The arithmetic-geometry tag has no wiki summary.

**12**

votes

**1**answer

570 views

### Is the set of surfaces over Spec Z with ample canonical sheaf empty

Main question. Does there exist a smooth projective morphism $X\to$ Spec $\mathbf Z$ of relative dimension two such that the canonical sheaf $\omega_{X_{\mathbf Q}}$ of the generic fibre $X_{\mathbf ...

**4**

votes

**0**answers

301 views

### Does the Riemann hypothesis for liftable varieties over a finite field imply the Riemann hypothesis for all varieties over a finite field

The Riemann hypothesis for varieties over a finite field has been proven by Deligne. Still I would like to ask the following question.
A variety $X$ over a finite field $k$ is liftable if there ...

**4**

votes

**0**answers

167 views

### Do regular noetherian schemes of dimension one only have finitely many etale covers of bounded degree

Let $X$ be a regular noetherian scheme of dimension one. Let $d$ be an integer.
Question. Are there only finitely many finite etale morphisms $Y\to X$ of degree $d$?
I want to exclude finite etale ...

**8**

votes

**0**answers

174 views

### Corresponding notion of unramified for motives (or de Rham cohomology)

The etale cohomolgoy of a variety $X$ over a number field $K$ is a Galois representation of $\mathrm{Gal}(\overline K/K)$ with some properties coming from $X$, e.g., it is unramified outside $S$ if ...

**1**

vote

**1**answer

311 views

### Hodge-Tate weights of etale cohomology

Let $K/\mathbb Q_p$ be a local field, $X/K$ a proper scheme with semi-stable reduction.
Question: What is the possible range of Hodge-Tate weights of the etale cohomology $H^i(X_{\overline K}, ...

**1**

vote

**0**answers

104 views

### Removing finitely many points from a Shimura curve

Let $X$ be a compact Shimura curve. If we remove finitely many points from this curve, do we neccessarily get a "non-compact Shimura curve"? I have some reasons to believe that the answer is negative, ...

**6**

votes

**1**answer

329 views

### How to calculate zeroth crystalline cohomology

I am just learning crystalline cohomology, so I understand the basic set-ups. But I can't really do any calculations.
For example, let's choose the base $S=W(k)/p^n$, and let $X$ be an affine scheme ...

**3**

votes

**0**answers

158 views

### Does the Albanese map satisfy Torelli's theorem

Let $M_h$ be the moduli space of canonically polarized varieties with Hilbert polynomial $h$. Let $M_h \to A_g$ be the Albanese map, with $g$ an integer which depends on $h$ and $A_g$ the moduli space ...

**4**

votes

**0**answers

164 views

### Is the moduli space of genus three smooth quartics affine?

Non-hyperelliptic curves of genus three are smooth quartics. Is the moduli space of such curves affine?
I think this follows from a more general result on smooth complete intersections, but I'm ...

**6**

votes

**1**answer

383 views

### If rational points are like entire curves, then what do algebraic points correspond to

I read somewhere that if $X$ is a projective variety of general type over a number field $K$, then rational points are an analogue of entire curves $\mathbf{C}\to X^{an}$ (with $X^{an}$ the ...

**3**

votes

**1**answer

270 views

### On the m-th power of the Hodge bundle and Arakelov's theorem

Let $S$ be a smooth projective curve over $\mathbf C$ and let $f:X\to S$ be a projective flat morphism with "semi-stable" fibres (i.e., the fibres are reduced and strict normal crossings divisors) and ...

**1**

vote

**1**answer

225 views

### Hilbert polynomial of $X\times P^1$

Let $X$ be a canonically polarized smooth projective geometrically connected variety over $k$ with Hilbert polynomial $h$.
What is the Hilbert polynomial of $X\times_k \mathbf{P}^1_k$? How does it ...

**1**

vote

**1**answer

283 views

### Tate Conjecture on decomposition of motives(?)

I apologize for the title. I myself conined it...
I am referring to Conjecture 1.2 (page 7) of Richard Taylor's paper Galois representations (Annales de la Faculte des Sciences de Toulouse 13 (2004) ...

**7**

votes

**1**answer

383 views

### For which fields does the isogeny theorem hold

Let $k$ be a field. We say that the isogeny theorem holds over $k$ if, for any abelian variety $A$ over $k$, there are only finitely many $k$-isomorphism classes of abelian varieties $B$ over $k$ ...

**3**

votes

**1**answer

140 views

### Does the Hilbert polynomial determine the weight of the Galois representation associated to a variety

Let $X$ be a curve or an abelian variety (over a finite field). Then the Galois representation associated to $X$ via the etale cohomology of $X$ (in degree $1$) is integral of weight $1$ and its ...

**2**

votes

**3**answers

338 views

### Useful notion of unramified Galois representation

Let $\mathbf C(t)$ be the field of rational functions and let $\overline{\mathbf C(t)}$ be an algebraic closure. Let $G$ be the Galois group of $\overline {\mathbf C(t)}$ over $\mathbf C(t)$.
Let ...

