Timeline for When is the image of a 2-dim l-adic representation associated to a modular form open
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15 events
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| Mar 25, 2021 at 18:20 | comment | added | David Loeffler | I believe this is a mistake in Kato's paper: the method genuinely does not work when the Galois image is quaternionic. | |
| Mar 25, 2021 at 17:44 | comment | added | user585094 | interesting, thank you! In Kato's paper "p-adic zeta functions of modular forms," it is asserted (12.8.2) that the image contains a conjugate of an open in SL2(Zp), and this fact is attributed to the paper of Ribet discussed in this thread. But perhaps the full strength of this assertion is not used. | |
| Mar 25, 2021 at 17:19 | comment | added | David Loeffler | No, that's not true: if the image is contained in a quaternion group (which can happen) then it cannot contain any nontrivial unipotent element, so it will not contain a conjugate of an open in SL2. | |
| Mar 25, 2021 at 17:07 | comment | added | user585094 | Is it at least true that the image contains an open subgroup of a conjugate of SL2 (Z_p)? | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Oct 14, 2015 at 21:49 | comment | added | Vesselin Dimitrov | @DavidLoeffler: I see. Thank you for explaining! I really had the abelian varieties in mind, rather than the modular forms .(And I did take the character to be trivial.) | |
| Oct 14, 2015 at 20:03 | comment | added | David Loeffler | @VesselinDimitrov You say "abelian varieties with real multiplications (which is what weight two non-CM eigenforms for Γ1(N) give rise to)." This is not true; there are examples of mod forms giving rise to abelian varieties with quaternion multiplication, for instance. Even when the ab var does have endomorphism algebra an order in a totally real field, this field isn't generally the same as the field of coefficients of the form, which will not be totally real when the character is nontrivial. | |
| Oct 14, 2015 at 15:33 | comment | added | Vesselin Dimitrov | Just to clear the point I tried to make but did not get quite right. In the weight two case, Faltings's isogeny and semisimplicity theorems allow for a clean, direct determination of the Lie algebra of the image, as mentioned in Ribet's review of Serre's book, in the context there of abelian varieties with real multiplications (which is what weight two non-CM eigenforms for $\Gamma_1(N)$ give rise to). The image itself (for all but finitely many places) will require more work, but I don't at the moment have access to Momose's paper to see how it is done. | |
| Oct 14, 2015 at 14:59 | vote | accept | user42690 | ||
| Oct 14, 2015 at 13:38 | comment | added | Joël | I have added a CW answer to the older question pointing out to this one. Perhaps David could accept it instead of the other, to avoid risks of confusion. | |
| Oct 14, 2015 at 9:35 | comment | added | Vesselin Dimitrov | I see, the determinental condition must be accounted for, so we can't say open image. But then, in the case I wrote about, it seems to me that we do get an open subgroup of $\mathrm{SL}_2(L_{\mathfrak{P}})$, with equality for all but finitely many $\mathfrak{P}$. Or I may be misreading page 4 of Ribet's review of Serre's book (Abelian $l$-adic representations). | |
| Oct 14, 2015 at 9:05 | comment | added | Myshkin | (The older question/answer should be edited with a link to this one, or some mention of the right answer) | |
| Oct 14, 2015 at 9:04 | comment | added | Vesselin Dimitrov | But if $A$ is an abelian variety whose endomorphism ring is an order in a totally real field of dimension $\dim{A}$, so in particular an abelian variety of $\mathrm{GL}_2$ type, defining a two dimensional $\mathfrak{l}$-adic representation, it does follow by Faltings that that representation has open image (Is this right?). So the statement on open image must be true for weight two non-CM normalized newforms for $\Gamma_1(N)$? | |
| Oct 14, 2015 at 8:34 | history | edited | David Loeffler | CC BY-SA 3.0 |
added 586 characters in body
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| Oct 14, 2015 at 8:29 | history | answered | David Loeffler | CC BY-SA 3.0 |