# Questions tagged [arithmetic-geometry]

Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.

**4**

votes

**0**answers

147 views

### Explicit elements of the first cohomology of modular curves

Let $M$ be a modular curve and $\pi:E\to M$ the universal elliptic curve. For a prime $\ell$, let $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$. I am wondering whether there are any explicit ...

**7**

votes

**1**answer

334 views

### index of smooth varieties

What are the simplest examples of smooth, projective varieties defined over the fraction field of an Henselian DVR of characteristic $0$ which have index $>1$?
EDIT: Also assume that the residue ...

**4**

votes

**2**answers

339 views

### Frobenius actions on de Rham cohomology, clarify questions on a paper of Kedlaya

I am reading the paper "$p$-adic cohomology: from theory to practice" by K. S. Kedlaya. I have several naive questions about section 2: Frobenius action on de Rham cohomology. As a physicist, I lack ...

**4**

votes

**0**answers

181 views

### Reference request: Formal Existence for stacks

Is there a formal existence Theorem for coherent sheaves on locally Noetherian Artin Stacks, in the spirit of Grothendieck's Formal GAGA?
Is it available for more general stacks?

**0**

votes

**1**answer

156 views

### Integral morphism between universally closed and separated schemes

Let $f : X\to Y$ be a morphism between schemes over $\text{Spec}(\mathbf{Z}_p)$.
Assume:
$f$ is integral
both $X$ and $Y$ are universally closed and separated over $\mathbf{Z}_p$
$f$ mod $p^n$ is an ...

**0**

votes

**2**answers

262 views

### Inverse limit of finite flat morphisms

Suppose that an inverse limit of finite flat morphisms $X_k\to S$ of qcqs schemes, with affine transition maps, is $X\to S$, such that a closed fiber of $X\to S$ is finite.
Is $X\to S$ finite?

**1**

vote

**1**answer

164 views

### Relative approximation of morphisms

Let $S$ a qcqs scheme, and let $f : X := \varprojlim_j X_j \to S$ be an inverse limit of qcqs schemes $f_j : X_j \to S$ with affine transition maps.
Suppose $f$ is (P). Is $f_j$ also (P) for all $j$ ...

**8**

votes

**1**answer

688 views

### Commutative algebra counterexample

Let $M$ be an $R[x]$-module, such that $M$ is finitely generated as an $R$-module.
Does there exist one such $M$, such that $M\otimes_{R[x]}R[x,x^{-1}]$ is not finitely generated as an $R$-module?

**12**

votes

**3**answers

477 views

### Some questions on the $p$-adic properties of special $L$-values

Warning: Some naive, speculative questions from a total non expert.
Let $\rho$ be a p-adic representation of the Galois group $Gal(\overline K/K)$ for a number field $K$. We can consider the Artin L-...

**13**

votes

**1**answer

425 views

### Arithmetic motivations for modularity in higher rank

The classical setting of modularity is that one can associate elliptic modular forms (or automorphic representations of GL(2)/$\mathbb Q$) to elliptic curves over $\mathbb Q$. This has far-reaching ...

**3**

votes

**1**answer

190 views

### Degeneracy maps and cusps

Let $N$ be a positive integer and $p$ a prime not dividing $N$. Let $X_0(N)$ be the modular curve (over $\mathbb{Q}$) associated to the congruence subgroup $\Gamma_0(N)$, which is the subgroup of $\...

**2**

votes

**1**answer

214 views

### Equivalence of vector bundles over $Spec(A_{\inf})$ and the punctured spectrum

I'm trying to understand the Lemma 4.6 of Bhatt-Morrow-Scholze's paper Integral $p$-adic Hodge Theory.
In the proof, for proving the restriction functor is fully faithful, it used a affine open cover $...

**4**

votes

**1**answer

138 views

### Gluing finitely presented quasi coherent sheaves

Let $X$ be a quasi-compact, separated scheme, and $\{\text{Spec}(A_i)\subset X\}_{i=1,\ldots, n}$ a finite affine open cover.
Suppose a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ is such ...

**7**

votes

**1**answer

358 views

### How is the Eichler-Shimura congruence related to L-functions?

My understanding is that the Eichler-Shimura relation expresses the Hecke operator $T_p$ in terms of the geometric Frobenius map. Specifically, $T_p = Frob + Ver$ for Frobenius map $Frob$ and it's ...

**4**

votes

**1**answer

521 views

### Proper mapping theorem

Let $Z\to X$ be a closed immersion of schemes. Assume $\mathcal{O}_Z$ and $\mathcal{O}_X$ both are coherent sheaves of $\mathcal{O}_Z$, resp. $\mathcal{O}_X$-modules.
In particular, the coherent ...

**2**

votes

**0**answers

210 views

### A question on direct limits of rings, and descent of ideals

Motivated by an étale cohomology calculation I am going to do, here is a question that should have positive and not too hard answer.
Let $A$ be a ring, $A_0$ a ring such that $A_0$ is equipped with ...

