Alex Youcis

Berkeley, CA

ayoucis.wordpress.com

Age: 25

I am a second year graduate student at the University of California, Berkeley.

My interests are generally in the field of arithmetic geometry. More specifically, I am interested in various aspects of the Langlands program. My advisor is Sug Woo Shin.

7h
accepted Families of abelian varieties on the line (or more generally simply connected varieties)
7h
comment Families of abelian varieties on the line (or more generally simply connected varieties)
Thanks so much! I'm going to accept Donu's answer because it answers all three of my questions, but I greatly appreciate your help.
7h
comment Do local Galois representations always lift?
You should ask this on overflow.
10h
comment Families of abelian varieties on the line (or more generally simply connected varieties)
Thanks Donu! This makes me very happy. One question: is Q2 obvious without using the cited fact that a $\mathbb{Q}$-VHS is constant? Namely, it's clear that this+Scmid's result imply Q2, but am I missing some obvious way of avoiding this result of Steenbrink and Peters?
10h
revised Families of abelian varieties on the line (or more generally simply connected varieties)
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11h
comment Families of abelian varieties on the line (or more generally simply connected varieties)
@PiotrAchinger Hey Piotr! Thanks, I'll look into this. I'm having trouble locating the original paper. Do you have a clue what the title is? Thanks!
11h
comment Families of abelian varieties on the line (or more generally simply connected varieties)
Also, in this paper it's not clear to me whether the following is true. In the case of elliptic schemes the following works: choose a trivilization $\alpha:\mathscr{E}[n]\xrightarrow{\approx}(\mathbb{Z}/N\mathbb{Z})^2$. This then defines a map $\mathbb{A}^1_\mathbb{C}\to Y(N)$ which extends to a map $\mathbb{P}^1_\mathbb{C}\to X(N)$. For $N\gg 0$ the genus of $X(N)$ is positive and so this map must factor through a point—thus the family is isotrivial. Do you know if there is a way to proceed using the geometry of $\mathcal{A}_{g,1}$ (or $\mathcal{A}_{g,n}$, or its spaces with level structure)?
11h
comment Families of abelian varieties on the line (or more generally simply connected varieties)
Hey ACL, thanks for the information! Do you have a belief that $\mathscr{A}$ should be isogenous to a Jacobian (as in the case of fields)? And is your statement 'I don't know whether there is a reference in the literature' mean "I think it's true" or "It is true—no one's bothered to type it up yet"? Also, do you have any opinion about the 'proof'I described about above?
11h
awarded Citizen Patrol
11h
revised Families of abelian varieties on the line (or more generally simply connected varieties)
added 1 character in body
12h
asked Families of abelian varieties on the line (or more generally simply connected varieties)
14h
comment Morphisms for good reduction are maps respecting filtration
@nfdc23 Ah, yes, I see. I was being silly! You're just saying that $T_p A=T_p \mathscr{A}$, where $\mathscr{A}$ is a model, and that we get a decomposition into an étale part a twist. Why is it not $M(1)$ though? Oh, because you wrote it cohomologically. OK, so I feel satisfies as soon as someone can come and verify my reduction. Thanks again nfdc23!
17h
revised Morphisms for good reduction are maps respecting filtration
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18h
answered Can a separable isogeny of elliptic curves have an inseparable dual?
21h
comment Examples of famous 'workhorse' theorems
Surely the Arthur-Selberg trace formula is an example of such a theorem.
23h
comment Morphisms for good reduction are maps respecting filtration
@nfdc23 Also, unless I'm being terribly silly (and I might) won't the canonical lift's endomorphism algebra contain the lift of Frobenius—how can it then be $\mathbb{Q}_p$? Finally, do you know conditions on $A$ that will make this true?
23h
comment Morphisms for good reduction are maps respecting filtration
@nfdc23 Hello nfdc23, thanks for your comment. A few questions. First, it seems as though you're answering the $\ell=p$ Tate's isogeny theorem part of my question—you agree then that this is an equivalent formulation of the original statement? Second, I'd appreciate if you could expand upon your comment. Why does the generic fiber of the canonical lift have this cohomology necessarily? I looked in some articles (e.g. Katz) and couldn't find anything.
1d
comment Kähler differentials in an inseparable field extension
I mean, the easiest way of doing this is to say that smoothness is equivalent to locally finite presentation+flat+locally free differentials of correct rank, but that seems somewhat cheating. Some more elementary ideas are as follows: 1) Reduce to purely inseparable. Reduce to primitive purely inseparable where the result is obvious. 2) Use connection between differentials and trace pairing, and the fact that the trace pairing is zero if extension is not separable.
1d
revised Morphisms for good reduction are maps respecting filtration
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1d
awarded Yearling
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