User nicolas b. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T12:54:52Z http://mathoverflow.net/feeds/user/14967 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107566/fourier-expansion-of-eisenstein-series-at-various-cusps Fourier expansion of Eisenstein series at various cusps Nicolas B. 2012-09-19T13:30:49Z 2012-09-19T23:33:48Z <p>Here is my setting: Let $E\in\mathcal{M}_k(\Gamma_0(N))$ be an Eisenstein series (of trivial Nebentypus) that is a normalized eigenform for all the Hecke operators at level $\Gamma_0(N)$. Assume that the Fourier expansion of $E$ lies in $\mathcal{O}_K[[q]]$ where $\mathcal{O}_K$ is the integer ring of some number field $K$. For any $\gamma\in\mathrm{SL}(2,\mathbf{Z})$ let us denote by $a_0(\gamma)$ the constant term of the Fourier expansion of $E|_k\gamma$. Here are my questions.</p> <ol> <li><p>Is it true that for some number field $L$ containing $K$ and the $N$-th roots of unity we have $a_0(\gamma)\in L$ and $E|_k\gamma\in a_0(\gamma)+q^{1/N}\mathcal{R}[[q^{1/N}]]$ where $S$ is a finite set of places in $L$ containing the finite places above $N$ and the archimedian ones and $\mathcal{R}$ denotes the ring of $S$-integers in $L$?</p></li> <li><p>Let $\lambda$ be a prime in $\mathcal{O}_K$ of residue characteristic $\ell\nmid N$. Since by assumption the Fourier expansion of $E$ lies in $\mathcal{O}_K[[q]]$, one may reduce $E$ modulo $\lambda$. Assume that the corresponding modular form over $\overline{\mathbf{F}}_l$ is cuspidal and let $\gamma\in\mathrm{SL}(2,\mathbf{Z})$. Assume moreover that $a_0(\gamma)$ is $\mathcal{L}$-integral for all $\mathcal{L}\mid\ell$. Is it true that $\ell$ divides the numerator of the norm of $a_0(\gamma)$?</p></li> </ol> <p>I believe the answer to both questions is yes, but I'd like to have a proof or reference. Thanks for your help! </p> http://mathoverflow.net/questions/64363/semistable-elliptic-curves-and-irreducible-galois-representations Semistable Elliptic Curves and irreducible Galois representations Nicolas B. 2011-05-09T08:36:32Z 2011-05-09T10:03:27Z <p>I am interested in the set of number fields $K$ having the property that for \emph{any semistable} elliptic curve $E$ defined over $K$, there exists a constant $c(E,K)>0$ such that $$p>c(E,K)\Longrightarrow \rho_{E,p}:\mathrm{Gal}(\overline{K}/K)\longrightarrow \mathrm{Aut}(E[p])\textrm{ is irreducible}.$$ Since such a constant $c(E,K)$ exists if and only if $\mathrm{End}_K(E)=\mathbf{Z}$, an equivalent formulation of the above property is~: for any elliptic curve $E/K$, we have $$\mathrm{End}_K(E)\not=\mathbf{Z}\Longrightarrow E\textrm{ has bad reduction at a finite place of }K.$$ There are lots of examples of such number fields (e.g. number fields which do not contain the Hilbert class field of some imaginary quadratic field), but I wonder whether there exists a nice characterization of the whole set.</p> <p>Many thanks in advance for your answers!</p> http://mathoverflow.net/questions/108169/modern-proof-of-serres-open-image-theorem Comment by Nicolas B. Nicolas B. 2012-09-27T08:45:54Z 2012-09-27T08:45:54Z Faltings' theorem is also useful in proving generalizations of Serre's open image theorem for elliptic curves to abelian varieties of higher dimension. You may take at look at Serre's letters to Ribet and Vigneras (if I remember correctly) in his collected works vol. 4. http://mathoverflow.net/questions/107566/fourier-expansion-of-eisenstein-series-at-various-cusps/107581#107581 Comment by Nicolas B. Nicolas B. 2012-09-21T19:30:55Z 2012-09-21T19:30:55Z OK. I got it. You definitely were of great help. Thanks a lot! http://mathoverflow.net/questions/107566/fourier-expansion-of-eisenstein-series-at-various-cusps/107581#107581 Comment by Nicolas B. Nicolas B. 2012-09-20T20:17:21Z 2012-09-20T20:17:21Z @Ramsey: Just a remark regarding the first question. When you say that &quot;all $q$-expansions have coefficients in $\mathcal{O}_K[1/N](\mu_N)$&quot; you actually mean the non-constant part of it, isn't it? (As stated in my question, the constant term need not be in $\mathcal{O}_K[1/N](\mu_N)$ I think). http://mathoverflow.net/questions/107566/fourier-expansion-of-eisenstein-series-at-various-cusps/107598#107598 Comment by Nicolas B. Nicolas B. 2012-09-20T09:20:18Z 