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7
votes
0answers
78 views

Variety acquiring rational point over any quadratic extension

Does there exist a variety $X$ over $\mathbb{Q}$ (or a number field) such that it has no rational points over $\mathbb{Q}$ but acquires points over any quadratic extension $\mathbb{Q}(\sqrt{d})$? If ...
2
votes
0answers
219 views

When an elliptic curve is a quotient of $\mathbb{G}_a$?

I want to know when an elliptic curve $E \rightarrow S$ is a quotient of $\mathbb{G}_a$. When $S$ is an analytic space, there is an exact sequence $$0 \rightarrow R^1 \mathbb{Z}^{\vee} \rightarrow ...
0
votes
0answers
99 views

Fermat's Theorem on p = a^2 + b^2 [migrated]

I have read that Fermat predicted that for an odd prime $p$, $p = a^2 + b^2$ iff $p = 1$ mod 4. I heard that such a criterion could be possible for a given integer $n$ like $p = a^2 + n b^2$ ...
7
votes
0answers
226 views

Can an abelian variety/Q have no points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial? This follows from the conjecture that the maximal ...
4
votes
1answer
136 views

Is Scholl construction of modular motives related to Deligne's construction of $\ell$-adic representations?

first of all, I need to declare my extreme ignorance on the topic of modular forms, so , please, does not assume that I know Deligne's construction in details. In ...
3
votes
1answer
157 views

Some clarifications regarding Deligne's paper on $\ell$-adic representations arising from modular forms

I've posted this question few days ago on math.stackexchange because it seems quite superficial. However, since I've got no responses at all, I'm posting it here. If the question is not suitable, ...
14
votes
0answers
231 views

Is the absolute Galois group of the rationals Hopfian?

Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?
39
votes
0answers
2k views

What is an étale theta function?

Let me start out by urging you to take seriously that whatever I write about the papers surrounding IUTT really are questions. If you would like to use it as a guide to the mathematics in any way, ...
9
votes
0answers
343 views

Paths in $\mathrm{Spec} \, \mathbb{Z}$ and Kim's proof of Siegel's theorem for $\mathbb{P}^1 \setminus \{0,1,\infty\}$

This is motivated by a basic number theory question I asked the previous day: Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$? I noted there that the answer to the ...
94
votes
1answer
4k views

What is a Frobenioid?

Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title. Recently, there has been a flurry of new discussion ...
3
votes
2answers
264 views

cohomological obstructions and rational points

Let $X$ a (nice) scheme over $\mathbb{Q}$. Are there cohomolgical obstructions answering the following questions: 1) is $X(\mathbb{Q})$ an empty set ? 2) is $X(\mathbb{Q})$ a finite (non empty) set ...
3
votes
1answer
164 views

When does a hyperelliptic Riemann surface admit a map of degree 3

Let $X$ be a hyperelliptic curve of genus $g>1$. For which $g$ does $X$ admit a map $X\to \mathbb P^1$ of degree $3$? I think a genus two curve $X$ admits a map of degree $3$. Proof: Pick $P$ ...
4
votes
2answers
261 views

References for general Hasse-Weil zeta function

Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case. I am ...
5
votes
0answers
111 views

Groupoid cardinality of DM stack and point counting on coarse moduli spaces

Let $X$ be a finite type DM stack over a finite field $k$ with a coarse moduli space $X_c$. (We only assume $X_c$ is an algebraic space and $X$ might have infinite inertia stack.) Under which ...
2
votes
1answer
169 views

Difference between Frobenii on Tate modules of special and generic fibre

Let $E$ be elliptic curve over $\mathbb Q$ and $p$ a prime of good reduction for $E$. Fix $\ell \neq p$. If $E_p$ is ordinary then we have Frobenius $F_p$ on $E_p$. Assume $F_p$ lifts to ...
2
votes
1answer
158 views

rational point of a curve [closed]

Let $X$ be a smooth projective curve over $\mathbb{Q}$. I heard (if I did not misunderstood) that the geometry of the complex points $X(\mathbb{C})$ (flat, hyperbolic case) dicts the shape (group ...
10
votes
1answer
208 views

Is there a solvable point on any variety over the field of complex rational functions?

Let $K = \mathbb{C}(T)$ be the field of complex rational functions in one variable, and let $V$ be a variety defined over $K$. Must $V$ have a solvable point? The variety $V$ is assumed ...
4
votes
1answer
137 views

Are quaternion algebras from Witt's theorem endomorphism rings of vector bundles?

