Timeline for Why are we defining character groups differently for additive and multiplicative group in Tate's thesis?
Current License: CC BY-SA 4.0
10 events
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| Jul 18, 2023 at 18:23 | comment | added | paul garrett | More generally, over non-archimedean local fields, unipotent (linear) groups are ascending unions of compact subgroups. (This is what makes Jacquet modules work so well.) Not so in the archimedean case. And not for $GL(1)$ over non-archimedean fields, either. | |
| Jul 18, 2023 at 17:20 | comment | added | Will Sawin | @KevinCasto Oh, sorry, my comment was completely garbled. It should be characters of the additive group, where the exponential map provides the example to show the archimedean fields behave differently. | |
| Jul 18, 2023 at 2:31 | comment | added | Rits | @MattYoung, thank you, it was a great and a clear answer. It also answers all my questions in the comment in the below answer! | |
| Jul 18, 2023 at 2:27 | vote | accept | Rits | ||
| Jul 18, 2023 at 0:59 | comment | added | Will Sawin | @KevinCasto But it is not true for the fields $\mathbb R$ and $\mathbb C$, which also appear in his work, as the identity function gives a quasi-character that's not a character. So maybe taking circle-valued multiplicative characters is a way of forcing the fields to behave like each other, in that $\mathbb C^*$-valued multiplicative characters have different natures for the archimedean and non-archimedean fields. | |
| Jul 17, 2023 at 23:00 | comment | added | Matt Young | This is true for $k$ any locally compact non-archimedean field. Any $x \in k$ is contained in a compact subgroup, such as the set of $y \in k$ with $|y| \leq |x|$. The image of any compact subgroup must be contained in $S^1$. | |
| Jul 17, 2023 at 22:46 | comment | added | Kevin Casto | Is it true for all of the fields that Tate is considering that the image of any continuous additive character $k^+ \to \mathbb C^*$ actually lands in $S^1$? | |
| Jul 17, 2023 at 21:59 | comment | added | LSpice | Great explanation, and right to the point. This same idea is, in some very broad sense, at the root of why diagonalizable and unipotent groups behave so differently. | |
| Jul 17, 2023 at 21:58 | history | edited | LSpice | CC BY-SA 4.0 |
Typo
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| Jul 17, 2023 at 20:29 | history | answered | Matt Young | CC BY-SA 4.0 |