2012-09-20T09:20:18Z Thank you Fil for your contribution! As pointed out by you and Ramsey, Katz' paper is the reference I should look at to get the results I was looking for. http://mathoverflow.net/questions/107566/fourier-expansion-of-eisenstein-series-at-various-cusps/107581#107581 Comment by Nicolas B. Nicolas B. 2012-09-20T09:04:30Z 2012-09-20T09:04:30Z @Ramsey: thanks for helping me to clarify the questions and your answers. http://mathoverflow.net/questions/107566/fourier-expansion-of-eisenstein-series-at-various-cusps/107581#107581 Comment by Nicolas B. Nicolas B. 2012-09-19T20:05:16Z 2012-09-19T20:05:16Z RIGHT! This is exactly what I mean! http://mathoverflow.net/questions/77728/can-local-duality-for-elliptic-curves-be-proven-with-big-rings Comment by Nicolas B. Nicolas B. 2012-09-19T19:49:34Z 2012-09-19T19:49:34Z @Chandan Singh Dalawat: Funny! Et ceux du Coll&#232;ge de France r&#233;pondaient des &quot;conneries&quot;, c'est &#231;a? (sorry but this joke only works in French!). http://mathoverflow.net/questions/107566/fourier-expansion-of-eisenstein-series-at-various-cusps/107598#107598 Comment by Nicolas B. Nicolas B. 2012-09-19T19:46:50Z 2012-09-19T19:46:50Z Hi Fil! Thank you for this reference. I think the theorems you mention don't directly answer my questions but the paper of Rapoport they refer to may contain the appropriate statements. I'll check this. http://mathoverflow.net/questions/107566/fourier-expansion-of-eisenstein-series-at-various-cusps/107581#107581 Comment by Nicolas B. Nicolas B. 2012-09-19T19:42:00Z 2012-09-19T19:42:00Z Thanks Ramsey for your very quick reply! So I guess Katz' paper you mention is the appropriate reference for the first statement, right? For my second question I was actually wondering (more generally) whether the reduction of $E|_k\gamma$ is the Fourier expansion of the reduction of $E$ at the cusp $\gamma\infty$. http://mathoverflow.net/questions/107566/fourier-expansion-of-eisenstein-series-at-various-cusps Comment by Nicolas B. Nicolas B. 2012-09-19T14:26:36Z 2012-09-19T14:26:36Z Right. Thanks. I think it's OK now. I apparently had a problem with subscripts. http://mathoverflow.net/questions/88039/generalization-of-singular-moduli Comment by Nicolas B. Nicolas B. 2012-02-10T09:13:52Z 2012-02-10T09:13:52Z For the case of abelian surfaces, Igusa has constructed invariants that generalize the $j$-invariants of elliptic curves. In particular, there is a generalization of Hilbert Class Polynomial using them. I am definitely not an expert, but I've found this paper that can help <a href="http://msp.berkeley.edu/ant/2011/5-4/p03.xhtml" rel="nofollow">msp.berkeley.edu/ant/2011/5-4/p03.xhtml</a> http://mathoverflow.net/questions/85484/algebraic-approaches-to-modular-forms Comment by Nicolas B. Nicolas B. 2012-01-12T13:21:28Z 2012-01-12T13:21:28Z Another good reference is the springer book \textit{A first course in modular forms} of Diamond and Shurman. Or if you wish something more algebraic, you can have a look at the paper of Diamond and Im entitled \textit{Modular forms and modular curves}. http://mathoverflow.net/questions/55288/how-to-show-modularity-of-an-elliptic-curve/55292#55292 Comment by Nicolas B. Nicolas B. 2011-11-23T08:50:04Z 2011-11-23T08:50:04Z There is also a paper of Dieulefait, Guerberoff and Pacetti (see <a href="http://arxiv.org/abs/0804.2302" rel="nofollow">arxiv.org/abs/0804.2302</a>) where they use the Faltings-Serre method to prove modularity of some elliptic curves over imaginary quadratic fields. http://mathoverflow.net/questions/77646/whats-known-about-complete-split-primes-in-qep Comment by Nicolas B. Nicolas B. 2011-10-18T20:50:25Z 2011-10-18T20:50:25Z @Chandan Singh Dalawat: I think that there is a mistake in the equation of the elliptic curve in your first comment. The one displayed is the curve labeled 37a1 in Cremona's table, but if I compute the $a_{\ell}$'s of $E$ for $\ell=4831, 22051,\ldots$, I don't find $2\pmod{7}$. Am I mistaken? http://mathoverflow.net/questions/77646/whats-known-about-complete-split-primes-in-qep Comment by Nicolas B. Nicolas B. 2011-10-10T07:37:45Z 2011-10-10T07:37:45Z Have you looked at Adelmann's LNM (vol. 1761) &quot;The Decomposition of Primes in Torsion Point Fields&quot;?