Here I consider cuspidal automorphic representations $\pi$ over the similitude group $\mathrm{GSp}(4,\mathbb{A}\_\mathbb{Q})$. Let $f$ be a non-zero vector in the representation $\pi$. I want to know if there is any reference/work on relating special values of the complex adjoint $L$-function $L(s,\pi,\mathrm{Ad})$ of $\pi$ to the Petersson norm $\langle f,f\rangle$.


I know there is an article of Atsushi Ichino ('On critical values of adjoint $L$-functions for $\mathrm{GSp}(4)$') on this topic. Yet this article assumes $\pi$ to be unramified over all the finite places and to be of a special type over the archimedean place. So can we remove or weaken these assumptions? Is there any work after this?


Any comment or suggestions would be welcome.



Special values of adjoint $L$-functions of automorphic representations of $\mathrm{GSp}(4)$ as Petersson norms
For $\mathrm{GL}\_2$, the relationship between the Petersson norm of a newform $f$ and its adjoint $L$-function is roughly a statement of the form
\[\frac{|a\_f(1)|^2}{\langle f, f\rangle} = \frac{c\_f}{\Lambda(1, \operatorname{ad} f)},\]
where $a\_f(n)$ denotes the first Fourier coefficient of $f$, $\Lambda(s,\pi)$ denotes the completed $L$-function (including the archimedean components, which are essentially products of gamma functions), and $c\_f$ is an explicit constant that essentially only depends on the behaviour of $f$ at the ramified primes that one can make completely explicit. (However, I don't think this explicit formulation has been written down anywhere in the literature, though it certainly can be done by, say, combining Lemma 4.2 of [this paper of mine](https://arxiv.org/pdf/1609.06740.pdf) with the results in Section 1 of [this paper of Gelbart and Jacquet](https://eudml.org/doc/82024).)


For $\mathrm{GSp}\_4$, the Fourier expansion is of the form
\[f(Z) = \sum\_{S} a\_f(S) e^{2\pi i \operatorname{Tr}(SZ)},\]
where the sum is over matrices $S = \begin{pmatrix} a & b/2 \\ b/2 & c \end{pmatrix}$ such that $a,b,c \in \mathbb{Z}$, $a > 0$, $b^2 - 4ac = D < 0$.


We should expect that $|a\_f(1\_2)|^2/\langle f, f\rangle$ is related to a special value of $L$-functions in some way. Unlike the $\mathrm{GL}\_2$ case, this special value of $L$-functions is (conjectured to be) much more complicated for $\mathrm{GSp}\_4$. This is a special case of Böcherer's conjecture, which in this setting states that
\[\frac{|a\_f(1\_2)|^2}{\langle f, f\rangle} = c\_f \frac{\Lambda(\frac{1}{2}, f) \Lambda(\frac{1}{2}, f \otimes \chi\_{-4})}{\Lambda(1, \operatorname{ad} f)}.\]
A particular case of this has recently been proved by [Furusawa and Morimoto](https://arxiv.org/pdf/1611.05567.pdf) (Theorem 1).


Note that $a\_f(S\_1) = a\_f(S\_2)$ if $S\_1 = A^t S\_2 A$ for some $A \in \mathrm{SL}\_2(\mathbb{Z})$. This means that for any imaginary quadratic field $K = \mathbb{Q}(\sqrt{D})$, noting that that there is a bijection between the class group $\mathrm{Cl}\_K$ and equivalence classes of matrices $S$, we may study
\[R\_f(K) = \sum\_{S \in \mathrm{Cl}\_K} a\_f(S).\]
Then Böcherer's conjecture in this setting states that
\[\frac{|R\_f(K)|^2}{\langle f, f\rangle} = c\_{f,D} \frac{\Lambda(\frac{1}{2}, f) \Lambda(\frac{1}{2}, f \otimes \chi\_D)}{\Lambda(1, \operatorname{ad} f)}.\]
Taking $D = -4$, so that $S \in \mathrm{Cl}\_K$ implies that $S \sim I\_2$, yields the previous formulation. One can additionally insert a class group character into the definition of $R\_f(K)$, which leads to different $L$-functions in the conjecture.


Should one wish to make the constant $c\_f$ explicit, one can look at the work of [Dickson, Pitale, Saha, and Schmidt](https://arxiv.org/pdf/1512.07204.pdf), where they have refined this conjecture to explicitly determine the constant $c\_f$. They still require several assumptions; see Theorem 1.12 of their paper for the precise statement.


In general, these kinds of conjectures/statements are easy to state at unramified primes, but take a lot of work to understand at ramified primes; this can be seen, for example, in the work of many people (Nelson, Nelson-Pitale-Saha, Hu, Collins, etc.) of determining the explicit expression for the inner product of three $\mathrm{GL}\_2$ newforms via the Watson-Ichino formula.


