Timeline for K3 surfaces that correspond to rational points of elliptic curves
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 31, 2014 at 15:32 | comment | added | Lev Borisov | @AbhinavKumar No, can you please send me a reference? This is more or less just a hobby for me at this point, so my knowledge of the literature is very spotty. | |
| Jul 31, 2014 at 13:55 | comment | added | Abhinav Kumar | @LevBorisov Are you aware of the work of Kudla and recent work of Darmon and others? It might have some connection to what you're looking for. | |
| Jul 31, 2014 at 2:07 | comment | added | Lev Borisov | @AbhinavKumar Indeed, theorem 7.6 of Dolgachev's paper gives some other description of the corresponding K3. Perhaps, it is more interesting than the Kummer K3, although either one may be suitable for the purposes of finding interesting points on $X_0(n)$ over $\bar{\mathbb Q}$. | |
| Jul 31, 2014 at 1:35 | comment | added | Lev Borisov | I am not interested in rational points on $X_0(n)^+$ as much as I am interested in rational divisors on it. Basically, I am after a more advanced version of the Heegner points -- some interesting points on $X_0(n)$ defined over $\bar {\mathbb Q}$ such that their images on elliptic curves of rank higher than $1$ give nontrivial traces. Note that Heegner points would only give traces that are torsion by Gross-Zagier formula. | |
| Jul 30, 2014 at 23:55 | comment | added | Abhinav Kumar | The field of definition of the point $P$ on $X_0(n)^+$ should probably correspond to the field of the definition of the corresponding divisor on the K3 surface. So saying that it comes from a $\bf{Q}$ point just means that the Picard group of the K3 can be fully realized over a small degree number field (here, probably the $2$-torsion field of the elliptic curves). | |
| Jul 30, 2014 at 23:35 | comment | added | Abhinav Kumar | It seems that the K3 is not the Kummer, but is is a double cover (i.e. related by a Shioda-Inose structure to the product abelian surface). See theorem 7.6 of that paper. | |
| Jul 30, 2014 at 21:00 | comment | added | Joe Silverman | Okay, that would be interesting, to geometrically relate the K3s and the elliptic curves. But I don't see how either is related to the point on $E$, and especially, how would the group law on $E$ relate to the $E_i$'s or the K3's. | |
| Jul 30, 2014 at 20:53 | comment | added | Lev Borisov | I don't think that these K3-s are all that hard to construct from the elliptic curves. My best guess is that if $E_1\to E_2$ is the corresponding cyclic isogeny, then the K3 is the resolution of $E_1\times E_2/<(-1,-1)>$, but I have not tried to verify it. | |
| Jul 30, 2014 at 19:06 | comment | added | Lev Borisov | Right. I am somehow hoping that by looking at a more complicated object, namely the associated K3 surface, one might discover something that is not visible at the elliptic curve level. | |
| Jul 30, 2014 at 18:56 | history | answered | Joe Silverman | CC BY-SA 3.0 |