All Questions
2,494 questions
2
votes
1
answer
499
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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
CommunityBot
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7
votes
1
answer
1k
views
Status of Ihara's lemma for Shimura curves over totally real fields?
What is the status of Ihara's lemma for Shimura curves over totally real fields?
In particular, why is it not implicit in the exact sequence of Rajaei, "On the levels of mod $l$ Hilbert modular forms" ...
CommunityBot
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1
vote
0
answers
190
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 ...
- 3,359
8
votes
1
answer
864
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 ...
- 1,031
3
votes
1
answer
569
views
Shafarevich's theorem for elliptic curves defined over function field of algebraic curve over algebraically closed field
Let $K$ be a number field and $S$ a finite set of places of $K$. Then Shafarevich's theorem states that there are only finitely many isomorphism classes of elliptic curves $E$ over $K$ with good ...
- 103
3
votes
1
answer
254
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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 ...
- 37k
9
votes
4
answers
3k
views
reduction of CM elliptic curves
Can someone indicate how to prove the following equivalences for a CM elliptic curve $E$:
(i) $p$ is inert in End($E$)
(ii) $E_p$ is supersingular
(iii) The trace of the Frobenius at $p$ is $0$ [...
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25
votes
0
answers
1k
views
Status of the Euler characteristic in characteristic p
In the introduction to the Asterisque 82-83 volume on `Caractérisque d'Euler-Poincaré, Verdier writes:
Enfin signalons que la situation en caractéristique positive est loin
d'être aussi ...
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6
votes
1
answer
352
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 ...
- 91.8k
17
votes
1
answer
6k
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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 ...
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12
votes
0
answers
267
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 ...
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5
votes
1
answer
312
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 $R^2f_{\...
- 3,993
13
votes
1
answer
819
views
Potential semi-stability of etale cohomology of etale covers.
Let $\mathbf{Q}_p$ denote the field of $p$-adic numbers.
Suppose that $K/\mathbf{Q}_p$ is a finite extension, and let $O_K$ denote the ring of integers of $K$. Suppose that $X$ is proper over $O_K$, ...
CommunityBot
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6
votes
1
answer
578
views
Vanishing cohomology of de-Rham Witt complex
Let $X$ be a smooth scheme over $\mathbb{F}_{p}$ for a prime number $p$. As far as I understand,
there is a surjective morphism from
$\Omega^\bullet_{W\mathcal{O}_X} \to W \Omega_{X}^\bullet$ which ...
- 1,043
0
votes
1
answer
228
views
Algebraic varieties in "mixed" affine spaces
Let $K\subset L$ be a field extension and let $K\subset F_1,F_2,...,F_n\subset L$ be proper intermediate fields. Consider the "mixed" affine space $\mathbb{A}_{(F_i)}:=\prod_{i=1}^n F_i$ instead of $\...
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18
votes
1
answer
552
views
Is there a known example of a curve X of genus > 1 over Q such that we know the number of points of X over the n-th cyclotomic field, for every n?
By Falting's theorem, these numbers are of course finite. Is there an example where we can explicitly compute them for every $n$?
Thank you!
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1
vote
1
answer
994
views
Comparison between Etale and Zariski topology on schemes
Let $Sch_{Zar}, Sch_{et}$ denote scheme with Zariski and Etale topology respectively. Is there a functor from $Sch_{et}$ to $Sch_{Zar}$ (or from $Sch_{Zar}$ to $Sch_{et}$) which preserves fiber ...
- 40.3k
7
votes
1
answer
943
views
Is Gouvêa-Mazur's "Infinite Fern" a fractal?
[EDIT]: Following Qiaochu Yuan's comment, it is better to clarify that I do not know what the right definition of a fractal in the following question should be. But a nice answer might contain such a ...
- 5,670
3
votes
1
answer
674
views
Why is the base change functor faithful
Let $L/k$ be a field extension of algebraically closed fields of characteristic zero. Let $U$ be a smooth quasi-projective variety over $k$.
I am trying to understand why the base-change functor from ...
- 63.1k
9
votes
1
answer
1k
views
Potentially good, semi-stable reduction => good reduction ?
Does a smooth proper variety having semi-stable reduction as well as potentially good reduction have good reduction ?
Note that over a $p$-adic field, this is true for the Galois representations in ...
- 149k
4
votes
1
answer
334
views
pro-$\ell$ etale fundamental group of a semi-abelian variety
Let $A$ be a semi-abelian variety over $K$, $\ell$ a prime number which is not equal to char($K$).
