In the 1st page of the introduction of Hazewinkel's Formal Groups and Applications book, there are two ways of constructing formal groups (law):

$\bullet$ Given a Lie group $G$, one can define a formal group $F$ around the identity $e \in G$. Further, one can associate a Lie algebra $\mathfrak{g}$ with every formal group $F$. That is, the formal group $F$ is a bridge between the Lie group $G$ and the Lie algebra $\mathfrak{g}$.

$\bullet$ Given a L-function $L(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^s},~a(n) \in \mathbb Z$, one can associate a formal power series $f_L(X)=\sum_{n=1}^{\infty}\frac{a(n)}{n}X^n \in \mathbb Q[[X]]$ and hence a formal group $F_L(X,Y)=f_L^{-1}(f_L(X)+f_L(Y))$.

Then Hazewinkel mentions that the formal groups $F$ and $F_L$ are not independent, that is, there is a relation between them.

$(1)$ What is the relation between $F$ and $F_L$? For arbitrary L-function, do we know the relation ?

As an example, Hazewinkel mentions that the formal group (formal completion) of an elliptic curve $E$ over $\mathbb Q$ gives some beautiful results concerning the zeta function of $E$. Regarding this, there is a famous work by T. Honda in the paper [Formal groups and zeta functions, Osaka J. Math. 5 (1968), 199–213.] (https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-5/issue-2/Formal-groups-and-zeta-functions/ojm/1200692167.full). For,

let $d$ be a discriminant of a quadratic number field $K=\mathbb Q(\sqrt d)$ and let $L'(s)=\sum_{n=1}^{\infty} \left(\frac{d}{n}\right)X^n$ be a Dirichlet L-function, where $\left(\frac{d}{n}\right)$ is Kronecker sysmbol. Then Honda says that the formal group associated with $L'$ is isomorphic to $F(X,Y)=X+Y+\sqrt d XY$ over the ring of integers of $K$.

$(2)$ What are the unsolved problems along this direction ?

$(3)$ Is there any generalization of Honda's work beyond quadratic extension of $\mathbb Q$? e.g., for cubic or any finite abelian extension

More specifically, suppose we are an arbitrary L-function; do we know the associated formal group (law)?

Note: I have been studied about formal groups over $p$-adc number field in my PhD works and so I want to continue my investigation in formal groups.

Your suggestions and guidance are much appreciated.

In the 1st page of the introduction of Hazewinkel's Formal Groups and Applications book, there are two ways of constructing formal groups (law):

$\bullet$ Given a Lie group $G$, one can define a formal group $F$ around the identity $e \in G$. Further, one can associate a Lie algebra $\mathfrak{g}$ with every formal group $F$. That is, the formal group $F$ is a bridge between the Lie group $G$ and the Lie algebra $\mathfrak{g}$.

$\bullet$ Given a L-function $L(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^s},~a(n) \in \mathbb Z$, one can associate a formal power series $f_L(X)=\sum_{n=1}^{\infty}\frac{a(n)}{n}X^n \in \mathbb Q[[X]]$ and hence a formal group $F_L(X,Y)=f_L^{-1}(f_L(X)+f_L(Y))$.

Then Hazewinkel mentions that the formal groups $F$ and $F_L$ are not independent, that is, there is a relation between them.

$(1)$ What is the relation between $F$ and $F_L$? For arbitrary L-function, do we know the relation ?

As an example, Hazewinkel mentions that the formal group (formal completion) of an elliptic curve $E$ over $\mathbb Q$ gives some beautiful results concerning the zeta function of $E$. Regarding this, there is a famous work by T. Honda in the paper [Formal groups and zeta functions, Osaka J. Math. 5 (1968), 199–213.] (https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-5/issue-2/Formal-groups-and-zeta-functions/ojm/1200692167.full). For,

let $d$ be a discriminant of a quadratic number field $K=\mathbb Q(\sqrt d)$ and let $L'(s)=\sum_{n=1}^{\infty} \left(\frac{d}{n}\right)n^{-s}$ be a Dirichlet L-function, where $\left(\frac{d}{n}\right)$ is Kronecker sysmbol. Then Honda says that the formal group associated with $L'$ is isomorphic to $F(X,Y)=X+Y+\sqrt d XY$ over the ring of integers of $K$.

$(2)$ What are the unsolved problems along this direction ?

$(3)$ Is there any generalization of Honda's work beyond quadratic extension of $\mathbb Q$, e.g., for cubic or any finite abelian extension?

More specifically, suppose we are given an arbitrary L-function; do we know the associated formal group (law)?

Note: I have been studied about formal groups over $p$-adc number field in my PhD works and so I want to continue my investigation in formal groups.

