Timeline for Complex L-functions for Hermitian modular forms?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 27, 2019 at 23:16 | comment | added | Puddles the turtle | Hecke operators relate to the local structure of the group. But $U(n,n)/K$ over (say) a prime that splits in $K$ looks like $\mathrm{GL}(2n)$, so the Hecke theory looks like that of $\mathrm{GL}(2n)$ for such primes. | |
| Jun 27, 2019 at 23:15 | comment | added | Puddles the turtle | Your comment is pretty vague. But you should really just learn the recipe of how all these things connect (Satake parameters, Langlands dual group, L-functions)are connected and then the conjectural story is just Lie theory. For example, the dual group of $U(n,n)$ is something close to $GL(2n)$ semi-direct product with $\mathrm{Gal}(K/\mathbf{\mathbf{Q}})$ acting by conjugate transpose. So $L$-functions related to representations of this group. Similarly, Satake paremeters are related to elements of this group. (When $n = 1$ this automorphism is inner, which is why things are simpler.) | |
| Jun 27, 2019 at 15:16 | vote | accept | Jon Aycock | ||
| Jun 27, 2019 at 9:07 | comment | added | Jon Aycock | How far can this U(n,n) <--> GSp(2n) analogy go? Does the Hecke algebra look pretty much the same for U(n,n) as it does for GSp(2n)? In particular, similar looking Satake parameters, with spin and standard L-functions for U(n,n) having Euler products looking the same as those for GSp(2n)? (I see a lot more information on Siegel than Hermitian modular forms, and would like to know what I can use!) | |
| Jun 26, 2019 at 23:25 | comment | added | Puddles the turtle | For $n = 2$, these degree $4$ and $5$ $L$-functions are the classical ones which are known to have analytic continuations. (Note $\mathrm{Gspin}(5) \simeq \mathrm{GSp}(4)$ by some accidental automorphism.) But there are other $L$-functions, for example, $\mathrm{GSp}(4)$ has a (unique) irreducible representation $\rho$ of dimension $91$, so there should also be an $L$-function $L(\pi,\rho,s)$ for a Siegel modular form. (It exists, but its analytic continuation is not known.) | |
| Jun 26, 2019 at 23:20 | comment | added | Puddles the turtle | It is exactly an example of this. If $G = \mathrm{GSp}(2n)$, then the dual group is $\mathrm{Gspin}(2n+1)$. The latter group has many representations, including a "spin" representation of dimension $2^n$ and a "standard" representation of dimension $2n+1$, so specifying that representation is part of the data defining an $L$-function for $\mathrm{GSp}(2n)$. | |
| Jun 26, 2019 at 21:02 | comment | added | Jon Aycock | Is the representation of the dual group you're alluding to like the spin representation in the case of Siegel modular forms? (As opposed to the standard representation?) Or is it different enough that the analogy there isn't useful? And for the follow-up, thank you for the insight! I thought we looked at U(1,1) instead of GL(2) because it was somehow necessary to get the similar picture with elliptic curves. | |
| Jun 26, 2019 at 19:04 | history | answered | Puddles the turtle | CC BY-SA 4.0 |