**6**

votes

**0**answers

209 views

### Galois action on functions on an adelic coset space

For a reductive group $G$ over a number field $K$, an automorphic representation for $(G, K)$ is an irreducible admissible constituent of the right regular representation of $G(\mathbb{A}_K)$ in the ...

**7**

votes

**0**answers

238 views

### Galois representations associated to the modular tower and automorphy

Consider $\mathcal{M} = \{M_{g,n}, \mu_{g,n}^{g’,n’}\}$, the system of moduli spaces of $n$-pointed smooth algebraic curves along with the basic maps amongst them coming from identifying and ...

**3**

votes

**0**answers

118 views

### Frobenius stratification of imperfect fields

Suppose $k$ is a (non-perfect) field of characteristic $p$, and $Fr$, its Frobenius map. I’d appreciate comments and references on the structure of the fitration/stratification of $k$ given by the ...

**2**

votes

**1**answer

256 views

### An explicit computation of the blow-up of curve over $\mathbb{F}_3$ at two points

I would like to work through computing the blow-up of a particular curve along a subvariety consisting of just two points, both of which are ordinary double points.
Let $$F(X,Y,Z) = X^4 + Y^4 - X Y^2 ...

**3**

votes

**0**answers

231 views

### Algebraic vs analytic de Rham cohomology

Let $X$ be a smooth projective variety over $\mathbf{C}$, $\Omega^{\bullet}_X$ its algebraic de Rham cohomology.
Let $p : X_{\rm an}\to X_{\rm Zar}$ the obvious morphism of sites.
We have $p^*\Omega^...

**8**

votes

**0**answers

308 views

### Bloch Ogus spectral sequence

Let $X$ be a smooth projective variety over $\mathbf{C}$, and $p : X_{\rm an}\to X_{\rm Zar}$ the obvious map of sites.
The Leray spectral sequence
$$H^r(X_{\rm Zar}, R^sp_*\mathbf{C})\Rightarrow H^{...

**14**

votes

**2**answers

680 views

### What are zeta functions good for?

I know a couple of answers to the above question:
They can be used for point counting over finite fields/estimating the distribution of primes in characteristic 0.
There are various conjectures/...

**5**

votes

**0**answers

260 views

### Geometric Frobenius on $\ell$-adic cohomology

Let $X$ be a smooth projective variety over a finite field $k$, and $F$ the geometric Frobenius on $H^*_{et}(X_{k^{sep}}, \mathbf{Z}_{\ell})$.
Is $F$ only rationally bijective, or integrally ...

**3**

votes

**0**answers

149 views

### On what varieties are the conjectures on $L$-functions true

In Peter Schneider's paper "Introduction to the Beilinson conjectures", he lists some hypothesis (conjectures) on the $L$-function associated to the pure motive $H^i(X)$ where $X$ is a smooth ...

**0**

votes

**0**answers

75 views

### Smoothness in formal GAGA

Suppose $X$ is a noetherian scheme over a complete noetherian local ring $(A,\mathfrak{m})$, and assume $X$ is projective.
If $\widehat{X}$, the $\mathfrak{m}$-adic formal completion, is smooth in ...

**4**

votes

**0**answers

284 views

### “Elementary” Proof that the divisor class group of varieties over finite fields is finite

Let $X$ be a geometrically integral (or geometrically reduced and geometrically connected) proper scheme over a finite field $k = \mathbb{F}_q$, so its Picard scheme exists and $\mathrm{Pic}^0_{X/k}$ ...

**7**

votes

**0**answers

223 views

### Reconstitution from reduction and tropicalization for $p$-adic varieties

For a variety $X$ over a $p$-adic field $k$, its reduction to a variety $X_p$ over residue field $k_p$ of $k$ and its tropicalization to $X_{tr}$ seem to me to be somewhat orthogonal processes.
Taken ...

**17**

votes

**0**answers

452 views

### Tropicalization of perfectoid spaces and their tilts

Does tropicalization exist in the world of perfectoid spaces? Since it does for Huber's adic spaces, I thought it might for perfectoid spaces too, yet I can't find any explicit references so far.
...

**7**

votes

**1**answer

243 views

### $p$-adic lifts of tropical varieties

What is currently known about lifts of tropical varieties to varieties over $\mathbb{Q}_p$ or its extensions? Starting with an appropriate rational polyhedral cone complex what are the obstructions ...

**9**

votes

**1**answer

202 views

### Liftable rational varieties

Is there an example of a rational smooth projective variety over a perfect field of characteristic $p$, that is not liftable to characteristic zero?

**4**

votes

**0**answers

190 views

### $p$-adic lifts of varieties over finite fields

Fix a finite field $k_p$ of characteristic $p$, as well as a $p$-adic lift of it, $k_0$. In other words $k_0$ is a $p$-adic field and $k_p$ its residue field.
Let $X_p$ be a non-singular variety over ...

**4**

votes

**0**answers

207 views

### Philosophical question on the role of motivic cohomology

As a physicist who has read almost nothing on the beautiful theories of motives and motivic cohomology, I have some philosophical questions on motivic cohomology. I apologize first for my naive ...

