# Special values of non-cm $L$-functions

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!

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 $$(r_1,r_3,r_5,r_7,r_9)=(48,-25,20,-25,48)\omega^-$$ $$(r_0,r_2,r_4,r_6,r_8,r_{10})=(22680/691,-14,9,-9,14,-22680/691)\omega^+i$$ $$\omega^-\omega^+=(4096/691)<\Delta,\Delta>$$

• 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. Oct 28, 2022 at 18:54
• typo: replace Eichler (not Shimura) by Eichler--Shimura. Oct 28, 2022 at 19:16
• 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 $\Delta$. Nov 11, 2022 at 12:32
• 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). Nov 11, 2022 at 12:36
• Thanks David, I was not aware of this precise form of B-K. 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:

https://en.wikipedia.org/wiki/Special_values_of_L-functions

• 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). Oct 28, 2022 at 12:48
• @DavidLoeffler Thanks! I edited the answer accordingly. Oct 28, 2022 at 12:53
• Thank you for responses Oct 28, 2022 at 17:37