User jnewton - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T07:28:15Z http://mathoverflow.net/feeds/user/1979 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74410/what-is-the-nature-of-the-locus-in-the-eigencurve-associated-to-some-conditions-o/74467#74467 Answer by jnewton for What is the nature of the locus in the eigencurve associated to some conditions on the associated automorphic representation (at $p$)? jnewton 2011-09-03T22:37:14Z 2011-09-03T22:37:14Z <p>Since the eigencurve only sees classical modular forms with non-zero $U_p$-eigenvalue, there are strong restrictions on the local factor at $p$ for the attached automorphic representations (e.g. you won't ever see supercuspidals). I think the situation is that if you fix a Nebentypus, with conductor $p^r$ at $p$, then the automorphic representations coming from classical eigenforms with non-zero $U_p$ eigenvalue must have local factor at $p$ either a) Principal series with conductor $p^r$ or b) Special of conductor $p$ (so $r=0$ in this case). (One can find these sorts of calculations in Casselman's article "On representations of $GL_2$ and the arithmetic of modular curves." in one of the Antwerp volumes.)</p> <p>Also, any classical point sufficiently close to (but not equal to!) one of the $p$-special classical points will actually be unramified principal series (by local constancy of the slope, since the slope of a weight $k$, $p$-special point has to be $(k-2)/2$), so every component will contain principal series points.</p> <p>As a complement to Michael's comment, bearing in mind Emerton's approach to constructing the eigencurve and his results on local-global compatibility in the $p$-adic Langlands programme, it might be natural to think about certain locally analytic representations of $GL_2(\mathbb{Q}_p)$ varying over the eigencurve - e.g. for the two $p$-stabilised $U_p$-eigenforms coming from a classical eigenform with level prime to $p$, one has two different locally analytic principal series representations over the two associated points on the eigencurve. One can also see the locally analytic Jacquet modules (as defined by Emerton) of these representations varying over the eigencurve.</p> <p>PS Maybe you're thinking of Alex Paulin's (http://math.berkeley.edu/~apaulin/) thesis for the $l \ne p$ case?</p> http://mathoverflow.net/questions/17774/does-the-image-of-an-p-adic-galois-representation-always-lie-in-a-finite-extensio/17777#17777 Answer by jnewton for Does the image of an p-adic Galois representation always lie in a finite extension? jnewton 2010-03-10T23:19:18Z 2010-03-10T23:19:18Z <p>A proof of the result you're after is contained at the beginning of section two of a recent paper of Skinner <a href="http://www.math.uiuc.edu/documenta/vol-14/10.html" rel="nofollow">here</a>. Skinner mentions that references for this fact seem to be rare.</p> http://mathoverflow.net/questions/14763/what-are-the-local-langlands-conjectures-nowadays-for-connected-reductive-groups/14769#14769 Answer by jnewton for What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field? jnewton 2010-02-09T14:35:28Z 2010-02-09T14:35:28Z <p>Here's a remark about the Satake isomorphism, although I don't think it's what you're looking for: Suppose $G$ is a split reductive group over $\mathbb{Z}$, then one formulation of Satake is an isomorphism </p> <p>$H(G(\mathbb{Q}_p),G(\mathbb{Z}_p)) \rightarrow \mathbb{C}\otimes K(Rep(\hat{G}))$</p> <p>where $H(G(\mathbb{Q}_p),G(\mathbb{Z}_p))$ is the Hecke algebra with respect to the maximal compact $G(\mathbb{Z}_p)$, and $K(Rep(\hat{G}))$ is the Grothendieck ring of the category of finite dimensional reps of the Langlands dual group $\hat{G}$ (a reductive group over $\mathbb{C}$).