Questions tagged [arithmetic-geometry]

Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.

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55 views

One-sided "neutral zones" of plane curves

Let $C \subset \mathbb{R}^2$ be an algebraic curve. We assume that for some $x_0 > 0$, the curve $C$ is given by $(x, f(x))$ where $f : [x_0, \infty) \rightarrow [0,\infty)$ is one-to-one and ...
3 votes
0 answers
175 views

Does every (ordinary) elliptic curve over a quite large prime field $\mathbb{F}_{p}$ have a lift to the function field with huge Mordell-Weil rank?

Ulmer (and then other authors) showed existence of elliptic curves $E$ over the function field $\mathbb{F}_{p}(t)$ (where $p$ is a prime) with huge Mordell-Weil ranks (i.e., depending on $p$). The ...
1 vote
0 answers
121 views

How Galois group acts on Tate-Shafarevich group?

Let $L/K$ be a quadratic number field extension. Let $\operatorname{Sha}(E/L)$ be Tate-Shafarevich group of elliptic curve $E/L$. How $\sigma \in \operatorname{Gal}(L/K)$ acts on $\operatorname{Sha}(E/...
0 votes
0 answers
62 views

$r$-dimensional Frobenius coin problem

I saw the following result in a paper of Stephenson (https://arxiv.org/pdf/1412.6911.pdf, Lemma $2.2$), which is a variant of the Frobenius coin problem. Let $n_1, . . . , n_p$ be $p$ non-negative ...
3 votes
1 answer
311 views

Finite subschemes of projective bundles

$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Proj{\mathbf{Proj}}$Let $S$ be a Noetherian scheme and $X$ finite $S$-scheme. The finite morphism $X \to S$ is projective in the sense of the ...
6 votes
1 answer
416 views

Why do Chern forms show up in Arakelov geometry?

Let $X$ be a regular, projective flat scheme over $\Bbb{Z}$, let $\bar{L}$ be a hermitian line bundle on $X$. In order to define the height of an integral closed subset $Y$ we define it on closed ...
5 votes
0 answers
147 views

Generalization of Deuring's theorem

Deuring has the following important theorem on elliptic curves over a finite field: Let $p$ be a prime at least $5$, and $N=p+1-a$ an integer with $|a|\leq 2\sqrt{p}$, then the number of elliptic ...
4 votes
2 answers
329 views

Is the value of the power series at 0.1 transcendental?

Let $$f(x)= \sum_{n=1}^{\infty}a_n x^n$$ where $a_n\in \{0,1\}$, and the $f(x)$ has a natural boundary. By the way, $$f(x)= \sum_{n=1}^{\infty}a_n x^n$$ is rational function or transcendental one on $\...
7 votes
1 answer
294 views

Faithful representations of integral models

I am reposting a question that I had asked on stackexachage three weeks ago. Let $G/\mathbb{Q}$ be a connected reductive group, and $\mathcal{G}/\mathbb{Z}$ be an integral model (i.e. flat affine ...
39 votes
1 answer
5k views

Clausen's modified Hodge Conjecture

In a recent talk at the University of Geneve, Dustin Clausen presented a "modified Hodge Conjecture". I found the abstract intriguing but couldn't find videos or notes available online. If I'...
0 votes
1 answer
142 views

Zariski dense in abelian scheme

Let $A \to S$ be an abelian scheme over an irreducible curve $S$ over complex numbers of relative dimension $g \geq 1$. Let $s: S \to A$ be a non-constant section and denote by $X:=s(S)$. Is it true ...
4 votes
1 answer
839 views

Does this conic have a rational point?

