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6
votes
0answers
136 views
Torelli-like theorem for K3 surfaces on therms of its ´tale cohomology
Is there a proof of a Torelli-like Theorem for a K3-surface over any field (non complex) in terms of its etale or crystalline cohomology?
For example: If $K\ne \mathbb{C} $ and $X\rightarrow ...
4
votes
2answers
175 views
Restricting the composition factors of subgroups of GL_m(Z/nZ)
For a positive integer $m$, let $\mathcal{A}(m)$ be the set of all integers $k \geq 5$ such that: there is a positive integer $n$ and a subgroup $G \subset \operatorname{GL}_m(\mathbb{Z}/n\mathbb{Z})$ ...
3
votes
0answers
76 views
Notion of good supersingular reduction for proper smooth variety over a $p$-adic field
Let $X$ be a proper smooth variety over a $p$-adic field $K$. Let $\mathcal{O}_K$ be the ring of integers of $K$ and $k$, its residue field. We say that $X$ has good ordinary reduction if there is a ...
2
votes
0answers
86 views
Semiabelian actions appearing in the toroidal campactification of a degenearting abelian varieties
Given a totally degenerated abelian variety $A_K$ (to make it easier) over a complete discrete valuation field $K$ with $R$, $\pi$ and $k$ the corresponding discrete valuation ring, uniformiser and ...
1
vote
1answer
148 views
Unexplained techniques in Demjanenko's “Sums of 4 Cubes”:
A "not well understood" proof, do you know if one knows by now the conceptual background?: http://www.math.u-bordeaux1.fr/~cohen/sum4cub.ps
1
vote
0answers
130 views
Katz's paper on Serre Tate local moduli
In katz's paper "Serre-Tate local moduli" chaper 3 has the following construction:
Let $A$ be a fixed ordinary elliptic curve defined over $k$ of char $p>0$. Consider the deformation of $A$ to ...
1
vote
0answers
137 views
Compactifications of group schemes
Let $G$ be a group scheme over a scheme $S$ which is the spectrum of a discrete valuation ring. Let $\eta$ (resp. $s$) be the generic (resp. closed) point. Assume that the generic fiber $G_{\eta}$ is ...
3
votes
1answer
74 views
Elliptic curves over global function fields and independence of l-adic representations
Serre has shown that the family of $\ell$-adic Galois representations of an elliptic curve defined over a number field $K$ is almost independent. More explicitly:
let $E/K$ be an elliptic curve and ...
5
votes
1answer
305 views
Serre-Tate 1964 Woods Hole notes
I am not sure if this is the right venue to ask this. Apologies in advance.
I would like to clarify the following. When people give as reference:
J.-P. SERRE and J. TATE.-Mimeographed notes from ...
0
votes
0answers
64 views
How to construct a reduction type that we need from a smooth curve
Let $X_{k}$ be a stable curve over a algebraically closed field $k$. We can find a complete DVR $R$ and deform $X_{k}$. Then we can obtain a stable curve $X$ over $R$ whose generic fiber $X_{\overline ...
3
votes
1answer
140 views
Relation between Lee and Yang' s “circle theorem”, zeta functions and Weil conjectures?
Ruelle mentions ( http://www.ihes.fr/~ruelle/PUBLICATIONS/%5B94%5D.pdf ) Lee and Yang' s "circle theorem", which comes from statistical mechanics and shall have not yet explored connections with zeta ...
7
votes
1answer
523 views
Are overlaps among {algebraic geometry, arithmetic geometry, algebraic number theory} growing?
From a naive outsider's viewpoint, just watching the MO postings
in those three fields scroll by, and hearing of breakthroughs in the news,
it appears there might be increasing overlap among the ...
4
votes
0answers
64 views
On the definition of LGP-monoids in IUT III
I have been trying to understand, without success, the definition of "LGP-monoids" on p. 80 of Mochizuki's IUT III and was wondering if anyone could provide some more explanation than what is given ...
4
votes
1answer
253 views
$R^2f_{\operatorname{et},*}\mathbb{G}_m$ vs $R^2f_{\operatorname{Zar},*}\mathbb{G}_m$
Let $S$ be the spectrum of a discrete valuation ring and $f:X\rightarrow S$ be a relative projective curve with generic fiber smooth and special fiber semistable. How much differ the sheaf ...
0
votes
0answers
81 views
How does one see level structure in the ell-adic Galois representation
This is probably a very easy question for those familiar with the arithmetic theory of abelian varieties because it is either possible or impossible for "trivial reasons".
Let $A$ be an abelian ...
