Timeline for On the local factor of Rankin-Selberg L-functions
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Sep 14 at 12:44 | comment | added | FeiHou | Dear professor, many many thanks for so warm-hearted explanations. Your time and help are highly appreciated! | |
| Sep 14 at 10:38 | comment | added | Will Sawin | @FeiHou Yes, that's what I mean. The simple answer is that your moral claim is not correct. It is an approximation valid in some cases but not always. If we fixed a convention for the absolute values of the $\alpha$ and $\beta$s in the unramified case, then I could replace the inverse by a complex conjugate divided by a suitable power of $p$, and it would be more correct in the $p\mid N$ case, because there would be no division by $0$, but not exactly right. One simply has to study the actual Galois representation or algebraic group representation to get the ansewr. | |
| Sep 14 at 1:38 | comment | added | FeiHou | On the other hand, morally the local factor for $p\mid N$ should just correspond to that for the case $p\nmid N$ with the local parameters $\{\alpha_f, \beta_f\}$ of $L(f,s)$ replaced by $\{\lambda_f(p), 0\}$. However, you have the local parameters appearing in the denominator, which probably contains zero in the denominator and seems a bit weird. It appears that one cannot get the corresponding local factors directly from the expression in the case of $p\nmid N$. | |
| Sep 14 at 1:21 | comment | added | FeiHou | ...Now, by your formula, the local factor for the Rankin-Selberg of the $L$-function with its dual at $p\mid N $ reads $$\prod_{i=1}^2\prod_{j=1}^2 \left(1-\frac{m^\prime_i}{m^\prime_j p^s} \right)^{-1}\left(1-\frac{m^\prime_i}{m^\prime_j p^{s+1}} \right)^{-1}.$$ So, is that what you mean for the case of $p\mid$ and nebentypus being trivial?? | |
| Sep 14 at 1:20 | comment | added | FeiHou | Dear prof. Sawin, great thanks for so detailed explanation. I still have one more puzzle on the case of $p\mid N$ and nebentypus being trivial, which needs your correction : For simplicity, one may write $m^\prime_1=\alpha\beta, m^\prime_2=\alpha\beta^{-1}$, where $\alpha=\lambda_f(p)$, and $\{ \beta, \beta^{-1} \}$ denote the the local parameters of $L(g,s)$. So, for $p\mid N$, $$L_p(f\times g,s)=\prod_{i=1}^2 \left( 1-\frac{m^\prime_i }{p^s} \right )^{-1}.$$... | |
| Sep 13 at 15:28 | history | answered | Will Sawin | CC BY-SA 4.0 |