Consider the conic $$C = \{X^2+uY^2+vZ^2=0\}\subset\mathbb{P}^2_{\mathbb{Q}(u,v)}$$ over the function field $\mathbb{Q}(u,v)$. Does $C$ have a $\mathbb{Q}(u,v)$-rational point?
2 votes
0 answers
137 views

Order $4$ element of Tate-Shafarevich group

Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$. Tate-Shafarevich group $\mathit{Sha}(E/\Bbb{Q})$ is defined as follows. $$\mathit{Sha}(E/\Bbb{Q})\stackrel{\text{def}}{=} \operatorname{Ker}...
5 votes
0 answers
183 views

Image of $\gamma-1$ on etale $(\varphi,\Gamma)$-modules

Let $p\geq 3$ be a prime, $D$ be an etale $(\varphi,\Gamma)$-modules over the classical period ring $A_{\mathbb{Q}_p}=\mathbb{Z}_p[\![T]\!][1/T]^{\widehat{\phantom{xx}}}_p$ and $\gamma$ be a ...
5 votes
1 answer
698 views

Symmetric powers, localisation and Frobenius

I am trying to understand the proof of lemma 2.2.18 in Lucas Mann's thesis. Its statement is surprising for me, because it talks about general rings which are not necessarily characteristic $p$, and ...
0 votes
0 answers
76 views

When can the valuative criterion of universal closure be checked on complete DVRs

Assume that I have a morphism of nice algebraic stacks $f : X \to Y$ that I want to show is universally closed. Suppose I have checked that for every complete DVR $R$ with algebraically closed residue ...
8 votes
1 answer
3k views

Where to start reading into p-adic non-abelian Hodge theory?

I'm curious about Faltings' "A $p$-adic Simpson correspondence". Do you know more detailed, introductory, expositions, surveys, texts of seminars on that? Edit: Annette Werner's survey &...
1 vote
0 answers
128 views

Characterization of Selmer group in terms of two descent

This question is about p 337 of Silverman's book ''The arithmetic of elliptic curves'', p 337, http://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf. Let $E:y^2=x^3+ax^2+bx$ and $E':Y^2=...
4 votes
1 answer
336 views

The notion of morphisms between two moduli problems in Katz-Mazur

I am reading Katz-Mazur Arithmetic Moduli of Elliptic Curves, and have some questions about the notion of morphisms between two moduli problems. What is the proper definition of morphisms between two ...
6 votes
1 answer
266 views

Definition of modular curve associated to $\Gamma(N)$

For a positive integer $N$, we define $$\Gamma(N)=\big \{ \begin{bmatrix} a & b \newline c & d\end{bmatrix}\in \operatorname{SL}_2(\mathbb{Z}): \begin{bmatrix} a & b \newline c & d\end{...
2 votes
0 answers
117 views

$K(S,2)=\{b \in K^{\times}/{K^{\times}}^2\mid v(b)≡0 \bmod2, \forall v \notin S\}$ and Selmer group

This question is essentially related to the theory of elliptic curves (Selmer group), but this question itself is just a field theoretic one. To calculate the Selmer group of given elliptic curve, we ...
6 votes
1 answer
467 views

Definition of locally symmetric space of reductive groups

This might seems like a bit of philosophical question and so maybe if I keep reading a bit more, I might get my answer. But, I ask nonetheless. In my attempt to study Shimura varieties, I came across ...
0 votes
1 answer
105 views

Is there a method to check if two sections of an elliptic surface are dependent over the endomorphism ring or not?

Let $E\!: y^2 = x^3 + a(t)x + b(t)$ be an elliptic curve over the function field $\mathbb{F}_{q}(t)$ over a finite field $\mathbb{F}_{q}$ of characteristic $5$ or greater. For simplicity, let $E$ be ...
4 votes
1 answer
375 views

Étale group schemes and specialization

If $A$ is an abelian variety over a finite field $\mathbf{F}_q$, then $A(\mathbf{F}_q)$ (resp. $A(\overline{\mathbf{F}}_q)$) is a finite (resp. infinite torsion) group, but $A(\mathbf{F}_q(t))$ is a ...
1 vote
0 answers
142 views

About the exact sequence $0\to $$Ш(E/\Bbb{Q})[\phi]\to Ш(E/\Bbb{Q})[2]\to Ш(E'/\Bbb{Q})[\hat{\phi}]$ in Silverman's book of elliptic curves

This question is about Silverman's book,$7$-th line from the bottom of $350$ page of 'The arithmetic of elliptic curves' (http://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf) . Let $E$ ...
2 votes
1 answer
229 views

