The arithmetic-geometry tag has no wiki summary.
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1answer
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curves with good reduction everywhere
It seems to be a folklore that for any genus $g$, there is a number field $K$ and a curve $X$ over $K$, such that $X$ has good reduction at all the places of $K$. Are any simple proofs of this?
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92 views
Effective Lang-Weil bounds for del Pezzo surfaces
Let $X$ be variety in $\mathbb{P}^N$ over $\mathbb{F}_q$ of dimension $n$ and degree $d$.
By the Lang-Weil bounds
$ |\# X(\mathbb{F}_q) - q^n| \le (d-1)(d-2)q^{n-1/2} + Cq^{n-1}$for a constant $C$ ...
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0answers
52 views
How should we understand the relative interior in Berkovich spaces
I'm reading Berkovich's book on analytic spaces. The notion of relative interior confuses me. Is there anyway to see how it "looks like"? For instance, if $r <1$, what is the relative interior of
...
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76 views
Arithmetic Geometry and Diophantine Approximations [on hold]
I am an Indian Phd. student in number theory. I have started working on computational number theory in India but i this area do not suit me. I am planning to move to Arithmetic Geometry. Recently i ...
5
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1answer
148 views
Shimura surfaces that do not contain a Shimura curve
Let $S$ be a Shimura surface i.e. a Shimura variety with $dimS=2$. Does $S$ necessarily contain a Shimura curve? I know that probably the answer is No, but do not have an explicit example. What is the ...
4
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1answer
175 views
Weil pairing, fixed field of a $p$-adic Galois representation
Let $A$ be an abelian variety over a $p$-adic field $K$. If $K(A_{p^\infty})$ is the field extension of $K$ obtained by adjoining the coordinates of all $p$-power division points of $A$. By the Weil ...
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2answers
800 views
Are most curves over Q pointless?
Fresh out of the arXiv press is the remarkable result of Manjul Bhargava saying that most hyperelliptic curves over $\mathbf{Q}$ have no rational points. Don Zagier suggests the paraphrase : Most ...
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237 views
Torelli-like theorem for K3 surfaces on terms of its étale 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
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2answers
180 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 $G \subset \operatorname{GL}_m(\mathbb{Z}/n\mathbb{Z})$ ...
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83 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 ...
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90 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 ...
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1answer
156 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
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0answers
132 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 ...
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0answers
141 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
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1answer
76 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
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1answer
316 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 ...
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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
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1answer
145 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 ...
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1answer
547 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 ...
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0answers
68 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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1answer
255 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 ...
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0answers
84 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 ...
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1answer
155 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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1answer
395 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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170 views
What makes the Cartier operator “tick”?
Let $C$ be a smooth curve over a finite field of characteristic $p$. Let $t$ be a local parameter at a point. If $f$ is a regular function on a neighbourhood of the point, one can write uniquely
$$f ...
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1answer
161 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 ...
5
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1answer
144 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 ...
2
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1answer
146 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 ...
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1answer
274 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 ...
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83 views
Density of integral (rational) points on affine varieties
In a series of papers due to Browning, Heath-Brown, and Salberger (some of which are joint work, some individual), they established that for any projective variety $X \subset \mathbb{P}^n$, they ...
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1answer
102 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 ...
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1answer
213 views
Belyi's theorem for function fields
Belyi's theorem states that every smooth projective algebraic curve $C$ defined over $\bar{\mathbb{Q}}$
admits a map $C\to\mathbb{P}^1$ ramified only over $0,1,\infty$.
Is there an analogue of this ...
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4answers
645 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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92 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 ...
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1answer
210 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...)
...
2
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1answer
104 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 = ...
6
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1answer
565 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 ...
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1answer
510 views
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 ...
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2answers
321 views
Examples of (Phi,Gamma)-modules
What is the (Phi,Gamma)-module of an elliptic curve over Z_p, expressed by a direct construction ?
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Natural construction of Hodge (Phi,Gamma)-modules
I am looking for a functor from varieties $X/\mathbf{Z}_p$ to $(\varphi,\Gamma)$-modules over the Robba ring over $\mathbf{Q}_p$ (overconvergent ones) that is contructed by differential methods ...
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2answers
166 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 ...
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78 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 ...
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2answers
212 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 ...
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0answers
108 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 $H^i(X,\Omega^{\bullet\geq j}_{X/\mathbb Z_p})\to H^i(X,\Omega^\bullet_{X/\mathbb ...
3
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1answer
128 views
pro-$\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 ...
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4answers
545 views
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 ...
3
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1answer
257 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 ...
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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 ...
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1answer
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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 ...
2
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1answer
148 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 ...
