Is there any literature on multivariable theta functions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T14:40:54Z http://mathoverflow.net/feeds/question/29136 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29136/is-there-any-literature-on-multivariable-theta-functions Is there any literature on multivariable theta functions? Simon Rose 2010-06-22T19:01:05Z 2010-06-22T23:13:13Z <p>The theta function of a lattice is defined to be $$\vartheta_\Lambda = \sum_{v\in\Lambda} q^{{\Vert v\Vert}^2}$$ which yields as a coefficient of <i>q<sup>k</sup></i> the number of vectors of norm-squared <i>k</i>.</p> <p>On the other hand, the Jacobi theta function is given by $$\vartheta(u,q) = \sum_{n=-\infty}^\infty u^{2n}q^{n^2}$$ and we have the obvious fact that if $\Lambda = \mathbb{Z}$ with its usual intersection form, then $\vartheta(1,q)$ is the theta function for that lattice.</p> <p>We also have the fact that $\vartheta_{\Lambda_1 \oplus \Lambda_2} = \vartheta_{\Lambda_1}\vartheta_{\Lambda_2}$, and so we can decompose our theta functions into products of theta functions of primitive lattices.</p> <p>Combining these facts, it is not entirely ridiculous to hope that there is some way to write, for a lattice of rank <i>k</i>, a theta function'' of the form $$\vartheta(u_1, \ldots, u_k, q)$$ such that $\vartheta(1, \ldots, 1, q)$ is the ordinary theta function of the lattice. In some sense, the <i>u</i>-variables keep track of the basis elements of the lattice which immediately raises the question as to well-definedness of such an idea; it is worth noting that for the lattice $\Lambda = \bigoplus_i\mathbb{Z}$ that this definition does make sense.</p> <p>So is there any literature on such objects? Do they make sense for lattices which are not just sums of copies of &#8484;? Do they have nice relations akin to those of normal theta functions?</p> http://mathoverflow.net/questions/29136/is-there-any-literature-on-multivariable-theta-functions/29146#29146 Answer by S. Carnahan for Is there any literature on multivariable theta functions? S. Carnahan 2010-06-22T20:49:45Z 2010-06-22T20:49:45Z <p>I'm not entirely sure what you're seeking, but Wikipedia has a short paragraph on <a href="http://en.wikipedia.org/wiki/Theta_function#Riemann_theta_function" rel="nofollow">Riemann theta functions</a>. They seem to be a suitable higher-dimensional generalization of the 2-variable Jacobi form you described.</p> <p>I'm not an expert in this subject, but my understanding is that a theta function is a section of a line bundle on the total space of a universal family of abelian varieties. That is, we have a smooth map $\pi: X_g \to A_g$, where $A_g$ is a moduli space of abelian varieties of dimension $g$ (maybe with a level structure and a polarization), and the fiber over a point on the parameter space describes the abelian variety being parametrized. I think Mumford's <em>Tata Lectures on Theta</em> describe this viewpoint in some detail.</p> http://mathoverflow.net/questions/29136/is-there-any-literature-on-multivariable-theta-functions/29165#29165 Answer by SandeepJ for Is there any literature on multivariable theta functions? SandeepJ 2010-06-22T23:13:13Z 2010-06-22T23:13:13Z <p>There are three ways to view theta functions</p> <ol> <li>as classical homomorphic functions in vector z and/or period matrix T</li> <li>as matrix coefficients of a representation of the Heisenberg and/or Metaplectic grp</li> <li>as sections of Line bundles on the Abelian variety and/or moduli space of the abelian variety</li> </ol> <p>Ram Murty's <a href="http://books.google.com/books?id=ZXvCc0zLtKUC" rel="nofollow">Theta functions - from the classical to the modern</a> discusses Weil's representation-theoretic interpretation of theta functions. See chapter 3 by Hoffstein on <em>Eisenstein series and theta functions on the metaplectic group</em>. It is the connection to the <a href="http://en.wikipedia.org/wiki/Metaplectic_group" rel="nofollow">metaplectic group</a> which gives rise to the functional equation of the multivariable theta function, which you will also find in the chapter on the Metaplectic group in vol 3 of Mumford's <em>Tata Lectures</em></p> <p>Bellman's <a href="http://www.amazon.com/Brief-Introduction-Theta-Functions-Bellman/dp/0030103606" rel="nofollow">Brief introduction to Theta functions</a> Section 61 alludes to theta functions in several complex variables.</p> <p>You may also want to search for material in books on Abelian varieties. For example, Baker's <a href="http://books.google.com/books?id=q4iHCNXXX_UC&amp;lpg=PP1&amp;pg=PR9#v=onepage&amp;q&amp;f=false" rel="nofollow">Abelian functions Chapter X</a> develops the theory based on the period matrix. Also Murty's book on <a href="http://books.google.com/books?id=1bJKqHqWgp4C&amp;lpg=PA3&amp;dq=Murty%2520Abelian%2520Varieties&amp;pg=PA3#v=onepage&amp;q&amp;f=false" rel="nofollow">Abelian varieties</a> and Polishchuk's <a href="http://www.amazon.com/Abelian-Varieties-Functions-Fourier-Transform/dp/0521808049" rel="nofollow">Abelian Varieties, Theta Functions and the Fourier Transform</a> </p> <p>Tyurin's <a href="http://www.amazon.com/Quantization-Classical-Quantum-Functions-Monograph/dp/0821832409/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1277247071&amp;sr=1-1" rel="nofollow">Quantization, Classical and Quantum Field Theory, and Theta Functions</a> might also be a useful reference, which I haven't browsed.</p> <p>See also: <a href="http://eom.springer.de/t/t092600.htm" rel="nofollow">Springer Encyclopedia of Math entry on theta functions</a></p>