**4**

votes

**0**answers

204 views

### Ring of Witt vectors and Fontaine's deRham period ring

The construction I am interested in is the following:
Let $V$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with algebraically closed residue field $k$. Let $K=Frac(V)$ and ...

**0**

votes

**0**answers

135 views

### What pure motives have Hodge realisations isomorphic to $\mathbb{Q}(0)$

Suppose $k$ is a number field and $\sigma:k \hookrightarrow \mathbb{C}$ is an embedding. If $\sim$ is an adequate equivalence relation, we can construct the category of pure motives with rational ...

**2**

votes

**0**answers

142 views

### Galois action on posets of number fields and $p$-adic fields

In the theory of group actions on posets one studies the action of a group $G$ on a poset $\mathcal{P}$ (via order-preserving permutations) partly through its derived action on the order complex of ...

**4**

votes

**1**answer

693 views

### Misunderstanding of Hodge conjecture

I ask a question on math stack exchange about Hodge conjecture and have not got any reply or comment.
https://math.stackexchange.com/questions/2704988/clarify-hodge-conjecture
so I decide to post ...

**5**

votes

**2**answers

265 views

### Does Beilinson's conjecture on values L-functions work for smooth projective varieties over a number field

In Nekovar's introductory paper "Beilinson's Conjecture"
http://math.stanford.edu/~conrad/BSDseminar/refs/BeilinsonintroII.pdf
The conjecture is formulated for smooth projective varieties over $\...

**10**

votes

**1**answer

394 views

### Arithmetic representation stability and Galois action

I am thinking about representation stability phenomena (as considered by Church, Farb, Ellenberg and others) in arithmetic settings where Galois action enters the picture, and am curious about their ...

**3**

votes

**0**answers

107 views

### Characterization of a meromorphic function as arithmetic zeta function

I'd like to know if there is a (conjectural) criterion for a meromorphic function on $\mathbb{C}$ to be the zeta function of an arithmetic scheme, i.e., a statement of the form
"If a meromorphic ...

**4**

votes

**0**answers

160 views

### Geometric fundamental group and algebraically closed residue field

my questions relates to the following talk of Tsuji:
https://www.youtube.com/watch?v=2brDj26phP0
At around 10:30 of the video, Tsuji is interrupted by a man stating that his construction does not ...

**1**

vote

**1**answer

161 views

### Alfred van der Poorten--rational functions paper

Does anybody has a copy of the following paper:
Alfred van der Poorten, Some facts that should be better known, especially about rational functions; Number Theory and Applications”, Richard A. Mollin ...

**3**

votes

**0**answers

172 views

### $L$-function of induced Galois representation

Suppose $L \subset K$ are number fields and let's choose and fix an algebraic closure $\overline{L}$ of $L$ such that $L \subset K \subset \overline{L}$, hence $H:=\text{Gal}(\overline{L}/K)$ is a ...

**7**

votes

**0**answers

268 views

### quasi-finite group schemes and associated Galois modules

Let $p$ be an odd prime.
Let $A$ be an abelian variety over $\mathbb{Q}$ and suppose that it has semistable reduction at $p$. Let $\mathcal{A}$ denote the Neron model of $A$ over $\mathbb{Z}_p$. Now ...

**2**

votes

**0**answers

154 views

### Set theoretic complete intersections in toric varieties

Is it expected that every smooth projective variety over the complex numbers, is a set-theoretic complete intersection into a smooth projective toric variety?
Is there an example of a smooth ...

**4**

votes

**1**answer

388 views

### Complete intersections in toric varieties

Let $X$ be a smooth projective variety over the complex numbers. Is $X$ a global complete intersection inside a smooth projective toric variety?

**1**

vote

**0**answers

155 views

### Algebraization of open formal subschemes

Let $\mathfrak{X}$ be a locally noetherian adic formal scheme over $\text{Spf}(A)$, with $A$ an $I$-adically complete and separated noetherian ring.
Suppose the mod $I$-fiber of $\mathfrak{X}$ is an ...

**2**

votes

**0**answers

188 views

### Neron Severi under specialization

Let $X$ be a smooth projective variety over $\mathbf{Q}$, and $\mathcal{X}$ a smooth projective model over $\mathbf{Z}[1/N]$ for $N$ large enough.
Call $\eta$ the generic point $\text{Spec}(\mathbf{Q}...

**2**

votes

**0**answers

119 views

### Genus Zero Diophantine Equations and Infinite Valuations

I'm interested in an explicit upper bound for the integral solutions of a certain genus zero curve $F(X,Y)=0$. I found some papers that address this problem:
[1] Solving genus zero diophantine ...

**4**

votes

**0**answers

213 views

### Motives up to homological equivalence

Let $X$ be a smooth projective variety over a field $k$ finitely generated over its prime field, and $M_{hom}(X)$ the category of motives modulo $\ell$-adic homological equivalence.
(1) Is $M_{hom}(...