Does the abelianization of geometrically pro-$\ell$ etale fundamental group $(\pi_{1}(A\otimes\...
- 565
3
votes
1
answer
115
views
Rational points on the curve y^p=f(x) in characteristic p
Let $K$ be a finite extension of $\mathbb{F}_q(t)$ and define the curve $C$ by
the equation $y^p=f(x)$ where $p=\mathbf{char} K$ and $f\in K[x]$.
What is the genus of $C$? When does it have infinitely ...
- 30.5k
4
votes
4
answers
4k
views
Roadmap to reach Arithmetic Geometry for a Physics Major
Hi Everybody! I am physics major but I read mathematics for myself. my main fields of interest are number theory and geometry. it seems that due to the works of A.Grothendieck, algebraic geometry must ...
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6
votes
2
answers
3k
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Examples of (Phi,Gamma)-modules
What is the (Phi,Gamma)-module of an elliptic curve over Z_p, expressed by a direct construction ?
CommunityBot
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40
votes
1
answer
14k
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Why is Faltings' "almost purity theorem" a purity theorem?
My understanding of purity theorems is that they come in several flavors:
1) Those of the form "this Galois representation is pure, i.e. the eigenvalues of $Frob_p$ are algebraic numbers all of whose ...
- 21.3k
3
votes
0
answers
298
views
What does Hodge theory tell us about simply connected surfaces of general type
Let $X$ be a smooth complex projective variety. We know that $\Omega^1_X$ has a non-zero section if and only if the abelianization of the fundamental group of X is infinite. This follows from Hodge ...
- 43
1
vote
1
answer
708
views
Can the Albanese map be anything?
Sorry for the vague title. This question is about the Albanese map from the variety $M$ of canonically polarized varieties to the set of abelian varieties. (The variety $M$ is not of finite type...)
...
- 6,253
2
votes
1
answer
470
views
Specialization of sections in an elliptic fibration
Let $\pi: X \rightarrow S$ be the Neron model of an elliptic curve over a dedekind domain (but probably any minimal elliptic fibration will suffice).
Let $\eta$ be the generic point of $S$, $K = S(\...
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1
vote
2
answers
577
views
References for period matrix of abelian variety
Hi, everyone.
I am looking for some references for period matrix of abelian variety over arbitrary field, if you know, could you please tell me?
For period matrix of abelian varieties, I means that ...
CommunityBot
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5
votes
2
answers
586
views
Quotient of a reductive group by a non-smooth subgroup
This is a continuation of my question Quotient of a reductive group by a non-smooth central finite subgroup.
Let $G$ be a smooth, connected, reductive $k$-group over a field $k$ of characteristic $p&...
CommunityBot
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5
votes
2
answers
335
views
Can one bound the Quadratic Points on Curves?
Let $C$ be a nonsingular projective curve defined over $\mathbb{Q}$, which does not admit a map of degree 1 or 2 to $\mathbb{P}^1$ or to an elliptic curve. It is then a consequence of Corollary 3 of [...
- 2,827
2
votes
1
answer
466
views
Minimal semistable model for K3-surfaces.
I wonder if a semistalbe K3 surface over a $p$-adic field has a minimal semistable model. I guess yes but I do not find any reference.
Also, if we have a semistable K3 surface with a log structure, ...
- 656
2
votes
0
answers
255
views
Lang isogeny for group stacks
Let $G$ be a commutative algebraic group stack over $\mathbb{F}_q$ (I don't really care about the precise definition: I'm secretly thinking about the Picard stack of a projective curve). To what ...
- 3,605
0
votes
0
answers
201
views
extending truncated Barsotti-Tate group
Let $X$ be a smooth projective curve defined over a finite field of char $p$, let $G[1]$ be a truncated Barsotti-Tate grop of level-1. My question is : can $G[1]$ be extended to a truncated ...
CommunityBot
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3
votes
0
answers
204
views
Hodge filtration over $\mathbb Z_p$
Let $p$ be a prime number.
Let $X\to\operatorname{Spec}\mathbb Z_p$ be smooth and proper. Is it true that
the map $H^i(X,\Omega^{\bullet\geq j}_{X/\mathbb Z_p})\to H^i(X,\Omega^\bullet_{X/\mathbb Z_p})...
- 2,842
3
votes
0
answers
186
views
Is Hasse-witt map isomorphism?
