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I have been reading Serre's paper on p-adic modular forms and there seems to be a connection between critical values of L-functions and weights of Eisenstein series in the following sense:

  • For the Riemann zeta function, the values of zeta functions at positive even integers $2k$ correspond to the Eisenstein series $$G_{2k} = 2\zeta(2k) + c_k\sum_{n\geq 1}\sigma_{2k-1}(n)q^n$$ and we see that the constant term is more or less the evaluation of $\zeta$ at the weight.

  • For the Kubota-Leopoldt p-adic L function $L_p(s,\omega^{1-u})$ where $(s,u) \in X = \mathbb Z_{p}\times \mathbb Z/(p-1)$ and $\omega$ the Teichmüller character, Serre constructs an Eisenstein series $E_k$ for $k = (s,u)\in X$ such that $$E_{k} = 2L_p(s,\omega^{1-u}) + \sum_{n\geq 1}\sigma^*_{k-1}(n)q^n.$$ Once again, the value of the L- function is visible as the constant term.

  • For $K$ a totally real number field of degree $r$, let $\zeta_K$ be it's Dedekind zeta function. For $m=2k$ an even integer, we can define a modular form of weight $k=rm$: $$f_m = 2^{-r}\zeta_K(1-m) + \sum_{\mathfrak a}\sum_{v\in \mathfrak d^{-1}\mathfrak a}(N\mathfrak a)^{m-1}q^{Tr(v)}$$ where $\mathfrak d$ is the discriminant and the sum is over integral ideals $\mathfrak a$. Yet again, one can see the value of the zeta function in the constant term. (I don't know this example very well so I might have made some mistakes here - please correct me).

I only have these examples so far but is there a general conjecture that explains this pattern? Can one always associate an Eisenstein series of some form to special values of L functions?

Is this a part of the Langland's conjectures/philosophy? Can one extend this to L-functions coming from modular forms or groups other than $GL_1$?

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    $\begingroup$ This is part of the Langlands philosophy; it falls into the Langlands-Shahidi method, which describes how $L$-functions arise as constant terms of Eisenstein series (on arbitrary reductive groups). This is true not just for critical values of $L$-functions, but for all values. The starting point for this observation (beyond just the Riemann zeta function and Dirichlet $L$-functions) is Langlands' book "Euler Products", but I learnt this all from Henry Kim's lectures in "Lectures on Automorphic $L$-Functions". $\endgroup$ – Peter Humphries Jul 27 '17 at 17:48
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(I disclaim expertise on p-adic L-function and p-adic automorphic forms things, though I did make some earlier contributions to rationality statements and local archimedean computations that helped p-adic developments in recent years.)

Although by now the catch-phrase "Langlands Philosophy" is often taken as an umbrella for almost anything having to do with automorphic forms, L-functions, and representation theory, in any case I think this is an insufficient explanation... especially as Langlands' conjectures and remarks came after many very interesting examples, and, further, the documented literal conjectures do not go quite so far as to explain every possible aspect of automorphic forms and such.

For example, the third illustrative example in the question is (modulo typos and details) the restriction of a (holomorphic) Hilbert modular Eisenstein series along the diagonal to an elliptic modular form. This was used by H. Klingen in the early 1960s to prove the appropriate rationality properties of (abelian) L-functions of totally real number fields. (G. Shimura talked about this in a course in the mid-1970s, and it was very striking to me. My old book on Hilbert modular forms gives this example, too.)

In that same vein, much of G. Shimura's work in the early 1970s, and later, proved special value results about various L-functions by using rationality properties of (holomorphic discrete series) automorphic forms. This line of thinking was involved in my own thesis under Shimura in the mid-70s, and affected my thinking subsequently: suitable pullbacks of holomorphic modular forms (Siegel, hermitian, ...?) are holomorphic, and have (numerical) Fourier coefficients in the same field as the original form. I. Satake's work on hermitian imbeddings of hermitian symmetric spaces in the mid-60s assured that these things work as best as could be hoped, even quite generally (but, sadly, not interacting much with exceptional groups E6 and E7...)

Another thread with old antecedents is the 1939 Rankin-Selberg integral representation, which was immediately aimed at giving analytic results to approach Ramanujan's conjectures (best known estimates, I think, prior to P. Deligne's 1974 proof of the last bit of the Weil conjectures), but also re-used by Shimura c. 1973 to prove essentially the full range of special values for suitable Hecke L-functions (after work of Damerell by different methods).

In the early 1980s, Piatetski-Shapiro and Rallis, myself, M. Harris, and a few others pursued the general idea (mentioned by P.-S. at the Budapest conference c. 1971) that certain configurations of subgroups could produce Euler products when various things (e.g., Eisenstein series) were integrated against each other. (My paper in the Shimura-conference volume, AMS 66, also on my web page, attempts to consider a sort of general case of this.) The Euler-product feature can easily occur without any special-value results.

For that matter, apparently E. Hecke and H. Maass were aware that certain $GL(1,k)$ periods of $GL(2)$ Eisenstein series $E_s$ (with quadratic extension $k$) produced essentially $\zeta_k(s)/\zeta(2s)$, and such things.

In the last 10-20 years, many people have demonstrated increasingly-sophisticated ways to evaluate periods of Eisenstein series (not just constant terms, as in Langlands-Shahidi) to produce Euler products. E. Lapid, O. Offen, G. Chinta, and many others (my apologies for not making a bigger list: lazy). H. Jacquet's relative trace formula sometimes helps in converting a not-quite-Euler-product situation to a visibly Euler-product situation.

The most-authentic "Langlands conjecture" version of such things is that (literally) (... and via a suitable form of "the fundamental lemma", see Loeser, Ngo, et al) the most general automorphic L-function (whatever this means exactly) should be equal to a/the standard L-function attached to a cuspform on $GL(n)$. There is no immediate conjecture about special values. (Deligne's 1978 ICM conjectures incorporated things known at the time, mostly due to Shimura, but also with some prescient interpolation.)

I don't think Langlands made conjectures about p-adic L-functions... but it is entirely believable to me that appealing additions to the original conjectures could be made that did refer to p-adic stuff. Given my limited appreciation of these things, I don't see that the long-ago conjectures really suggested so much about many recent developments, but those recent developments (Wiles, Wiles-Taylor, Taniyama-Shimura-etal...) are compatible with the vague general idea that many things are related to automorphic forms, and in non-obvious ways.

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