The tag has no wiki summary.

learn more… | top users | synonyms

18
votes
2answers
1k 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 ...
8
votes
0answers
328 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
votes
2answers
226 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
121 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 ...
4
votes
0answers
137 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
189 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
188 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
154 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
100 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 ...
6
votes
1answer
372 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 ...
4
votes
1answer
199 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 ...
10
votes
1answer
874 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 ...
6
votes
0answers
90 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
263 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
1answer
173 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 ...
15
votes
1answer
447 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!
9
votes
0answers
305 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 ...
1
vote
1answer
203 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 ...
6
votes
1answer
254 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 ...
3
votes
1answer
178 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 ...
7
votes
1answer
366 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 ...
3
votes
1answer
105 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 ...
10
votes
2answers
369 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 ...
3
votes
4answers
930 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 ...
1
vote
0answers
126 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 ...
1
vote
1answer
251 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
votes
1answer
147 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
votes
1answer
616 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 ...
16
votes
1answer
1k 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 ...
5
votes
2answers
371 views

Examples of (Phi,Gamma)-modules

What is the (Phi,Gamma)-module of an elliptic curve over Z_p, expressed by a direct construction ?
3
votes
0answers
156 views

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 ...
5
votes
2answers
198 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 ...
0
votes
0answers
89 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 ...
0
votes
2answers
264 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 ...
3
votes
0answers
134 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 ...
4
votes
1answer
156 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 ...
9
votes
4answers
774 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
votes
1answer
297 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 ...
3
votes
0answers
134 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 ...
0
votes
1answer
226 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 ...
2
votes
1answer
157 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 ...
1
vote
2answers
466 views

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, ...
3
votes
0answers
187 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 ...
1
vote
1answer
134 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 ...
2
votes
0answers
88 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. ...
3
votes
1answer
269 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 ...
3
votes
3answers
528 views

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 ...
0
votes
1answer
178 views

When is the determinant of the push-forward of an ample line bundle ample

Let $f:X\to S$ be a "nice" morphism of "nice" schemes. Let $L$ be an ample line bundle on $X$. When is $\det f_\ast L$ also ample? A "nice" morphism could be anything from "finite type separated" to ...
5
votes
1answer
253 views

Inertia subgroup in the ordinary reduction case when $p=2$

Dear MO, Let $K/\mathbb{Q}_2$ be a finite extension, and let $E/K$ be an elliptic curve with good ordinary reduction, and such that $\mathbb{Q}_2(j(E))=K$. Let ...
6
votes
0answers
129 views

simple proof of relation between H^1 crystalline and Dieudonne module?

Hi, Let $k$ be a perfect field of characteristic $p > 0$. Let $A/k$ be an abelian variety. Then the first crystalline cohomology group of $A$ with respect to $W(k)$ (= Witt vectors) is canonically ...