Timeline for Which even lattices have a theta series with this property?
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| Jun 4, 2019 at 6:21 | comment | added | pregunton | The genus consists of these two lattices only, they are the unique even unimodular lattices in 16 dimensions. Two lattices having the same theta series are called "isospectral" (the term is used e.g. in this monograph by Conway); I'm not sure if isospectral lattices must always be in the same genus, maybe this follows from the Siegel-Weil formula but as I said I do not really know much about modular forms. | |
| Jun 4, 2019 at 2:50 | comment | added | Kimball | @pregunton For the 16-d examples, how large is the genus? What do the theta series look like for the other lattices in this genus? | |
| Jun 3, 2019 at 16:01 | comment | added | pregunton | Thanks for your comments! I'm glad to see I was not too far off, at least in the even-dimensional case. I agree that the 16-d examples seem to be a bit anomalous, since they are neither the only class in their genus nor have any obvious multiplicative structure (that I know of). Perhaps there is some notion of "class up to isospectrality" that we could use to find an uniform proof, though I don't hold out much hope. | |
| Jun 3, 2019 at 8:34 | history | answered | Kimball | CC BY-SA 4.0 |