Your suggestions and guidance are much appreciated.

In the 1st page of the introduction of Hazewinkel's Formal groups and Applications book, there are two ways of constructing formal groups (law):

$\bullet$ Given a Lie group $G$, one can define a formal group $F$ around the identity $e \in G$. Further, one can associate a Lie algebra $\mathfrak{g}$ to every formal group $F$. That the formal group $F$ is a bridege between the Lie group $G$ and Lie algebra $\mathfrak{g}$.

$\bullet$ Given a L-function $L(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^s},~a(n) \in \mathbb Z$, one can associate a formal power series $f_L(X)=\sum_{n=1}^{\infty}\frac{a(n)}{n}X^n \in \mathbb Q[[X]]$ and hence a formal group $F_L(X,Y)=f_L^{-1}(f_L(X)+f_L(Y))$.

Then Hazewinkel mentions that the formal groups $F$ and $F_L$ are not independent, that is, there is a relation between them.

$(1)$ What is the relation of $F$ and $F_L$? For arbitrary L-function, do we know the relation ?

As an example, Hazewinkel mentions: formal group (formal compltion) of a elliptic curve $E$ over $\mathbb Q$ gives some beautiful results concerning the zeta function of $E$. Regarding this, there is a famous work by T. Honda in the paper Formal groups and zeta functions, Osaka J. Math. 5 (1968), 199-213.. For,

let $d$ be a discriminant of a quadratic number field $K=\mathbb Q(\sqrt d)$ and let $L'(s)=\sum_{n=1}^{\infty} \left(\frac{d}{n}\right)X^n$ be a Dirichlet L-function, where $\left(\frac{d}{n}\right)$ is Kronecker sysmbol. Then Honda says, the formal group associated to $L'$ is isomorphic to $F(X,Y)=X+Y+\sqrt d XY$ over the ring of integers of $K$.

$(2)$ What are the unsolved problems along this direction ?

$(3)$ Is there any generalization of Honda's work beyond quadratic extension of $\mathbb Q$? e.g., for cubic or any finite abelian extension

More specification, suppose we are an arbitrary L-function, do we know the associated formal group (law)?

Note: I have been studied about formal groups over $p$-adc number field in my PhD works and so I want to continue my investigation in formal groups.

Your suggestion/guidance is much appreciated.

In the 1st page of the introduction of Hazewinkel's Formal Groups and Applications book, there are two ways of constructing formal groups (law):

$\bullet$ Given a Lie group $G$, one can define a formal group $F$ around the identity $e \in G$. Further, one can associate a Lie algebra $\mathfrak{g}$ with every formal group $F$. That is, the formal group $F$ is a bridge between the Lie group $G$ and the Lie algebra $\mathfrak{g}$.

$\bullet$ Given a L-function $L(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^s},~a(n) \in \mathbb Z$, one can associate a formal power series $f_L(X)=\sum_{n=1}^{\infty}\frac{a(n)}{n}X^n \in \mathbb Q[[X]]$ and hence a formal group $F_L(X,Y)=f_L^{-1}(f_L(X)+f_L(Y))$.

Then Hazewinkel mentions that the formal groups $F$ and $F_L$ are not independent, that is, there is a relation between them.

$(1)$ What is the relation between $F$ and $F_L$? For arbitrary L-function, do we know the relation ?

As an example, Hazewinkel mentions that the formal group (formal completion) of an elliptic curve $E$ over $\mathbb Q$ gives some beautiful results concerning the zeta function of $E$. Regarding this, there is a famous work by T. Honda in the paper [Formal groups and zeta functions, Osaka J. Math. 5 (1968), 199–213.] (https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-5/issue-2/Formal-groups-and-zeta-functions/ojm/1200692167.full). For,

let $d$ be a discriminant of a quadratic number field $K=\mathbb Q(\sqrt d)$ and let $L'(s)=\sum_{n=1}^{\infty} \left(\frac{d}{n}\right)X^n$ be a Dirichlet L-function, where $\left(\frac{d}{n}\right)$ is Kronecker sysmbol. Then Honda says that the formal group associated with $L'$ is isomorphic to $F(X,Y)=X+Y+\sqrt d XY$ over the ring of integers of $K$.

$(2)$ What are the unsolved problems along this direction ?

$(3)$ Is there any generalization of Honda's work beyond quadratic extension of $\mathbb Q$? e.g., for cubic or any finite abelian extension

More specifically, suppose we are an arbitrary L-function; do we know the associated formal group (law)?

Note: I have been studied about formal groups over $p$-adc number field in my PhD works and so I want to continue my investigation in formal groups.

Your suggestions and guidance are much appreciated.