Flat scheme-theoretic closure

Suppose $R$ is a discrete valuation ring with fraction field $K$. Let $X\subset \mathbf{P}^n_{C_K}$ be a closed subscheme, flat over $C_K$, a smooth projective curve over $K$. Let $C_R$ be a flat ...
0 votes
0 answers
113 views

Rank growth of elliptic curve $E:y^2=x^3-17$ in quadratic number field

Let $E:y^2=x^3-17$ be an elliptic curve. It is known that rank$(E/\Bbb{Q})=0$. (For example, prop $6.5$, $362$p in Silverman's book 'The arithmetic of elliptic curves') Over $K=\Bbb{Q}(i)$, what is ...
3 votes
0 answers
109 views

The degree map is a positive definite quadratic form

It is known that if $E_1$ and $E_2$ are elliptic curves over some field $K$ then the degree map $\deg: Hom(E_1,E_2) \to \mathbb Z$ is a positive definite quadratic form. A reference for this is III.6....
6 votes
2 answers
812 views

Is it possible to classify all Weil cohomologies?

Weil cohomologies seem to be "natural" and useful cohomology theories. Wikipedia lists Betti, De Rham, 𝓁-adic étale, and crystalline cohomologies as examples of Weil cohomology. Do we have ...
1 vote
0 answers
202 views

What is a definition of $A(P_v)$ in the definition of Brauer-Manin obstruction?

This is a question related to the definition of Brauer-Manin obstruction. Let $K$ be a number field. $X/K$ be an algebraic variety over $K$. Let $O_{X,P}$ be a local ring of $X$ at $P$. Let $Br(X)=\...
6 votes
1 answer
164 views

$\mathbb{Q}$-rank of the space of angles of pythagorean triples

A pythagorean triple is a triple of integers $(a,b,c)$ with $a^2 + b^2 = c^2$. Given a triple, $(a/c, b/c)$ is a point on the unit circle, so we may associate to it the normalized angle $$\theta_{a,b} ...
2 votes
2 answers
407 views

Existence of rational points on generalized Fermat quintics

Do there exist integers $(x,y,z)\neq (0,0,0)$ such that $$ (a) \quad 2x^5+3y^5=6z^5 $$ $$ (b) \quad x^5+3y^5=7z^5 $$ Here is a short motivation. Equation $ax^d + by^d=cz^d$ is trivial for $d=1$, ...
5 votes
1 answer
442 views

Existence of rational points on some genus 3 curves

Do there exist a pair of rational numbers $(x,y)$ such that $$ (a) \quad x^4+x^3+y^4+y-1=0 $$ $$ (b) \quad x^4+x^3+y^4+y^2-1=0 $$ Magma function IsLocallySoluble returns that both equations are ...
0 votes
0 answers
152 views

A question on the Hilbert-Kamke problem

The Hilbert-Kamke problem consists in studying the integral solutions of the Diophantine system $$ x_1^i + \dots + x_s^i = n_i \text{ for } 1\leq i\leq k $$ with $x_i\geq 0$ for $i = 1,\dots,k$. I am ...
0 votes
1 answer
197 views

Are degrees and ramification degrees preserved upon passing to the smooth compactification?

Let $\phi :C_1\to C_2$ be morphism of projective singular curve. Let $\tilde{C}_1$ and $\tilde{C}_2$ be their smooth compactification. Then $\phi$ extends to $\tilde{\phi} : \tilde{C}_1\to \tilde{C}...
1 vote
0 answers
203 views

$p$-adic étale cohomology group of open smooth varieties

Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $X$ be a smooth variety over $K$. Dr. Yamashita announced that he had proved the Galois representation of $p$-adic étale cohomology group $H^*_{\...
2 votes
1 answer
145 views

What is the sum operation on torsors induced by Weil uniformization?

