p[[T]]$ under the Iwasawa isomorphism. Fix any $a \in \mathbf{Z}_p^{\times}$. Then the measure $\alpha \circ (a)$ corresponds to the power series $F_{\alpha}((1+T^{a})-1) \in \mathbf{Z}_p[[T]]$ under the Iwasawa isomorphism.

My question is: what exactly does $F_{\alpha}((1+T^{a})-1)$ mean? That is, for a general $a \in \mathbf{Z}_p^{\times}$, the expression $(1+T^{a})-1$ is a power series (not necessarily a polynomial). So when we write $F_{\alpha}((1+T^{a})-1)$, we are plugging in a power series into another power series.

Does it make sense to do this? Even once we expand out all the brackets and group all like terms, wouldn't we still have to add infinitely many coefficients for each $T^n$ term? So we'd have to worry about convergence issues for the coefficients. Is it known that the coefficients for each sum would always converge?

In page 16 of these notes on $p$-adic $L$-functions, it makes the following claim:

Let $\alpha$ be a $p$-adic measure on $\mathbf{Z}_p$ which corresponds to a power series $F_{\alpha}(T) \in \mathbf{Z}_p[[T]]$ under the Iwasawa isomorphism. Fix any $a \in \mathbf{Z}_p^{\times}$. Then the measure $\alpha \circ (a)$ corresponds to the power series $F_{\alpha}((1+T)^{a}-1) \in \mathbf{Z}_p[[T]]$ under the Iwasawa isomorphism.

My question is: what exactly does $F_{\alpha}((1+T)^{a}-1)$ mean? That is, for a general $a \in \mathbf{Z}_p^{\times}$, the expression $(1+T)^{a}-1$ is a power series (not necessarily a polynomial). So when we write $F_{\alpha}((1+T)^{a}-1)$, we are plugging in a power series into another power series.

Does it make sense to do this? Even once we expand out all the brackets and group all like terms, wouldn't we still have to add infinitely many coefficients for each $T^n$ term? So we'd have to worry about convergence issues for the coefficients. Is it known that the coefficients for each sum would always converge?

In page 16 of these notes on $p$-adic $L$-functions, it makes the following claim:

Let $\alpha$ be a $p$-adic measure on $\mathbf{Z}_p$ which corresponds to a power series $F_{\alpha}(T) \in \mathbf{Z}_p[[T]]$ under the Iwasawa isomorphism. Fix any $a \in \mathbf{Z}_p^{\times}$. Then the measure $\alpha \circ (a)$ corresponds to the power series $F_{\alpha}((1+T^{a^{-1}})-1) \in \mathbf{Z}_p[[T]]$ under the Iwasawa isomorphism.

My question is: what exactly does $F_{\alpha}((1+T^{a^{-1}})-1)$ mean? That is, for a general $a \in \mathbf{Z}_p^{\times}$, the expression $(1+T^{a^{-1}})-1$ is a power series (not necessarily a polynomial). So when we write $F_{\alpha}((1+T^{a^{-1}})-1)$, we are plugging in a power series into another power series.

Does it make sense to do this? Even once we expand out all the brackets and group all like terms, wouldn't we still have to add infinitely many coefficients for each $T^n$ term? So we'd have to worry about convergence issues for the coefficients. Is it known that the coefficients for each sum would always converge?

In page 16 of these notes on $p$-adic $L$-functions, it makes the following claim:

Let $\alpha$ be a $p$-adic measure on $\mathbf{Z}_p$ which corresponds to a power series $F_{\alpha}(T) \in \mathbf{Z}_p[[T]]$ under the Iwasawa isomorphism. Fix any $a \in \mathbf{Z}_p^{\times}$. Then the measure $\alpha \circ (a)$ corresponds to the power series $F_{\alpha}((1+T^{a})-1) \in \mathbf{Z}_p[[T]]$ under the Iwasawa isomorphism.

My question is: what exactly does $F_{\alpha}((1+T^{a})-1)$ mean? That is, for a general $a \in \mathbf{Z}_p^{\times}$, the expression $(1+T^{a})-1$ is a power series (not necessarily a polynomial). So when we write $F_{\alpha}((1+T^{a})-1)$, we are plugging in a power series into another power series.

Does it make sense to do this? Even once we expand out all the brackets and group all like terms, wouldn't we still have to add infinitely many coefficients for each $T^n$ term? So we'd have to worry about convergence issues for the coefficients. Is it known that the coefficients for each sum would always converge?

In page 16 of these notes on $p$-adic $L$-functions, it makes the following claim:

Let $\alpha$ be a $p$-adic measure on $\mathbf{Z}_p$ which corresponds to a power series $F_{\alpha}(T) \in \mathbf{Z}_p[[T]]$ under the Iwasawa isomorphism. Fix any $a \in \mathbf{Z}_p^{\times}$. Then the measure $\alpha \circ (a)$ corresponds to the power series $F_{\alpha}((1+T^{a^{-1}})-1) \in \mathbf{Z}_p[[T]]$ under the Iwasawa isomorphism.

My question is: I am confused about what exactly $F_{\alpha}((1+T^{a^{-1}})-1)$ means. Because for a general $a \in \mathbf{Z}_p^{\times}$, the expression $(1+T^{a^{-1}})-1$ is a power series (not necessarily a polynomial). So when we write $F_{\alpha}((1+T^{a^{-1}})-1)$, we are plugging in a power series into another power series.

Does it make sense to do this? Even once we expand out all the brackets and group all like terms, wouldn't we still have to add infinitely many coefficients for each $T^n$ term? So we'd have to worry about convergence issues for the coefficients. Is it known that the coefficients for each sum would always converge?

In page 16 of these notes on $p$-adic $L$-functions, it makes the following claim:

Let $\alpha$ be a $p$-adic measure on $\mathbf{Z}_p$ which corresponds to a power series $F_{\alpha}(T) \in \mathbf{Z}_p[[T]]$ under the Iwasawa isomorphism. Fix any $a \in \mathbf{Z}_p^{\times}$. Then the measure $\alpha \circ (a)$ corresponds to the power series $F_{\alpha}((1+T^{a^{-1}})-1) \in \mathbf{Z}_p[[T]]$ under the Iwasawa isomorphism.

My question is: what exactly does $F_{\alpha}((1+T^{a^{-1}})-1)$ mean? That is, for a general $a \in \mathbf{Z}_p^{\times}$, the expression $(1+T^{a^{-1}})-1$ is a power series (not necessarily a polynomial). So when we write $F_{\alpha}((1+T^{a^{-1}})-1)$, we are plugging in a power series into another power series.

Does it make sense to do this? Even once we expand out all the brackets and group all like terms, wouldn't we still have to add infinitely many coefficients for each $T^n$ term? So we'd have to worry about convergence issues for the coefficients. Is it known that the coefficients for each sum would always converge?