**0**

votes

**0**answers

203 views

### Vanishing of motivic cohomology with finite coefficients in negative degrees

I wonder whether the "finite coefficient version" of Beilinson-Soule conjecture i.e. the following statement holds or not.
STATEMENT:
Let $X$ be a smooth and projective scheme over a finite field ...

**1**

vote

**0**answers

136 views

### Special values of zeta functions and extensions of base fields.

Let $X$ be a scheme of finite type over a finite field $k=\mathbb{F}_{q}$ of $q$ elements.
Then, one can define the zeta function $Z_{X/k}(T)$ of $X$ ovet $k$ as $\prod_{x\in ...

**2**

votes

**2**answers

450 views

### Classification of quasi-split unitary groups

Let $U$ be a unitary group defined with respect to an extension $E/F$ of non-archimedean local fields, and assume it is realised with respect to a pair $(V,q)$, where $V$ is an $n$-dimensional vector ...

**6**

votes

**1**answer

240 views

### How do you compute the primes of bad reduction?

Suppose that I am given a subscheme $Y$ of $\mathbf{P}^n_{\mathbf{Z}}$, flat over $\operatorname{Spec}\mathbf{Z}$ and with smooth generic fiber $Y_{\mathbf{Q}}$, defined by the vanishing of some ...

**3**

votes

**1**answer

219 views

### Torsion of elliptic curves is finite

Let $S$ be an integral 1-dimensional scheme with function field $K$.
Let $E$ be an elliptic curve over $K$. The torsion of $E$ over $K$ is not necessarily finite. As an example consider an elliptic ...

**12**

votes

**6**answers

1k views

### SAT and Arithmetic Geometry

This is an agglomeration of several questions, linked by a single observation: SAT is equivalent to determining the existence of roots for a system of polynomial equations over $\mathbb{F}_2$ (note ...

**2**

votes

**0**answers

345 views

### $\sigma$-conjugate iff $p$-adically close

First some notations. Let $p$ be a prime, $k$ a perfect field of characteristic $p$, $W=W(k)$ the ring of Witt vectors over $k$, $\sigma : W \rightarrow W$ the Frobenius, $R$ a commutative ...

**1**

vote

**2**answers

361 views

### Equation for simple Jacobian of a genus two curve

Let $X$ be a curve of genus two over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian of $X$.
Can we write down an explicit equation for the abelian surface $J$?
I know $X$ can be ...

**2**

votes

**1**answer

135 views

### Detecting sections on an arithmetic variety

Let $S$ be Spec $O_K$ with $O_K$ the ring of integers of a number field $K$.
Let $X\to S $ be an arithmetic variety, i.e., an integral smooth quasi-projective $S$-scheme with generic fibre $X_\eta$ ...

**4**

votes

**1**answer

308 views

### Properties of subvarieties of a simple abelian variety

Let $A$ be a simple abelian variety over a field $k$. (For simplicity, we assume char $k =0$.)
Let $X$ be a smooth projective geometrically connected variety over $k$ of positive dimension.
Suppose ...

**3**

votes

**0**answers

273 views

### An example of almost etale extension

In the paper of Faltings' "p-adic Hodge theory", Faltings showed an example of almost etale extension before he proved the almost purity theorem. The example is following:
Let $k$ be a perfect field ...

**1**

vote

**1**answer

166 views

### global section of vector bundle and reduction

Let $k$ be an algebraically closed field of char $p\neq 0$, $W_2(k)$ the witt vector of length 2. $C_1$ a smooth projective curve over $W_2(k)$, and $H_1$ a vector bundle over $C_1$. We denote $C_0$ ...

**6**

votes

**1**answer

214 views

### Representability of sheaf of Ext^1 of a Néron model by $\mathbb{G}_m$

Let's work over a trait $S=\mathrm{Spec}R$, where $R$ is a dvr with fraction field $K$, residue field $k$. Given an abelian variety $A_K$ with semi-stable reduction, let $A$ over $S$ be its Néron ...

**12**

votes

**2**answers

1k views

### Learning path for the proof of the Weil Conjectures

Assume you are an algebraic geometry advanced student who has mastered Hartshorne's book supplemented on the arithmetic side by the introduction of Lorenzini - "An Invitation to Arithmetic Geometry" ...

**6**

votes

**1**answer

267 views

### Are ranks of Jacobians over number fields unbounded?

Fix a number field $K$.
Is the rank of $J(K)$ unbounded, where $J$ ranges over the Jacobians of all smooth, projective, geometrically connected curves over $K$?
Does there exist an integer $g$ such ...

**14**

votes

**1**answer

854 views

### Does the moduli space of genus three curves contain a complete genus two curve

Inspired by the question
Does the moduli space of smooth curves of genus g contain an elliptic curve
and its amazing answers, I ask (pure out of curiosity) whether the moduli space $M_3$ of (smooth ...