Let $k$ be an algebraically closed field, $G$ a reductive group, and $C$ a curve. The algebraic version of the Weil uniformization theorem (see e.g. arXiv:1511.06271v2) says that groupoid of $G$-...
0 votes
0 answers
74 views

Extreme elliptic curves from good $abc$-triples

It is a well-known fact that the $abc$-conjecture of Masser and Oesterle and Szpiro's conjecture are equivalent. For the convenience of the reader I will write down the statements for both: $abc$-...
3 votes
1 answer
267 views

Given a smooth hyperplane section Y of a variety X there exists a Lefschetz pencil of hyperplane sections of X containing Y

Let $X$ be a variety contained in $\mathbb{P}^N$ and let $Y$ be a smooth hyperplane section of $X$. I have read in page 54 of Voisin's book "Hodge theory and complex algebraic geometry II" ...
1 vote
0 answers
251 views

Implicit function theorem and compactification of algebraic curve

Let $C$ be a singular curve defined over a local field $K$. Let $\tilde{C}$ be its smooth compactification(maybe this is not normalization). Why $\tilde{C}(K)\neq \emptyset$ implies ${C}(K)\neq \...
6 votes
1 answer
735 views

B. W. Jordan's thesis on arithmetic of Shimura curves

I'm looking for Bruce W. Jordan's thesis: On the diophantine arithmetic of Shimura curves. Thesis, Harvard University, 1981. I could not find the pdf at the following site. https://www.math.harvard....
3 votes
1 answer
156 views

How to compute Selmer set?

Let $X$ be an affine variety and $G$ an affine algebraic group (for example $\operatorname{PGL}_n$). How do I compute the Selmer set $$ \operatorname{Sel}_\zeta(\mathbb{Q},G) = \{\tau \in H^1(\mathbb{...
3 votes
0 answers
130 views

A Galois equivariant Weil cohomology theory with coefficients in the rational numbers and a variation of the Tate/Hodge conjecture

A well-known example of Serre shows that there can be no Weil cohomology theory with $\mathbb Q$ coefficients for schemes over $\mathbb F_{p^2}$. However, this example is no obstruction to a Weil ...
2 votes
1 answer
204 views

Finite flat pullback of the diagonal

Let $X, Y$ be smooth projective connected complex varieties of the same pure dimension $d$ and $f : X\to Y$ a finite flat surjective morphism. Let $\Delta_X$ be the closed subscheme of $X\times X$ ...
4 votes
1 answer
230 views

Relation between rational Tate module and universal cover of a p-divisible group

We can associate two $\mathbb Q_p$ vector spaces to a $p$-divisible group, and I'm a little confused about the relation between these two groups. First of all, I think part of my problem is that when ...
13 votes
3 answers
1k views

Is the map on étale fundamental groups of a quasi-projective variety, upon base change between algebraically closed fields, an isomorphism?

$\DeclareMathOperator\Spec{Spec}$Let $k \subset L$ be two algebraically closed fields of characteristic $0$. Let $U \subset \mathbb P^n_k$ be a smooth quasi-projective variety and let $U_L$ denote the ...
5 votes
0 answers
323 views

Calculating étale fundamental groups from the usual fundamental group

$\newcommand{Spec}{\operatorname{Spec}}$Let $X$ be a connected affine smooth variety over $\mathbb{Q}$, with a point $x\in X(\Spec(\mathbb{Q})$. For any algebraically closed field $K$ of ...
11 votes
1 answer
448 views

Do people prefer working on $\mathrm{GSp}$ and $\mathrm{GU}$ rather than $\mathrm{Sp}$ and $\mathrm{U}$, and why?

I am a new learner of Iwasawa theory and currently reading the famous paper by Skinner-Urban in 2014, and the following-up works by many other people. When reading these papers, I found that some ...
6 votes
2 answers
555 views

Mordell curves with large rank

An elliptic curve is (for the purpose of this question) a cubic algebraic curve defined by an equation (short Weierstrass equation) of the form $$\displaystyle E_{a,b} : y^2 = x^3 + ax + b, a, b \in \...
4 votes
1 answer
232 views

Cycles contained in ample enough hypersurfaces

Let $X$ be an irreducible smooth projective variety of pure dimension $d$ over the complex numbers and $Z\subset X\times X$ a codimension $d$ irreducible smooth closed subvariety. Is there a smooth ...

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