Let $k$ be a field with char $k \neq 2$. For $a,b \in k^{\times}$, let $(a,b)$ denote the quaternion algebra with $i^2=a$ and $j^{2}=b$, and let $C(a,b)$ denote the projective plane conic given by ...
19
votes
1answer
822 views

When is “independence of l” known?

My question is for which varieties over local fields is "independence of l" known for etale cohomology. Say $X/{\mathbb Q}_p$ is a complete non-singular variety and $W_l$ is the (complex) ...
5
votes
1answer
156 views

The generic fiber pullback for $p$-divisible groups in characteristic $p$

Let $R$ be a discrete valuation ring with the field of fractions $K$ and the residue characteristic $p$. If $K$ is of characteristic $0$, then a celebrated theorem of Tate says that the pullback ...
4
votes
1answer
239 views

Torsors in the analytic topology versus torsors in the etale topology

Let $S= \mathbb A^1_{\mathbb C}$ be the affine line, and let $G$ be a smooth connected reductive group over $S$, e.g., $G = \mathbb G_m, \mathrm{SL}_n$ or $SO_n$. Is every analytic $G$-torsor over ...
11
votes
0answers
492 views

Applications of $p$-adic Hodge theory

I am trying to learn $p$-adic Hodge theory. I found some materials explaining main theorems (or aspects) of the theory. However, I could not find references which explaining how to use the theory. ...
6
votes
1answer
196 views

Weil height of an Abelian Variety with everywhere (potentially) good reduction

Background: Suppose that $E$ is an elliptic curve over $\mathbb{Q}$ with everywhere (potentially) good reduction. there are many ways to define the height of $E$, and I will be concerned with the ...
3
votes
1answer
252 views

“Weight-mondoromy” for open varieties

Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy ...
5
votes
4answers
557 views

Good lecture notes/books on Jacobian of hyperelliptic curve

I want to understand what the Jacobian variety is from an algebraic (or arithmetic?) perspective. I want to know: What is the definition of the Jacobian? Widely know facts about it. Why the ...
8
votes
2answers
506 views

Adjoining torsion points from abelian varieties

Let $L/\mathbb{Q}$ be the field generated over $\mathbb{Q}$ by all of the (projective) coordinates of all of the torsion points of all abelian varieties defined over $\mathbb{Q}$. Is $L$ algebraically ...
33
votes
4answers
3k views

Fermat's last theorem over larger fields

Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite. Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite? Here ...
1
vote
2answers
198 views

Computing the nonsingular projective model of a plane curve

Is there an implemented algorithm available in standard software systems (Sage, Magma, Macaulay, etc.) that will compute the nonsingular projective model of a plane curve over $\mathbb Q$?
2
votes
1answer
144 views

Trivial Weil-Châtelet group

Does there exist an elliptic curve over a number field $K$ such that $WC(E/K)\cong H^1(G_K, E)$ is trivial?
0
votes
1answer
204 views

The number of solutions of a Diophantine equation [closed]

Is $\lim_{n \rightarrow \infty} |\{(x,y) \in \mathbb{Q}(\zeta_n)^2 : y^3 = x^3 + x + 1\}| < \infty ?$ where $\zeta_n$ is a primitive $n$-th root of unity. That is, I am asking whether the number ...
5
votes
0answers
177 views

On a claim of Deligne about representations of Weil-Deligne groups

In Deligne's article 'Les constantes des equations fonctionelles des fonctions L' http://publications.ias.edu/sites/default/files/Number20.pdf, we find the following claim: Proposition 8.9 (ibid.): ...
1
vote
0answers
144 views

Are there good properties of the divided power completion map?

Let $Y \to X$ be a closed immersion of smooth schemes over, say, the ${\rm Spec}(\mathbb{Z}_p)$. The completion map $$X_{/Y}\to X$$ is an ind-closed immersion (sometimes called pseudo-closed ...
10
votes
2answers
290 views

Is there a proof of Warning's Second Theorem using p-adic cohomology?

Let $\mathbb{F}_q$ be a finite field, $n \in \mathbb{Z}^+$, and $f_1,\ldots,f_r \in \mathbb{F}_q[t_1,\ldots,t_n]$ with $\operatorname{deg}(f_i) = d_i$. Put $d = \sum_{i=1}^n d_i$ and suppose $d< ...
8
votes
2answers
439 views

Finite etale atlas for Deligne-Mumford stacks

Let $X$ be a smooth finite type separated connected Deligne-Mumford stack over $\mathbb C$. Does there exist a finite etale morphism $Y\to X$ with $Y$ a scheme? What if $X$ is an algebraic space ...
5
votes
0answers
107 views

