User potap - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T14:04:57Z http://mathoverflow.net/feeds/user/23215 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112258/express-weierstrass-g-2-and-g-3-in-terms-of-theta-functions-of-the-periods Express Weierstrass' g_2 and g_3 in terms of theta-functions of the periods potap 2012-11-13T06:22:57Z 2012-11-13T13:39:30Z <p>If E is a complex elliptic curve defined as the quotient of C over a lattice generated by w_1 and w_2, then it can be also written in Weierstrass form y^2=4*x^3-g_2*x-g_3. The coefficients g_2 and g_3 can be computed as well-known Eisenstein sums, however, there is a better expression in terms of Jacobi theta-functions of w_1 and w_2 (and as a consequence, an expression for j-invariant via w_2/w_1). Unfortunately, I could not find that expression in the books on my bookshelf, like Koblitz, Milne, Silverman etc, and the only place on the Internet where I found such formulas, were two wiki-articles: <a href="http://en.wikipedia.org/wiki/Weierstrass" rel="nofollow">http://en.wikipedia.org/wiki/Weierstrass</a>'s_elliptic_functions and <a href="http://en.wikipedia.org/wiki/Theta_function" rel="nofollow">http://en.wikipedia.org/wiki/Theta_function</a>. These formulas, however, contradict each other, and neither of them, when run on computer, gives rational values for elliptic curves with complex multiplication and class number 1. Can anybody give a reliable link to correct formulas?</p> http://mathoverflow.net/questions/110408/real-elliptic-curves Real elliptic curves potap 2012-10-23T10:13:50Z 2012-10-23T10:13:50Z <p>Take an elliptic curve E as C/Z^2. Then z -> (P(z),P'(z)), where P is the Weierstrass P-function, embed this curve into CP^2 as the cubic y^2=4*x^3-g2*x-g3. The group structure on E corresponds to the geometric group law on the cubic in CP^2. Now suppose g2 and g3 are real and consider the real part of that cubic. Topologically, it may consist of two circles, e.g. when g2=4, g3=0. The preimage of this real curve in E must be a subgroup of C/Z^2. But it is immediately evident that C/Z^2 cannot contain a subgroup consisting of two disjoint circles. Can anyone solve this contradiction?</p> http://mathoverflow.net/questions/769/exhibit-an-explicit-bijection-between-irreducible-polynomials-over-finite-fields/95215#95215 Answer by potap for Exhibit an explicit bijection between irreducible polynomials over finite fields and Lyndon words. potap 2012-04-26T01:36:48Z 2012-04-26T01:36:48Z <p>The correspondence invented by Golomb relies on the choice of a primitive element a in the field of order q^n. Then, to each Lyndon word L=(l_0,l_1,...,l_{n-1}) one assigns the primitive polynomial having as a root the element a^{m(L)} where m(L) is the integer sum of l_i*q^i over i=0,1,...,n-1. </p> http://mathoverflow.net/questions/112258/express-weierstrass-g-2-and-g-3-in-terms-of-theta-functions-of-the-periods/112276#112276 Comment by potap potap 2012-11-14T04:45:32Z 2012-11-14T04:45:32Z @Robert: Shame on me! I turned back on my chair and immediately found a paperback copy of that book in Russian translation. How come I did not think about it? (Imagine that I know Herb personally and I have even participated in a party at his home...) Thanks a lot! http://mathoverflow.net/questions/112258/express-weierstrass-g-2-and-g-3-in-terms-of-theta-functions-of-the-periods/112282#112282 Comment by potap potap 2012-11-13T14:53:44Z 2012-11-13T14:53:44Z Thanks a lot. I managed to find a dejaview file of that book in Russian. This is exactly what I needed! The best. http://mathoverflow.net/questions/112258/express-weierstrass-g-2-and-g-3-in-terms-of-theta-functions-of-the-periods/112276#112276 Comment by potap potap 2012-11-13T14:35:57Z 2012-11-13T14:35:57Z I found a pdf file on the Internet. The book is VERY good, but it does not give any formulas for $j$ in terms of theta-functions of $\tau$. http://mathoverflow.net/questions/112258/express-weierstrass-g-2-and-g-3-in-terms-of-theta-functions-of-the-periods/112264#112264 Comment by potap potap 2012-11-13T14:13:11Z 2012-11-13T14:13:11Z Yes I know. However, you can take Weierstrass's $P$ and $P'$ for any lattice and write the relation between them which will depend both on $\omega_1$ and $\omega_2$ (the equivalence class of the curve will be defined by the ratio alone, of course). http://mathoverflow.net/questions/112258/express-weierstrass-g-2-and-g-3-in-terms-of-theta-functions-of-the-periods/112276#112276 Comment by potap potap 2012-11-13T13:39:23Z 2012-11-13T13:39:23Z Thanks a lot. I'll try to find that book. If all else fails, I'll look nt up on Amazon. http://mathoverflow.net/questions/112258/express-weierstrass-g-2-and-g-3-in-terms-of-theta-functions-of-the-periods/112264#112264 Comment by potap potap 2012-11-13T10:06:24Z 2012-11-13T10:06:24Z Thank you. But I think that $g_2$ cannot be expressed in terms of $\tau$ alone: it depends on both periods $\omega_1$ and $\omega_2$ and can change under homotheties! Do you mean that $\omega_1=1$ and $\omega_2=\tau$? By &quot;better&quot; I mean computational complexity, as I'm doing computer experiments. http://mathoverflow.net/questions/110408/real-elliptic-curves Comment by potap potap 2012-10-23T12:17:04Z 2012-10-23T12:17:04Z Yes you are right. Thanks a lot! My computer experiments didn't show that: I must write a better program to see this phenomenon.