Values of the j-function - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T11:14:23Z http://mathoverflow.net/feeds/question/36226 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36226/values-of-the-j-function Values of the j-function Aeryk 2010-08-20T21:30:46Z 2010-08-21T00:48:37Z <p>In general, how do you compute the algebraic values of the modular j-function at quadratic imaginary points? (In other words, how do you compute the algebraic values of singular moduli?)</p> <p>For instance, the Mathematica website (http://mathworld.wolfram.com/j-Function.html) gives the standard nine integral examples that result when the class number \$h_k=1\$, but then it gives 18 examples for when the class number is 2 without any specific references. How does one compute these? More importantly, can you also do it for higher degree cases? Or even just find the defining degree-\$h_k\$ polynomial?</p> http://mathoverflow.net/questions/36226/values-of-the-j-function/36232#36232 Answer by Richard Borcherds for Values of the j-function Richard Borcherds 2010-08-20T22:13:08Z 2010-08-20T22:13:08Z <p>One crude but effective method is to compute all the h conjugates a<sub>j</sub> numerically to high precision, from which you can find the polynomial &Pi;(x-a<sub>j</sub>) they are the roots of using the fact that it has integral coefficients, (h=class number, and the values a<sub>j</sub> are the values of j at the imaginary quadratic integers with the same discriminant.)</p> <p>Alternatively see the paper <a href="http://www.reference-global.com/doi/abs/10.1515/crll.1985.355.191" rel="nofollow">On singular moduli</a> by Gross and Zagier, which gives an explicit expression for the values of j as products of many small algebraic integers. </p> http://mathoverflow.net/questions/36226/values-of-the-j-function/36243#36243 Answer by SandeepJ for Values of the j-function SandeepJ 2010-08-21T00:48:37Z 2010-08-21T00:48:37Z <p>In case the Gross-Zagier paper doesn't meet your needs, you can also refer to the following </p> <ol> <li>Harold Baier <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.111.5103" rel="nofollow">Efficient computation of singular moduli</a> </li> <li>Noriko Yui <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.730" rel="nofollow">On The Singular Values Of Weber Modular Functions</a></li> </ol> <p>and of course David Cox's book <a href="http://www.cs.amherst.edu/~dac/primes.html" rel="nofollow">Primes of the form \$x^2 + Ny^2\$, Section 3.12</a></p> <p>The bad thing about the j-function is that it is a level 1 modular function so the coefficients of its defining polynomial are going to explode with increasing degree. Its easier to compute the singular moduli using a modular function of some higher level (e.g. Weber func has level 48) as demonstrated in the papers mentioned above as well in Cox's book.</p>