Fix a level $N \geq 3$ and denote by $\Gamma(N)\subset SL(2,\mathbb{Z})$ the subgroup of matrices which are congruent to the identity modulo $N$. The open modular curve $Y(N)$ corresponding to $\Gamma(...
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4
votes
0
answers
136
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A subring of the Serre Swinnerton -Dyer ring of level N modular power series
Suppose ell is prime and (N,ell)=1. Consider those power series over Z that are expansions at infinity of modular forms for gamma_0 (N) of weight a multiple of ell-1. I'll say that an element of (Z/...
- 5,422
11
votes
4
answers
2k
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Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)
A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$.
An arithmetic subgroup is called congruence if it contains a subgroup of type $\Gamma(N)$ for ...
- 9,399
0
votes
1
answer
294
views
a question of Galois cohomology
Let $R$ be a complete DVR with algebraically closed residue field $k$ and fractional field $K$ , $PGL(2)$ the automorphic group of projective line over $\overline K$.
My question is:
When $H^{1}(Gal(...
- 5,019
2
votes
1
answer
239
views
morphism between two elliptic curves over a local field
Let $X,Y$ be two elliptic curves over $K:=K(R)$ which have good models over $R$, Char($K$)=$0$, where $R$ complete DVR with algebraically residue field $k$.
If $L$ is a finite extension of $K$, such ...
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2
votes
2
answers
1k
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are moduli stacks deligne-mumford stacks in general
Let M be your favorite moduli stack over the field of complex numbers.
Is it reasonable to expect M to be a Deligne-Mumford stack?
I know this is true for the moduli space of curves of genus g, ppav'...
- 3,857
4
votes
1
answer
369
views
Gauss mapping in finite characteristic
Suppose that $X\subset\mathbb P^n$ is a $d$-dimentional smooth projective variety (not a linear subspace) over an algebraically closed field. If $\gamma\colon X\to\mathrm{Gr}(d,\mathbb P^n)$ is Gauss ...
- 1,939
4
votes
0
answers
290
views
Is Normalization of log smooth scheme smooth?
Let $f:Y\rightarrow X$ be a finite flat morphism between smooth schemes over $Spec k$, where $k$ is a perfect field. Let $D$ be an irreducible and smooth divisor of $X$, $U=X\setminus D$ the ...
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4
votes
3
answers
4k
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reduction types of elliptic curves
Let $E/K$ be an elliptic curve, where $K$ is a complete local field with residue field $k$ and char$(k) = p$. I'm trying to make sense of Kodaira symbols and Tate's algorithm.
My current ...
CommunityBot
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1
vote
1
answer
257
views
arithmetic group over function fields and its fundamental domain
Let $G$ be a semi-simple algebraic group defined over a global function field $K$.
Let $S$ be a finite set of places of $K$. For a place $v$ of $K$ let $K_v$ be the completion under $v$. We take $K_S=\...
- 52.9k
2
votes
0
answers
220
views
Bound for the degree of the field of definition for a closed point of a variety
While attempting to prove some existence theorem for matrices over $\mathbb{F}_{2^k}$ I've come across the following problem concerning fields of definition for closed point of, say affine, varieties.
...
- 351
3
votes
1
answer
626
views
Heegner Points on $X_0(N)$ when some primes dividing $N$ are inert in the imaginary quadratic field
If $K = \mathbb{Q}(\sqrt{-D})$ is a imaginary quadratic field with discriminant $-D$, then we get Heegner points on $X_0(N)$ as long as there exists $\mathfrak{n} \subset \mathcal{O}_K$ such that $\...
- 471
2
votes
0
answers
606
views
Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces
Is there any version of Kawamata-Viehweg vanishing theorem that holds for excellent surfaces? I will be very glad if there is one such, but if not then is it at least true for a surface over a non ...
- 165
6
votes
1
answer
485
views
Local Norm Mapping for Abelian Varieties
Let $A/K$ be an abelian variety defined over a nonarchimedean local field $K$ of characteristic $0$ and let $L$ be a finite extension of $K$. Consider the norm map $$A(L)\xrightarrow{N_{L/K}}A(K)$$ I ...
- 639
4
votes
0
answers
814
views
Adjunction Formula for Weil Divisors on a Normal Variety X
Let $X$ be a normal variety over an algebraically closed field $k$ of characteristic $p>0$ and $S$ be a prime Weil divisor on $X$ which is normal too. Now if $K_X+S$ is NOT $\mathbb{Q}$-Cartier, ...
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