User jeff yelton - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T12:23:29Z http://mathoverflow.net/feeds/user/24757 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/121765/field-of-definition-of-isogenies-of-abelian-varieties field of definition of isogenies of abelian varieties Jeff Yelton 2013-02-14T02:31:04Z 2013-02-14T02:31:04Z <p>Let $A$ be an abelian variety over a field $k$, and let $N$ be a finite subgroup of $A$. Suppose that $N$ is also defined over $k$, or at least that all Galois automorphisms fixing $k$ leave $N$ invariant. Then I've heard it stated that there is an abelian variety $B / k$ and an isogeny $\phi: A \rightarrow B$ with kernel $N$, such that $\phi$ is defined over $k$. I can't seem to find this stated anywhere (except in Silverman, where it is stated for elliptic curves as an exercise) and am not sure how to prove it. Does anyone know how to prove this and/or where to cite it in the literature?</p> http://mathoverflow.net/questions/101269/generators-of-principal-congruence-subgroups-and-sage-package generators of principal congruence subgroups and Sage package Jeff Yelton 2012-07-03T23:38:42Z 2012-07-04T06:09:09Z <p>I'm trying to find an explicit minimal set of generators for principal congruence subgroups of $\mathrm{SL}_{2}(\mathbb{Z})$, $\Gamma(N)$ for $N$ all powers of $2$. I know the question has been asked before as to how to find a minimal set of generators for congruence subgroups of special linear groups in the $n = 2$ case, and it was mentioned that there is an algorithm for computing this using Farey symbols. There is a package for Sage written by Chris Kurth which I would like to download, but it seems that I can't find a working link to it. I guess my main questions are as follows:</p> <p>1) Can anyone tell me how to get this KFarey package on Sage? (Unfortunately, it's probably impractical for large $N$...)</p> <p>2) Does anyone have any other practical idea as to how to find a minimal set of generators for each $\Gamma(2^{n})$? In particular, if anyone happened to know the answer even for $\Gamma(4)$, it would be greatly helpful to me in the short term.</p> <p>3) (In case explicit generators cannot easily be found) does anyone know how to compute the abelianization of each $\Gamma(2^{n})$?</p> <p>Thanks very much!</p> <p>Jeff</p> http://mathoverflow.net/questions/100322/elements-of-the-pure-braid-group-seen-as-galois-automorphisms/100817#100817 Answer by Jeff Yelton for elements of the pure braid group seen as Galois automorphisms Jeff Yelton 2012-06-27T22:35:35Z 2012-06-27T22:35:35Z <p>Okay, I see this, and thank you for the answers. I may never be able to describe the maximal extension field, but fortunately, I probably don't need that. Does anyone have any idea how I may compute the Galois action on particular elements of field extensions? For instance, I'm interested in knowing how the generators of $P_{3}$ act on such elements as $\sqrt{\sqrt{t_{1} - t_{3}} - \sqrt{t_{2} - t_{3}}}$. I'm not sure I have the topological intuition to be able to deduce this, especially since we're looking at dimension > 1 spaces. It would be very helpful in general to know how $P_{n}$ acts on square roots of sums of elements whose behavior under $P_{n}$ we already know (as in the example I just mentioned).</p> http://mathoverflow.net/questions/100322/elements-of-the-pure-braid-group-seen-as-galois-automorphisms elements of the pure braid group seen as Galois automorphisms Jeff Yelton 2012-06-22T03:16:57Z 2012-06-27T22:35:35Z <p>Let $Y_{n}$ be what I'm calling the ordered configuration space, the topological space of all ordered subsets of $\mathbb{C}$ of cardinality $n$. This can be viewed as $$\mathrm{Spec}(\mathbb{C}[t_{1}, ... , t_{n}, {(t_{i} - t_{j})^{-1}}_{1 \leq i &lt; j \leq n}]).$$</p> <p>The (topological) fundamental group of this is the pure braid group $P_{n}$. So by a general version of the Riemann Existence Theorem, the elements of $P_{n}$ must be Galois automorphisms of the maximal Galois extension of the function field $\mathbb{C}(t_{1}, ... , t_{n})$ which is unramified except at the primes $(t_{i} - t_{j})$. How may I describe algebraically the Galois automorphisms corresponding to, say, the generators of $P_{n}$?</p> <p>I have a guess (something along the lines of each generator $A_{i,j}$ of $P_{n}$ sending $(t_{i} - t_{j})^{1/n} \mapsto \zeta_{n}(t_{i} - t_{j})^{1/n}$, except that seems to make the Galois group abelian) but am not sure how to prove anything systematically, and I can't find this explicitly described in any source. Thank you for any help you can give.</p> http://mathoverflow.net/questions/121765/field-of-definition-of-isogenies-of-abelian-varieties Comment by Jeff Yelton Jeff Yelton 2013-02-14T03:30:08Z 2013-02-14T03:30:08Z Do you mean, in his &quot;varietes abeliennes&quot; book? http://mathoverflow.net/questions/101269/generators-of-principal-congruence-subgroups-and-sage-package Comment by Jeff Yelton Jeff Yelton 2012-07-05T03:50:06Z 2012-07-05T03:50:06Z Yes, I suppose I should be able to think of $\Gamma(2^{n}) / &lt;-1&gt;$ as the fundamental group of the associated Riemann surface and then the abelianization should be the first homology group, which would be $\mathbb{Z}^{2g}$?