Let $K$ a totaly real number field, $\mathcal{O}_K$ its ring of integre and $h$ the narrow class number of $K$. Let $\mathbf{f}$ a collection $(f_1, ..., f_h)$ of Hilbert cusp forms $f_\lambda$ $(\lambda =1, ..., h)$ of weight $k=(k_1,\dots,k_n)$ with respect to $\Gamma_{\lambda}(\mathcal{N}),$ $$R_{\mathbf{f}}(s)=\sum_{\mathfrak{m}\subseteq\mathcal{O}_K}\frac{C(\mathfrak{m},\mathbf{f})^2}{\rm{N}(\mathfrak{m})^s}$$ the Rankin-Selberg zeta function, associated to $\mathbf{f}.$ can someone describe the analytic proprities (meromorphic continuation, functional equation, poles and residues) of $R_{\mathbf{f}}(s).$
4
-
2$\begingroup$ Are you sure you are expecting to attach an L-function to an $h$-tuple of Hilbert modular forms? This would be remarkable, if you could do it... but I wonder whether you mean something else. Why class number, for example? And what does $C(\mathfrak m,\mathbf f)$ mean? $\endgroup$– paul garrettCommented Mar 23, 2017 at 15:36
-
$\begingroup$ @paulgarrett, I will add some details, to the question to be more clear. $\endgroup$– MedCommented Mar 23, 2017 at 15:53
-
$\begingroup$ I'm sorry, I'm still confused: why a tuple of Hilbert modular forms? Are you thinking of a modular form as a tuple, one for each real place of the totally real field? $\endgroup$– paul garrettCommented Mar 23, 2017 at 16:04
-
1$\begingroup$ @paulgarrett: Shimura uses a semi-adelic setup. The connected components of the adelic quotient correspond to the narrow ideal classes (by strong approximation for $\mathrm{SL}_2$), and $f_\lambda$ is the restriction of the adelic form to the $\lambda$-th component. So the adelic form can be identified with an $h$-tuple of classical forms (each transforming with respect to its own congruence subgroup). $\endgroup$– GH from MOCommented Mar 23, 2017 at 16:04
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
2
(Thanks again to GH-from-MO for explaining the convention in the relevant paper.) I think this is easier to understand from a (genuinely) adelic viewpoint, since that makes the proof(s) be the same as for the classic Rankin-Selberg story from 1939, in the same way that Iwasawa-Tate's viewpoint on Hecke L-functions makes Hecke's general case nearly identical to Riemann's original argument. One does not have to explicitly mention class numbers and so on.
Namely, as is in principle well-known, for cuspform $f$ (holomorphic or not) one integrates $|f|^2$ against a suitable Eisenstein series, whose meromorphic continuation and functional equation in the small case of $GL_2$ come from Poisson summation. The "uneven" weights, at least if they are all of the same parity, are easily accommodated by fairly obvious choices of vectors in the principal series from which the Eisenstein series is formed. Since it's $|f|^2$, the parities are all "even", so there is no obstacle.
A genuine adelic set-up also presents all the Euler factors as local integrals against local Whittaker functions... whose computations are essentially identical to the classical case, at least at good primes. The archimedean integrals are the same.
answered Mar 23, 2017 at 16:13
-
$\begingroup$ Dear Paul thank you so much for your answer. $\endgroup$– MedCommented Mar 23, 2017 at 16:16
-
1$\begingroup$ Yes. One can find a quick and fairly explicit treatment (all very standard of course) on pages 25-26 of Blomer-Harcos: Twisted L-functions over number fields and Hilbert's eleventh problem, Geom. Funct. Anal. 20 (2010), 1-52. $\endgroup$ Commented Mar 23, 2017 at 16:22
$\begingroup$
$\endgroup$
0
I think the answer to your question is contained in Proposition 3.2 of Shimura: The critical values of certian Dirichlet series attached to Hilbert modular forms, Duke Math. J. 63 (1991), 557-613.
answered Mar 23, 2017 at 16:00