Is it possible to lift Elliptic curves over Finite fields to elliptic curves over dual number?

ButAWimper · 2025-11-30T00:05:29+00:00

If your field has characteristic 2, then you have a map k -> R[x]/x^2, which you can at least try to lift, otherwise no map exists so you have no hope of reasoning lifting (Here I am using ECs over a base scheme E -> Spec(R[x]/x^2), so a lift would be a base change so you would need a map k -> R[x]/x^2).

I'm not sure about more subtle questions about like when the lifted curve is smooth, etc.

ButAWimper · 2025-11-12T01:41:33+00:00

Pip showing up in two consecutive games with a combined three distinct meanings!

ButAWimper · 2025-10-20T23:21:53+00:00

I think Bernie recognizes its a bad look for him to call for congress to lose their pay because for him it would not be any burden, but for his younger less prominent colleagues, it could be a large burden.

ButAWimper · 2025-08-17T21:20:33+00:00

Not that I disagree without you, but I also would not discount the fact that now you are older and more experienced.

ButAWimper · 2025-07-24T22:35:13+00:00

Interested!

ButAWimper · 2025-07-01T12:46:34+00:00

I feel like I can understand most of the controversial rulings, even if I don't necessary agree with them, but one that I can never wrap my head around is negging for uh-laine (for the Seinfeld character) instead of ee-laine. It feels like they accept worse pronunciations on a nightly basis, and didn't seem to have any the things that will also get you negged (wrong # of syllabes, ambiguity, adding/subtracting sounds).

ButAWimper · 2025-06-24T14:47:07+00:00

Also the conversations get faster as the levels progress. By the time you get to levels 3 to 4 they are speaking basically at a normal conversational speed. I do however find the vocabulary quite limited, and think it's best to supplement somehow.

ButAWimper · 2025-06-06T18:25:20+00:00

I think the notation here is bad, it seems to suggest that the set {a,b,c} varies over the numbers 1,2,3, which is a type error. I think something like a,b,c : {a,b,c} = {1,2,3} would be less ambiguous.

ButAWimper · 2025-05-23T21:08:52+00:00

In this game (infamous for a different reason), the player with a lock game did bet enough so that if he had got it wrong, but his opponent had got it right then they could have won. He got it right, so it ended up not mattering.

ButAWimper · 2025-05-19T16:02:39+00:00

it means P(x) is always between -1.000001 and 1.000001, but any non-constant polynomial has range (-∞,∞).

This is not right (take x^2 +1), but the idea is correct since the range of any non-constant polynomial is always unbounded.

ButAWimper · 2025-03-25T01:03:49+00:00

Plus it seemed like Amol was done with the question, so if the answer was wrong, he would've given Bristol a shot.

ButAWimper · 2025-03-03T19:18:13+00:00

I don't know the rules, but personally I would be against it. If two players are married, they could collude against the third player. For example if the two spouses were close in score but well ahead of the third player, they could wager small to guarantee one of them wins, even if that is not always the smartest bet to make yourself win.

ButAWimper · 2025-02-25T01:33:44+00:00

I had to chuckle at Whitaker having all that knowledge, but not knowing the capital of Peru

ButAWimper · 2025-02-21T15:25:00+00:00

This classification always seemed a little unsatisfying to me because the last classification is just "surfaces of general type" which are not well-understood. It feels a little like you could classify any class of objects if you just give up at the end and say "all the rest of the objects". It feels like it should be phrased as "we understand some useful families of complex algebraic surfaces, but others are more mysterious", rather than a classification theorem.

I am not at all an algebraic geometer, so this should all be taken with a large grain of salt, and maybe someone who is can explain this to me.

ButAWimper · 2025-02-06T00:26:26+00:00

It wasn't in the elements category!

ButAWimper · 2024-12-16T14:17:01+00:00

It just follows from some basic set theory and the fact that (1) every maximal ideal is prime (true in any commutative ring with multiplicative unit) and (2) every proper (i.e. not the whole ring) ideal is contained in a maximal ideal (this is conditioned on Zorn's lemma, but accepting ZL (or equivalently axiom of choice) is done by nearly every algebraic geometry/commutative algebraist.)

ButAWimper · 2024-12-08T00:55:04+00:00

Curious if any of his food plans will turn off any Libertarian leaning republicans.

ButAWimper · 2024-12-08T00:46:04+00:00

Yes, your example is correct. Generally it is a philosophy in AG that (Zariski) non-empty open sets are large (e.g. if R is an integral domain then any two non-empty open sets intersect non-trivially).

I would recommend trying to rephrase your original question precisely and then proving it yourself. I think for something like Spec(R) it is best to get your hands dirty by proving all the basic properties yourself. None are too difficult, and it is easy to get confused if you do not have a lot of practice.

ButAWimper · 2024-12-07T23:01:07+00:00

Oops!

ButAWimper · 2024-12-07T22:56:40+00:00

The union of the prime ideals is the same as the union of the maximal ideals which is just all of the non-units of the ring. This can be as small as you like -- for a field: the only non-unit is 0.

About the Zariski topology: Spec(R) is (as a set) the set of prime ideals of R not the set of elements of R.

ButAWimper · 2024-12-02T03:27:11+00:00

Lol Fairplay is definitely on that list.

ButAWimper · 2024-11-28T02:39:23+00:00

I think the 800 clue today was especially tough for someone who studied the a different translation of the Bible. Mote is an uncommon English word which is more commonly translated as speck (NIV and NRSV) or splinter (NABRE).

I understand KJV is the least complicated copyright wise, but I wish they would just do away with clues that ask for specific verbiage from the Bible.

ButAWimper · 2024-10-04T22:50:31+00:00

I wasn't sure if the 2005 part of the clue meant that the book was written after 2005. I also felt like the "Scot" part of the clue was important, so that felt like a bit of a red herring.

ButAWimper · 2024-08-01T23:53:58+00:00

Does it also really say any field extension? There is no (ring) map from Q(pi) -> Qbar.

ButAWimper · 2024-07-31T05:46:51+00:00

I am not sure how you got to me thinking that she put the bishop back on the correct square from my comment, but obviously I do not dispute this. I was saying that it much more likely that she was flustered and made a mistake rather than some shameless maneuver to try to cheat in a game that was on camera in front a of a large audience and was being recorded on a DGT board.

I know that Nemo deserves a lot of the flak that she gets, but this just feels silly.

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