Timeline for Elliptic curves whose $2,3,5$-parts of Sha are large
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 19, 2022 at 14:10 | comment | added | Sylvain JULIEN | Possibly related, while I wouldn't gamble on it: arxiv.org/abs/2209.08088 | |
| Sep 16, 2022 at 7:48 | comment | added | Chris Wuthrich | Delaunay heuristics will say that the proportion of elliptic curves with sha of size $(2\cdot 3\cdot 5)^{2k}$ is positive for all $k$, right? So you expect it to exist in that sense, no? But explicitly constructing one seems hard to me. A cyclic isogeny of degree 30 might have helped but they don't exist over $\mathbb{Q}$. | |
| Sep 15, 2022 at 20:43 | comment | added | Stanley Yao Xiao | @ChrisWuthrich the use of the word "family" is admittedly ambiguous. I am simply looking for a sequence $\{E_n\}_{n \in \mathbb{N}}$ of elliptic curves with the property that $\lim_{n \rightarrow \infty} \min \{|\text{Sel}_2(E_n)|, |\text{Sel}_3(E_n)|, |\text{Sel}_5(E_n)|\} = \infty$. "Large" family refers to a family which has positive density. | |
| Sep 15, 2022 at 20:29 | comment | added | Chris Wuthrich | What do you mean by "family"? And "large family"? | |
| Sep 15, 2022 at 18:32 | comment | added | Stanley Yao Xiao | @ChrisWuthrich I want to think about the problem of bounding the rank uniformly in large families, and one approach is to think about families where at least one of 2, 3, or 5 Selmer is bounded in size. These particular primes arose because we have a good understanding of the co-regular spaces parametrizing the Selmer elements. | |
| Sep 15, 2022 at 7:02 | comment | added | Chris Wuthrich | I ave no troubles believing that such a family exists. But I wonder what you want to do with it? | |
| Sep 15, 2022 at 7:01 | comment | added | Chris Wuthrich | It is only known in general that the $n$ torsion part is finite. The primary parts are known to be finite only if the analytic rank is zero or one (over the rationales). | |
| Sep 15, 2022 at 0:58 | history | asked | Stanley Yao Xiao | CC BY-SA 4.0 |