For the sake of simplicity, assume $f$ is a non-cm eigenform of weight $k$ on the group $\mathrm{SL}(2, \mathbb{Z})$. Are there any known results or conjectures regarding any special values of the associated $L$-function $L(f, n)$ for any integers $n$ for weight $k > 2$? If so, what can be said about the rational factor appearing said special values? In the case of elliptic curve $L$-functions, BSD predicts the factor of $\#E(\mathbb{Q}_{\mathrm{tors}})^{2}$ appearing in the denominator for $L(E, 1)$. I'd like to know if something similar in the form of known results or conjectures exist for $L$-functions satisfying the criteria above. I do believe they are quite mysterious, and I have been rather curious about Ramanujan's $L$-function $L(\Delta, s)$ for $s = 11$ in particular. This is something I'm far from an expert in!
2 Answers
Since the OP is interested in particular by $L(\Delta,11)$, the relevant theorems come into the framework of Deligne's theory of special points and special values, while Bloch-Kato would be for $s>11$. Here the theorems are due to Shimura and especially Manin, which for $\Delta$ (and for general eigenforms similarly) states the following: set $r_j=(-2i\pi)^{-j-1}j!L(\Delta,j+1)$.
There exist real numbers $\omega^+$ and $\omega^-$ such that $$(r_1,r_3,r_5,r_7,r_9)=(48,-25,20,-25,48)\omega^-$$ $$(r_0,r_2,r_4,r_6,r_8,r_{10})=(22680/691,-14,9,-9,14,-22680/691)\omega^+i$$ $$\omega^-\omega^+=(4096/691)<\Delta,\Delta>$$
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1$\begingroup$ Any good book on modular forms should have this, including my own with F. Str\"omberg. Just a comment: the results for $r_1$ up to $r_9$, exactly those without $691$, are quite easy and I believe due to Eichler (not Shimura). The result for $\omega^-\omega^+$ is also easy, standard Rankin--Selberg. The more subtle result for $r_0$ and $r_{10}$ is due to Manin. $\endgroup$ Oct 28, 2022 at 18:54
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1$\begingroup$ typo: replace Eichler (not Shimura) by Eichler--Shimura. $\endgroup$ Oct 28, 2022 at 19:16
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1$\begingroup$ I'm afraid the claim "Bloch-Kato would be for s > 11" is incorrect here. The rational numbers $(48, -25, 20, \dots)$ etc aren't just random, and part of the Bloch--Kato conjecture is that for each prime $p$, the $p$-parts of these numbers are telling you something about the order of some $p$-adic Selmer group associated to $\Delta$. $\endgroup$ Nov 11, 2022 at 12:32
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1$\begingroup$ The general picture is that for a (non-zero) critical L-value, Deligne's conjectures tell you the special value up to $K^\times$ where $K$ is the coefficient field, but Bloch--Kato refines this to actually predict the special value up to a unit on $K$ (thus up to sign when $K = \mathbb{Q}$, as here). $\endgroup$ Nov 11, 2022 at 12:36
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1$\begingroup$ Thanks David, I was not aware of this precise form of B-K. $\endgroup$ Nov 11, 2022 at 20:34
For higher weight newforms, it seems that the conjecture you might want to look into is the Bloch-Kato conjecture. Here are a few links that might help you get started. The first is a survey article from 2003 (although a lot of progress has been made since then, this could be a good starting point):
https://jtnb.centre-mersenne.org/item/?id=JTNB_2003__15_1_179_0
Here is a fun paper on the distribution of zeros of certain polynomials (period polynomials) whose coefficients are built using the critical values you wish to study:
https://www.pnas.org/doi/10.1073/pnas.1600569113
Here is a high-level overview that requires a bit of background in algebraic number theory, representation theory of finite groups, and group cohomology:
https://www.claymath.org/sites/default/files/bellaiche.pdf
Here is a related Wikipedia article:
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1$\begingroup$ This is a generally great answer, but the first of the two Wikipedia links is not relevant (that is about the "other" Bloch--Kato conjecture). $\endgroup$ Oct 28, 2022 at 12:48
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1$\begingroup$ @DavidLoeffler Thanks! I edited the answer accordingly. $\endgroup$ Oct 28, 2022 at 12:53
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