Timeline for A constructive proof of the theorem of the cube
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 6, 2023 at 0:27 | comment | added | R. van Dobben de Bruyn | @DamianRössler maybe your comments should be turned into an answer! | |
| May 12, 2022 at 17:48 | comment | added | Damian Rössler | For an explicit formula, see p. IX of Breen, Fonction thêta et théorème du cube, LNM 980. On an elliptic curve, the theta function can be expressed in terms of the Weierstrass $\sigma$-function and the Dedekind $\eta$-function. For this, see p. 37 of Polishchuk, Abelian varieties, Theta functions and the Fourier transform. | |
| May 12, 2022 at 12:56 | comment | added | Damian Rössler | On an elliptic curve (more generally, an abelian variety), a divisor is the zero scheme of a theta function on the universal covering of the curve. The theorem of the cube asserts that a certain linear combination of pull-backs of a theta function by various maps (see the Lemma in R. van Dobben de Bruyn's answer) is a function invariant under the action of the lattice acting on the universal covering. This function gives you an analytic description of the function $f$ you seek (note that it might be hard to describe $f$ otherwise, unless you want a description in Weierstrass coordinates). | |
| May 9, 2022 at 20:18 | answer | added | R. van Dobben de Bruyn | timeline score: 8 | |
| May 9, 2022 at 17:24 | comment | added | Dimitri Koshelev | I already know the divisor $D$. I want to write down the function $f$ with this divisor. | |
| May 9, 2022 at 9:17 | comment | added | Tom Ducat | Only knowing $f_0,f_1,f_2$ does not provide enough information to uniquely determine the divisor $D$, so is the question actually meant to be "given $f_0,f_1,f_2$, how to write down an explicit divisor $D$ with $D\cdot V_i=\operatorname{div}(f_i)$"? | |
| May 7, 2022 at 17:21 | history | asked | Dimitri Koshelev | CC BY-SA 4.0 |