Timeline for Complex plane mod lattice to elliptic curve correspondence generalization
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 22, 2022 at 15:05 | comment | added | Jeremy Rouse | The problem of inversion of abelian integrals in general seems somewhat challenging, depending on what you're looking for. The paper here might be a good start. | |
| Jul 22, 2022 at 15:04 | comment | added | Jeremy Rouse | The link was to the paper "On the distribution of lengths of short vectors in a random lattice" by Seungki Kim which was published in Math. Z. in 2016. (You can find the paper on the author's newer website here.) | |
| Jul 21, 2022 at 2:20 | comment | added | Тyma Gaidash | @JeremyRouse The “here” link does not work. Shifting to another topic, do you know any sources giving an explicit form, even if “rather painful”, of the inverse of an Abelian integral in terms of Riemann theta functions? Thanks for your time. | |
| Jul 10, 2015 at 14:26 | comment | added | Jeremy Rouse | The paper here might be a good start for some prior results and some ways to think about questions like this, but I'm not entirely sure what exactly you're interested in investigating. | |
| Jul 10, 2015 at 13:55 | vote | accept | Samuel Reid | ||
| Jul 10, 2015 at 13:55 | comment | added | Samuel Reid | Exactly! I was using this to construct the moduli space of lattices and then wanted to investigate extremized quantities in moduli subspaces. Do you have a reference for any review/open problems related to this? I was wanting to explore this connection very deeply for my masters thesis. | |
| Jul 10, 2015 at 1:13 | comment | added | Jeremy Rouse | It is known that $\mathbb{C}^{g}/\Lambda$ can be realized as a subvariety of a projective space if and only if $\Lambda$ has a Riemann form. Is it possible that instead of working with abelian varieties, what you really want is to study the moduli space of lattices up to homothety? If so, the right object is ${\rm SL}(n,\mathbb{R})/{\rm SL}(n,\mathbb{Z})$, where $n$ is the rank. | |
| Jul 9, 2015 at 20:50 | comment | added | Samuel Reid | So, there is no hope for understanding this correspondence for free $\mathbb{Z}$-modules of rank 3? What if I embed $\Lambda = \bigoplus_{i=1}^{3} \omega_{i}\mathbb{Z}$ into a free $\mathbb{Z}$-module of rank 4, where the 4th dimensional basis vector is trivial? Then could I take $\mathbb{C}^{2}/\Lambda$? I need the odd-dimensional case, and I'm not sure how this generalizes to that, although thank you for the response and references. | |
| Jul 9, 2015 at 20:17 | history | answered | Jeremy Rouse | CC BY-SA 3.0 |