Timeline for Principal ideal domains with finitely many units
Current License: CC BY-SA 4.0
22 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 14, 2021 at 11:59 | vote | accept | Mare | ||
| Apr 9, 2021 at 18:04 | comment | added | David E Speyer | So they get $\tfrac{q^{a-g+1}-1}{q-1} \geq \tfrac{q^a - 2g q^{a/2}+1}{a}$. It turns out that the optimal choice is $a = 2g-1$ (just big enough for RR to apply), and then this forces $g \leq 4$. | |
| Apr 9, 2021 at 18:03 | comment | added | David E Speyer | But also, given any $\mathbb{F}_{q^a}$ point of $X$, summing up its $\mathrm{Gal}(\mathbb{F}_{q^a}/\mathbb{F}_q)$ conjugates gives a degree $a$ divisor defined over $\mathbb{F}_q$, which must be equivalent to $a x_{\infty}$ by the class number one hypothesis. Once $a$ is large enough for Riemann-Roch to kick in, the number of effective divisors equivalent to $a x^{\infty}$ is $\tfrac{q^{a-g+1}-1}{q-1}$. | |
| Apr 9, 2021 at 18:02 | comment | added | David E Speyer | Oh, this is nice too! They get the same reduction to $q=2$ or $3$ that I do. Then they look at $\# X(\mathbb{F}_{q^a}$ in two ways. By Weil, this is $\geq q^a - 2 g q^{a/2} +1$. | |
| Apr 9, 2021 at 16:10 | comment | added | David E Speyer | Huh, this paper takes a totally different approach. I used to like my approach better, but now I have discovered I miscomputed a term in the $q=2$, $g$ even case and my solution doesn't work. I'll go see how they handle it. | |
| Apr 9, 2021 at 16:09 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 191 characters in body
|
| Apr 9, 2021 at 14:31 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 305 characters in body
|
| Apr 9, 2021 at 13:24 | comment | added | pregunton | I just found this paper which lists eight cases of function field with class number 1, including the four in your answer. | |
| Apr 9, 2021 at 12:47 | comment | added | David E Speyer | Remarks: I did go back and check that the curves are unique; they are. | |
| Apr 9, 2021 at 11:21 | history | bounty ended | Mare | ||
| Apr 9, 2021 at 10:31 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 2 characters in body
|
| Apr 8, 2021 at 16:39 | history | edited | David E Speyer | CC BY-SA 4.0 |
edited body
|
| Apr 8, 2021 at 15:37 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 320 characters in body
|
| Apr 8, 2021 at 15:27 | history | edited | David E Speyer | CC BY-SA 4.0 |
deleted 1 character in body
|
| Apr 8, 2021 at 15:19 | history | edited | David E Speyer | CC BY-SA 4.0 |
deleted 5 characters in body
|
| Apr 8, 2021 at 15:11 | history | edited | David E Speyer | CC BY-SA 4.0 |
deleted 5 characters in body
|
| Apr 8, 2021 at 15:03 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 1631 characters in body
|
| Apr 8, 2021 at 14:07 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 422 characters in body
|
| Apr 8, 2021 at 14:02 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 422 characters in body
|
| Apr 8, 2021 at 13:49 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 246 characters in body
|
| Apr 8, 2021 at 13:42 | history | edited | David E Speyer | CC BY-SA 4.0 |
added 40 characters in body
|
| Apr 8, 2021 at 13:23 | history | answered | David E Speyer | CC BY-SA 4.0 |