**2**

votes

**1**answer

314 views

### Grothendieck's section conjecture and base change: restricting sections

Let $X$ be a smooth projective geometrically connected curve over $\mathbf{Q}$ of genus at least two. Fix an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$ and let $G_{\mathbf{Q}}$ be the ...

**7**

votes

**1**answer

280 views

### Varieties with infinitely many etale covers and rational points

Let $X$ be a (smooth projective geometrically connected) variety over a field $k$.
Consider the set Et$(X,k)$ of finite etale covers $Y\to X$ over $k$, with $Y$ geometrically connected over $k$.
...

**7**

votes

**3**answers

414 views

### “Good reduction” for singular varieties

A projective nonsingular variety $X$ over a number field $K$ has the notion of good reduction at places $p$ of $K$. Informally, $X$ has good reduction modulo $p$ if $X$ remains nonsingular when ...

**21**

votes

**1**answer

878 views

### Do all curves have Néron models

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$.
Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$?
By a Néron model, I mean ...

**8**

votes

**2**answers

362 views

### Can one ignore primes lying over $l$ in the Fontaine-Mazur conjecture? Counterexamples?

The Fontaine-Mazur conjecture predicts that an $l$-adic Galois representation of a number field is 'geometric' if it is unramified outside a finite set of primes and is De Rham for primes lying over ...

**4**

votes

**0**answers

248 views

### Dieudonné modules over rings of charateristic zero

Dear Colleagues,
would appreciate if you could recommend references, if such a theory exits, for the following question.
Let $A$ be an Abelian scheme over $\text{Spec}(R)$, where $R$ is a subring of ...

**1**

vote

**1**answer

305 views

### Does a curve over a number field have a finite etale cover of given degree

Let $X$ be a (smooth projective geometrically connected) curve over a number field $K$ of genus $g\geq 2$. Let $d\geq 2$ be an integer.
Does there exist a curve $Y$ over $K$ with a finite etale ...

**1**

vote

**1**answer

241 views

### What is the reduction of this hyperelliptic curve

Let $K$ be a number field and $E/K$ an elliptic curve with equation $Y^2Z = X^3 +AXZ^2+BZ^3$ in $\mathbf{P}^2_K$, where $A,B\in K$.
Let $S$ be non-empty finite set of finite places of $K$ and suppose ...

**3**

votes

**2**answers

257 views

### Defining isogenies over smaller fields

I'm having some issues with abelian varieties and fields of definition. This already became clear in my previous question on Jacobians. Here's another question. If somebody can explain some nice facts ...

**6**

votes

**2**answers

402 views

### Jacobians defined over smaller fields

Let $L/K$ be an extension of number fields.
Let $X$ be a curve over $L$ which can not be defined over $K$. Let $J(X)$ be the Jacobian of $X$ over $L$.
In general, the Jacobian $J(X)$ probably ...

**8**

votes

**1**answer

370 views

### Modularity of higher dimensional abelian varieties

In another question I asked about strategies for giving an effective version of the Shafarevich conjecture for abelian varieties over $\mathbb{Q}$.
For elliptic curves, one can give a proof using ...

**9**

votes

**5**answers

1k views

### The significance of modularity for all Galois representations

On pg. 1 of the slides of a talk, Henri Darmon wrote:
Question: What is an interesting Diophantine equation?
A “working definition”. A Diophantine equation is interesting
if it reveals or ...

**2**

votes

**0**answers

295 views

### Is the geometry of a variety determined by the counts of rational points?

In Diophantine Geometry: An introduction, Hindry and Silverman write "Geometry Determines Arithmetic" (pg. 2) and "Geometry Governs Arithmetic" (pg. 474).
On pg. 211 of the same book, the authors ...

**2**

votes

**1**answer

223 views

### Is There a Mayer-Vietoris Spectral Sequence of Motivic Cohomology for Closed Coverings?

For etale cohomology, there is a spectral sequence of the following form ("Mayer-Vietories spectral sequence for closed covers"):
$E_{1}^{p,q}=\oplus_{i_{0}< \cdots < i_{p}} H_{ Y_{i_{0} \cdots ...

**18**

votes

**3**answers

1k views

### Over which fields does the Mordell-Weil theorem hold?

According to a well-known theorem of Mordell, the group of rational points $E(\mathbf{Q})$ of an elliptic curve $E/\mathbf{Q}$ is finitely generated. Weil generalized this theorem to abelian varieties ...

**1**

vote

**0**answers

149 views

### Fields over which cubic hypersurfaces are rational

All cubic hypersurfaces having at least one double point are birational to some $P^n$ over an algebraically closed field. How does the statement change as I pass to non alg closed fields? Does it hold ...

**5**

votes

**1**answer

353 views

### is Hasse principle a birational invariant?

...it is probably a very trivial question, but I am a beginner in arithmetics.

**6**

votes

**1**answer

345 views

### Honda-Tate in families

Let $k$ be a finite field, say with $q=p^a$ elements. Honda-Tate theory states that there is a bijection between isogeny classes of simple abelian varieties over $k$ and ...