</p> <p>But now write $K$ for $\mathbb{F}_p((T))$, $A$ for $\mathbb{F}_p[[T]]$. We also have a Satake isomorphism for the Hecke algebra $H(G(K),G(A))$, with exactly the same target as our earlier isomorphism! So the Satake isomorphism doesn't seem to see any difference between p-adic fields and function fields, and indeed one can view the categorified Satake isomorphism of, say, <a href="http://arxiv.org/abs/math.RT/0401222" rel="nofollow">Mirkovic-Vilonen</a> as saying something for the p-adic field case as well.</p> http://mathoverflow.net/questions/969/does-ribets-level-lowering-theorem-hold-for-prime-powers/6285#6285 Answer by jnewton for Does Ribet's level lowering theorem hold for prime powers? jnewton 2009-11-20T17:06:54Z 2009-11-20T17:06:54Z <p>There's some slides from a talk by Ian Kiming <a href="http://www.uni-due.de/~hx0037/CMF/resources/KimingTalk.pdf" rel="nofollow">here</a> which discuss this question. He states a theorem (on slide number 8) corresponding to the existence of the map from a Hecke algebra at level N/(p^u) (where p^u is the largest power of p dividing N) to Z/ell^n Z. As buzzard says, it's not clear that this map will lift, but Kiming speculates that if you allow the weight of your modular form to vary you can find a char 0 lift. </p> http://mathoverflow.net/questions/110023/level-lowering-for-weight-1-forms Comment by jnewton jnewton 2012-10-19T09:53:16Z 2012-10-19T09:53:16Z Dear Robert, if you assume that $a_p(f)^2 \ne \chi(p)$ mod $p$ then I think you can deduce level lowering in weight one from level lowering in weight $p$ using Corollary 13.11 of Gross's companion forms paper. At least, this will give you a mod $p$ weight one form of level $N/q$ - I don't know how easy it is to show that this form will lift to characteristic zero when the form of level $N$ you started with lifts, which seems to be the setting of your question. The weight one case of Gross's result was generalised to the Hilbert setting recently by Gee and Kassaei (arXiv:1206.6631). http://mathoverflow.net/questions/103909/non-classical-specializations-of-hida-families/103915#103915 Comment by jnewton jnewton 2012-08-15T09:09:48Z 2012-08-15T09:09:48Z I'm not sure I understand these remarks - what do you mean by &quot;falling back to some classical specialization&quot;? The question is asking whether the specialisation of a Hida family at a non-classical weight is overconvergent, so I don't see the relevance of Coleman's result (which tells us about overconvergent forms of classical weight). http://mathoverflow.net/questions/103909/non-classical-specializations-of-hida-families Comment by jnewton jnewton 2012-08-04T22:17:26Z 2012-08-04T22:17:26Z Lemma 1 of Buzzard-Taylor &quot;Companion forms and weight 1 forms&quot; confirms your expected answer for integral weights. Perhaps one can adapt the proof to work for general weight weight $\kappa$? http://mathoverflow.net/questions/74410/what-is-the-nature-of-the-locus-in-the-eigencurve-associated-to-some-conditions-o/74467#74467 Comment by jnewton jnewton 2011-09-06T13:59:30Z 2011-09-06T13:59:30Z Dear Michael, yeah I guess I should have mentioned my own work too! http://mathoverflow.net/questions/47407/does-sl-3r-embed-in-sl-2r/47420#47420 Comment by jnewton jnewton 2010-11-26T14:37:28Z 2010-11-26T14:37:28Z Invariant Basis Number: <a href="http://en.wikipedia.org/wiki/Invariant_basis_number" rel="nofollow">en.wikipedia.org/wiki/Invariant_basis_number</a> http://mathoverflow.net/questions/46469/can-one-compare-integral-structures-on-de-rham-and-crystalline-cohomology Comment by jnewton jnewton 2010-11-18T15:50:33Z 2010-11-18T15:50:33Z Thanks for the clarification. http://mathoverflow.net/questions/46469/can-one-compare-integral-structures-on-de-rham-and-crystalline-cohomology Comment by jnewton jnewton 2010-11-18T13:12:59Z 2010-11-18T13:12:59Z Won't applying Fontaine's $\mathbb{D}_{\rm cris}$ to $H^i_{\mathrm{et}}(X_{\overline{\mathbb{Q}}_p}, \mathbb{Z}_p)$ forget the integral structure? (since $B_{\rm cris}$ is a $\mathbb{Q}_p$-algebra) http://mathoverflow.net/questions/33269/fontaine-mazur-for-gl-1 