Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [arithmetic-geometry]

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

4
votes
0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
2answers
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
1answer
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
1answer
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
3answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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}(...