Can the hyperbolic core of a curve over $\mathbb Q$ be defined over $\mathbb Q$ as an algebraic stack

Here is a question I've been wondering about for a while. Currently it is mere curiosity and I do not have any direct applications in mind. Let $X$ be a smooth quasi-projective geometrically ...
2
votes
0answers
107 views

subschemes of abelian scheme over artinian basis

Let $R$ be an artinian thickening of a field $k$. Denote with $S=Spec(R)$. Let $A$ be an abelian scheme over $S$. Let $X$ be a closed, reduced, equidimensional subscheme of the special fiber $A_k$. I ...
8
votes
2answers
349 views

“Forms” of quadrics

The theory of Severi-Brauer varieties is well-known. Let $k$ be a (perfect) field. There may exist varieties not isomorphic to $\mathbf{P}^n$ over $k$, which are isomorphic to $\mathbf{P}^n$ over ...
0
votes
0answers
88 views

Computing a projection of a $p$-adic plane curve

Given a prime $p$ and a polynomial equation $f(x,y)=0$ with rational coefficients, I would like to obtain a precise description of the set of all numbers $y\in\mathbb Q_p$ such that the equation has a ...
3
votes
0answers
114 views

Singularities in mixed characteristic

Let $R$ be a regular local ring in mixed characteristic. Moreover, I assume that $R$ is the local ring of a point on a smooth $\mathbb Z_p$-scheme and that $R/pR$ is regular. ($\mathbb Z_p$ is the ...
0
votes
1answer
88 views

Action of $(\mathbb{Z}/2g\mathbb{Z})$ on quadratic forms on $\mathbb{Z}/2\mathbb{Z}$-vector space

Let $\mathbb{Z}/2\mathbb{Z}$ the 2 elements field, with additive notation. I need some clarifications on the relation between quadratic forms on a $\mathbb{Z}/2\mathbb{Z}$-vector space (say, of ...
5
votes
2answers
233 views

Are Anderson $T$-motives motives for the function field analogy?

this question is related to this one Geometry for Anderson's motives?, though the previous one doesn't answer exactly my question. Let $\mathbb{C}_{\infty}$ be the function field analog of ...
9
votes
1answer
580 views

Why is the section conjecture important?

As in the title, I want to know the reason for importance of the section conjecture. Of course, the statement of conjecture is important as itself, even I cannot fully grasp the soul of it. However, ...
12
votes
0answers
253 views

Artin L-function and Zeta function of twisted Dirac operator

If one thinks of a Frobenius as an element in the fundamental group of an arithmetic curve and of a Galois representation $\sigma$ as a flat connection on the curve, then the definition of the Artin ...
2
votes
2answers
277 views

Reference for Skinner-Urban on the Iwasawa main conjecture for $GL_2$

Does anyone know the existence of an expository paper or a report discussing the work of Skinner-Urban "The Iwasawa main conjecture for $GL_2$"? I am interested in partucular in the case of elliptic ...
3
votes
0answers
106 views

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= ...
8
votes
1answer
836 views

What is the arithmetic Nullstellensatz?

The only precise statement (coming from a reliable source) of the "arithmetic Nullstellensatz" I can find is in Gowers's book, stating that two polynomials with integral coefficients have the same ...
4
votes
1answer
233 views

What is known about the Brauer group of an arithmetic surface?

Let $X$ be an arithmetic surface over $\mathbb{Z}$, that is we have $\pi: X\rightarrow Spec(\mathbb{Z})$, $X$ is integral, two-dimensional and regular and $\pi$ is projective and flat. What is known ...
1
vote
0answers
79 views

Points on the intersection of an affine quadric and cubic over a finite field

Are there absolute constants $N$ and $B$ such that the following is true? Let $p>B$ be a prime. Let $q(x_0,\dotsc,x_n)$ and $c(x_0,\dotsc,x_n)$ be homogeneous of degree $2$ and $3$ with ...
8
votes
1answer
209 views

Power operations and Lambda-structure-like lifts of Frobenius in $E_\infty$-geometry?

A $\Lambda$-structure on a commutative ring $R$ is a ring endomorphism wich restricts to the $p$-Frobenius homomorphism after localizing at $(p)$. One may think of this as a "flow" $\Phi \colon ...
0
votes
1answer
205 views

Obstruction and 1st order infinitesimal deformations of Generalized Elliptic Curves (Deligne-Rapoport)

We consider the deformation theory of a generalized elliptic curve $(C_0,+)$ over a field $k$. Let $D$ be the deformation functor. And now we only consider the case that $C_0$ is irreducible as in ...