The arithmetic-geometry tag has no wiki summary.

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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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**0**answers

123 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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**1**answer

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 ...

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**1**answer

177 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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### 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**answer

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### 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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**1**answer

223 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 ...

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**1**answer

324 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 ...

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**1**answer

234 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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**1**answer

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### 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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**1**answer

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### 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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**2**answers

454 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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**4**answers

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### 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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145 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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281 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...)
...

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### 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 = ...

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**1**answer

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### 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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**1**answer

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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 ...

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**2**answers

476 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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218 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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103 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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301 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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141 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 ...

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**1**answer

166 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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### 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 ...

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**1**answer

340 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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142 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 ...

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239 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 ...

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**1**answer

160 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**answers

516 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, ...

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201 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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145 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 ...

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### 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.
...

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**1**answer

329 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 ...

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816 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 ...

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**1**answer

238 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 ...

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**1**answer

264 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 ...

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167 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 ...

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256 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 ...

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**1**answer

179 views

### Are period domains ever contractible

Which simply-connected period domains are contractible?
Examples. Siegel upper-half space? Poincare upper-half plane? Universal cover of a Shimura variety?
Are these contractible?

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**1**answer

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### scheme of generalizations

Hi,
I have the following problem. Let $\mathcal{O}$ be a valuation ring and $S=Spec(\mathcal{O})$, denote with $s$ the closed point and with $\eta$ the generic one. Let $X\rightarrow S$ be a proper, ...

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233 views

### Isogenous elliptic curves have same conductor

Let $E/K, E'/K$ be elliptic curves defined over a number field $K$. Let $\phi: E \to E'$ be a non-constant isogeny defined over $K$. Why must the conductors be equal?
I know that this is an ...

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**1**answer

217 views

### Tricks to produce examples of hypersurfaces with index greater than $1$

Recall, index of an algebraic scheme $X$ is defined to be the greatest common divisor of the degrees of the space of zero cycles on $X$. I am interested in examples of hypersurfaces in ...

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### Status of Beilinson conjectures?

(I hesitate to post that question here, but I received on answer on FB:)
Does anyone know how the current status of work on them is? And how the possible generalizations etc. which one thinks ...

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### Possible reasons why the image of 0 in the modular parametrization would always be O for a family of quadratic twists of elliptic curves

Let $E$ be an elliptic curve and $E_n$ be its quadratic twist by $n$. Let $\phi_n: X_0(N_n) \to E_n$ be the normalized modular parametrization of $E_n$ ($\infty \to O$)
For some particular curves ...

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### The etale cohomology``ring" structure of torsion sheaves on varieties

For a topological manifold $M$, one can speak of the cohomology ring structure $H^*(M, k)$ where $k$ is a ring. If one replace $M$ by an arithmetic schemes $X$ over a base ring $S$, and replace $k$ by ...

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342 views

### Simple field extension and rational points

Let $F$ be an infinite field and $f$ a homogeneous form on $F$ such that $f$ has no non-trivial zero in $F$. Let $F'$ be a finite extension of $F$ such that $f$ has a non-trivial zero in $F'$. Is it ...

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### computing the order of the image of 0 under the modular parametrization map for an elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve of rank 0, with modular parametrization $\phi: X_0(N) \to E$. Let $\Omega_0$ be the least positive real period. A paper I'm reading (Yoshida, Some variants of ...

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### Overconvergent/infinitesimal site, base change and six operations

This question is about 6 operations formalism for 'crystalline' cohomology theories - more specifically the infinitesimal cohomology of smooth $\mathbb{C}$-varieties, and the overconvergent cohomology ...