Timeline for Trace map on rational points of elliptic curves
Current License: CC BY-SA 4.0
13 events
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| May 20, 2021 at 18:31 | comment | added | François Brunault | If $L/K$ is cyclic then this $H^1$ can be computed as Tate cohomology and there is no reason for it to be zero or nonzero. E.g. if $E(L)=E(K)$ and there is no torsion then $H^1$ will be zero. PS. In the comments, you need to write @(...) if you want us to get notified. E.g. I'm adding @ChrisWuthrich in this comment. | |
| May 13, 2021 at 20:35 | comment | added | user176661 | Thank you guys, your comments have been very helpful. It seems it is much more involved than I thought. Btw how about $H^1$, is it known that $H^1 (G_{L/K}, E(L))$ is never 0? | |
| May 13, 2021 at 17:56 | comment | added | François Brunault | @ChrisWuthrich Thanks for the references. I'm curious whether it is possible at all to modify ETNC so that (loosely speaking) the equivariant $L$-function ''determines'' the Galois module structure of the Mordell-Weil group. In any case it seems quite involved. | |
| May 13, 2021 at 15:22 | comment | added | Chris Wuthrich | I have an unfinished paper in a drawer related to the question asked above. One shouldn't use etnc. Usually local conditions can be used to determine the surjectivity of the trace map and $E(L)\otimes\mathbb{Z}_p$ as a $\mathbb{Z}_p[G]$-module. | |
| May 13, 2021 at 15:19 | comment | added | Chris Wuthrich | One can get a link between etnc and the $G$-module of $E(L)$. However, all the work on etnc to make it explicit for elliptic curves (see Bley and Burns-Macias Castillo) impose conditions that force $E(L)\otimes\mathbb{Z}_p$ to be projective (and hence the $\hat H^0$ above is trivial. However it can't describe the $G$-module completely in the non-projective case as there is just not enough information in one element of $\mathbb{R}[G]$ compared to the vast number of possible lattices (in the torsion-free case) when $G$ is not a tiny group. | |
| May 13, 2021 at 13:52 | comment | added | François Brunault | @ChrisWuthrich The ETNC is an identity in (an elaborated version of) $\mathbb{R}[G]^\times/\mathbb{Z}[G]^\times$. Assuming we can compute the Galois module structure of the Tate-Shafaverich group (!), periods and so on, ETNC will give us the determinant of the height pairing in the above group. Does this give any information about the $G$-module $E(L)$? In a different direction, ETNC does give information about integral Galois modules, see Burns-Flach, "Motivic L-functions and Galois module structures" and their subsequent articles. | |
| May 13, 2021 at 9:34 | comment | added | Chris Wuthrich | @François Brunault I fear it is not that easy. You are right that the order of vanishing should give the rank, but that does not help with the finer question of what the Galois module is and the surjectivity of the trace will depend on that. If the extension is cyclic of prime degree $p$ and the rank grows from $1$ to $p$ and no torsion is present, there are two possible modules. The etnc will link the value of $L'(E,\chi,1)$ to the arithmetic, but only if all the other involved terms vanish it could distinguish the two cases. | |
| May 13, 2021 at 8:56 | comment | added | François Brunault | In theory the orders of vanishing of $L(E \otimes \rho,s)$ at $s=1$ should give you (only conjecturally) the $G$-module structure of $E(L) \otimes \mathbb{Q}$, and possibly the values (or derivatives) at $s=1$ will give $E(L)$ as $G$-module but I haven't checked. | |
| May 13, 2021 at 8:52 | comment | added | François Brunault | For a fixed $E/K$, if you know generators of $E(L)$ then you will be able to determine the $G$-module structure, and compute the image of the trace map by linear algebra. Also, you can consider the $L$-function $L(E/L,s)$ (base change of $E$ to $L$). It factors as a product of copies of $L(E \otimes \rho,s)$ where $\rho$ are the Artin representations of $G$. There is a generalisation of BSD conjecture for these $L$-functions (equivariant Tamagawa number conjecture for $L(E/L,s)$). | |
| May 13, 2021 at 8:31 | comment | added | François Brunault | $E(L)$ is a $G$-module of finite type and one may ask which $G$-modules arise in this way. Already for $G=\{1\}$ and $K=\mathbb{Q}$, this is highly non-trivial as it asks what are the possible Mordell-Weil groups of elliptic curves. | |
| May 13, 2021 at 7:59 | comment | added | Chris Wuthrich | If $G$ is the Galois group then the failure to to be surjective is measured in $\hat H^0(G, E(L))$. So this will depend on the Galois module structure of $E(L)$. A lot of things can happen. If the rank doesn't grow, it can happen (only rarely) that the map is surjective, but usually it isn't. Instead if the rank grows it is much harder to know. Even the local situation for a cyclic extension is complicated. | |
| May 13, 2021 at 7:54 | history | edited | YCor |
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| May 13, 2021 at 7:20 | history | asked | user176661 | CC BY-SA 4.0 |