Laurent Berger
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 Mar 30 comment Elliptic curves and the $\ell$-adic image of the decomposition group The image may be hard to compute, but the Lie algebra of the image is known. See the appendix to Serre's book "Abelian ell-adic representations and elliptic curves". Mar 17 comment Does X(13) have potentially good reduction at 13? @znt on a mac it actually works the same as with an ipad/iphone: press and hold a key and all the possible modifications of the letter will eventually be displayed for you to choose from Feb 26 comment Which groups are Galois over some p-adic field? See this MO question mathoverflow.net/questions/172569/local-inverse-galois-problem Feb 11 comment Uniformizer for splitting field of p^{1/p^n} over p-adics I am pretty sure that the answer is "no, it's too messy". If however somebody does have an answer, then we could use it to study the field of norms of the union of all those fields, and I'd be very interested in that. Feb 7 comment When does the radius of convergence of the product of two $p$-adic power series increase? Have you looked at "Rank one solvable p-adic differential equations and finite Abelian characters via Lubin–Tate groups" by Andrea Pulita? The abstract starts with "We introduce a new class of exponentials of Artin–Hasse type, called π-exponentials". Jan 27 comment Hodge-Tate representations A simpler example would be a non-trivial extension of $Q_p$ by $Q_p(-1)$. Apr 8 comment Algebraic integer with conjugates on the unit circle The same is true of $\alpha^k$ and the characteristic polynomials of the $\alpha^k$ have coeffts that are bounded indept of $k$. A lot of them must be equal to each other, so $\alpha^k = \alpha^\ell$ for some $k$ and $\ell$. Mar 1 comment Can an abelian variety/Q have no non-trivial points over Q_sol? @Pablo sorry, my mistake! Mar 1 comment Can an abelian variety/Q have no non-trivial points over Q_sol? By definition, an abelian variety over a field K has a rational point over K, so in your question, you presumably mean a homogenous space for your abelian variety. Jan 27 comment Number of common solutions of polynomial system See for instance the introduction to arxiv.org/abs/1408.3224 Jul 26 comment Linear map with two “incompatible” representations Very nice, thank you! Jul 24 comment If $G$ is compact, $H \leq G$ open, $V$ an irreducible $H$-rep, is $\text{Ind}_H^G$ semisimple? The continuity of the action does not imply that V is finite diml. Jul 13 comment is there a p-adic implicit function theorem? See page 73 of the latest edition of Serre's book. Jul 13 comment Power series defined by Witt vectors / Teichmüller representatives of p-adics ... "Quiconque s’est intéressé aux corps locaux sait bien qu’une extension très ramifiée du corps $Q_p$ des nombres $p$-adiques ressemble à s’y méprendre à un corps de séries formelles à coefficients dans son corps résiduel. C’est sans doute Marc Krasner qui a tenté le premier de formuler ce phénomène abondamment utilisé depuis en théorie de Hodge p-adique [...]" (Fontaine, Bourbaki 1057). Jul 13 comment Power series defined by Witt vectors / Teichmüller representatives of p-adics If you take $K$ to be ramified and play the same game, then as you increase the ramification, your two fields are "more and more isomorphic". This observation of Krasner is the basis for the theory of the "field of norms" and more recently the theory of "perfectoid spaces". Here is what Fontaine says about this in his recent Bourbaki exposé... Jul 13 comment What is the lowest-weight non-cyclotomic Galois representation in $\overline{\mathcal M}_{g,n}$? I think that Gaëtan Chenevier is thinking about these things as well, you could ask him directly. May 2 comment How to correct an error in a submitted paper? Same here - I recently had a paper rejected on the basis of three reports, one of which was a very angry report which was not based on the version of the paper that I'd sent to the journal. My paper, however, stayed rejected after this was pointed out :-( Feb 6 comment Status of local Langlands conjecture over positive characteristic Laumon pas Laumont! Nov 26 comment maximal abelian extension of quadratic extension of $\mathbb Q_p$ You can always take $f(T)=\pi T + T^q$ where $\pi$ is a uniformizer and $q$ is the cardinality of the residue field. Any power series satisfying the conditions that you wrote will do. Oct 16 comment De Rham cohomology of formal groups @Jon Beardsley: thank you!