Comment by jnewton jnewton 2010-07-25T15:50:10Z 2010-07-25T15:50:10Z Oh and I guess the proposition in section 6 of Fontaine-Mazur is relevant to the question? http://mathoverflow.net/questions/33269/fontaine-mazur-for-gl-1 Comment by jnewton jnewton 2010-07-25T15:02:15Z 2010-07-25T15:02:15Z The terminology used in Fontaine and Mazur's paper is that &quot;geometric&quot; means &quot;almost everywhere unramified and potentially semistable at places |p&quot; and an irreducible p-adic Galois representation is said to &quot;come from algebraic geometry&quot; if it is isomorphic (up to Tate twist) to a subquotient of the etale cohomology of an algebraic variety. Their conjecture is then that an irreducible p-adic rep comes from algebraic geometry if and only if it is geometric. http://mathoverflow.net/questions/31538/non-abelian-class-field-theory-and-fundamental-groups Comment by jnewton jnewton 2010-07-13T16:11:00Z 2010-07-13T16:11:00Z So I guess what I wrote above is just saying that the automorphic reps corresponding to Artin reps are those with infinitesimal character zero at $\infty$. By the way, I don't think these will be discrete series at $\infty$ - for GL_2(R) they're either limit of discrete series (the holomorphic weight one case) or principal series (the Maass form case). http://mathoverflow.net/questions/31538/non-abelian-class-field-theory-and-fundamental-groups Comment by jnewton jnewton 2010-07-13T15:35:12Z 2010-07-13T15:35:12Z @Minhyong: My impression was that the automorphic representations corresponding to Artin reps are characterised by their local factors at archimedean places e.g. for GL_2/Q, if the Artin rep is odd it comes from a holomorphic modular form of weight 1, if it is even it comes from a non-holomorphic Maass form with Laplacian eigenvalue 1/4. I guess the local factors at arch. places should match under local Langlands with arch. Weil group reps which factor through the local Galois groups, given by the actions of complex conjugations on the Artin rep. http://mathoverflow.net/questions/31538/non-abelian-class-field-theory-and-fundamental-groups Comment by jnewton jnewton 2010-07-12T17:08:05Z 2010-07-12T17:08:05Z I guess the (strong) Artin conjecture tells you that given $L/K$ unramified, then non-trivial irreducible n-dimensional (complex) representations of $Gal(L/K)$ arise from cuspidal automorphic representations of $GL_n(\mathbb{A}_K)$ of `level 1' (i.e. unramified at all finite places) and certain type at the archimedean places. So if we can rule out the existence of any of these, we can conclude that there's no non-trivial unramified $L$ - this may well be difficult to do though. http://mathoverflow.net/questions/23002/analogue-of-shimura-curves-in-the-symplectic-case/23008#23008 Comment by jnewton jnewton 2010-05-13T14:00:01Z 2010-05-13T14:00:01Z So I guess the natural candidate for the inner form in the global question would be a unitary group in two variables over a quaternion algebra D/F split at just one infinite place - a brief look at chapter 8 in Milne's notes on Shimura varieties made me believe it should be possible to do this and get a PEL Shimura variety, but I'm not sure. http://mathoverflow.net/questions/15235/homeomorphism-onto-a-closed-subset-of-a-scheme-that-isnt-a-closed-immersion/15236#15236 Comment by jnewton jnewton 2010-02-14T08:46:45Z 2010-02-14T08:46:45Z In the second example, is $X$ the normalisation of $X'$ (as in the third example)? http://mathoverflow.net/questions/14959/extending-cusp-forms Comment by jnewton jnewton 2010-02-12T11:24:25Z 2010-02-12T11:24:25Z So I guess implicit in Kevin's reformulation is that the functoriality map from auto reps of $\tilde{G}$ to auto reps of $G$ is something like apply restriction of cusp forms and take an irreducible auto subrepresentation... is it known that there's a unique such subrepresentation?