# Questions tagged [arithmetic-geometry]

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

1,912
questions

3
votes

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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

0
answers

87
views

### Subschemes of finite flat group schemes

Let $S=\text{Spec}(A)$ for $A$ a Noetherian local ring. Let $G$ be a finite locally free $S$-group scheme.
Assume first that $A$ is complete and with finite residue field of characteristic $p>0$ ...

0
votes

2
answers

208
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 $\...

6
votes

1
answer

216
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 ...

0
votes

1
answer

118
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 ...

6
votes

1
answer

715
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

112
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

126
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 ...

0
votes

0
answers

63
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 ...

0
votes

1
answer

96
views

### Rational points on genus 3 curves defined by short equations

(a) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3-y=x^4-x.
$$
(b) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3+y=x^4+x.
$$
If not a complete answer, I would be happy to ...

3
votes

0
answers

82
views

### Height pairing and the Néron model of an elliptic curve

I have a question on Joseph Silverman's book ``Advanced topics in the arithmetic of elliptic curves’’ (1999 printing). I asked him; he answered that he doesn't know off-hand and suggested that I put ...

2
votes

0
answers

95
views

### Families of quadratic forms over a function field

I have a family of quadratic form in two variables $q_{t}(x,y) = ax^2 + bxy+cy^2$ where $a,b,c\in\mathbb{C}(u,v)$ are rational functions depending on a parameter $t\in\mathbb{P}^1$.
I would like to ...

35
votes

1
answer

2k
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'...

4
votes

1
answer

294
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 ...

1
vote

0
answers

105
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=...

6
votes

1
answer

186
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

104
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 ...

0
votes

1
answer

61
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 ...

6
votes

1
answer

340
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 ...

4
votes

1
answer

311
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 ...

2
votes

1
answer

138
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 ...

1
vote

0
answers

127
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 $E$ ...

4
votes

0
answers

73
views

### Determinants of perfect complexes and Hilbert polynomials

Let $X$ be a smooth projective variety over an algebraically closed field $k$, and let $K^{\bullet}$ be a perfect complex of $\mathcal{O}_X$-modules.
It is possible to define a canonical line bundle $\...

0
votes

0
answers

85
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 ...

-2
votes

0
answers

121
views

### Weierstrass equation of smooth projective model of $ay^2=x^4-b$

Let $a,b$ be a rational number.
Let $C$ be an smooth projective model of $ay^2=x^4-b$.
$C$ is elliptic curve. I want to know the Weierstrass equation of $C$ in $\Bbb{P}^2$.
For example $a=2,b=17$,
...

3
votes

0
answers

94
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

1
answer

150
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} ...

1
vote

0
answers

187
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)=\...

2
votes

2
answers

382
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

386
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 ...

3
votes

0
answers

138
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

192
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

176
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

120
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

67
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

225
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" ...

5
votes

1
answer

665
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....

1
vote

0
answers

227
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 \...

2
votes

0
answers

143
views

### Gysin maps for singular varieties

Let $X$ be an integral projective variety of pure dimension $n$ over an algebraically closed field and $Z\subset X$ a closed irreducible subvariety of pure codimension $c$.
Is there a functorial (...

3
votes

0
answers

110
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

158
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 $X\times X$ ...

5
votes

0
answers

243
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

360
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

494
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

220
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 ...

5
votes

1
answer

182
views

### System of two linear Diophantine equations

Let $n\in\mathbb{N}$ be a positive integer. Denote by $f(n)$ the number of integral solutions of the following system
$$
\left\lbrace\begin{array}{l}
\sum_{i=1}^nx_i = 3n; \\
\sum_{i=1}^n (2i-1)x_i = ...

0
votes

0
answers

84
views

### Relation between divisibility problem of Shafarevich group and group structure of $Ш(E/K)$

For abelian variety $A/K$,
divisibility problem (i.e. $\forall n≧1$, $Ш(A/K)⊂p^nH^1(G_K,A)$ holds for fixed prime $p$?) was asked by Cassels in 1962 and even now discussed.
On the other hand, once ...

1
vote

1
answer

134
views

### Cohomology classes fixed by algebraic automorphism subgroups

Let $X$ be a smooth projective complex variety and $t\in H^{2p}(X,\mathbf{Q})(p)$ a rational cohomology class.
Assume that there exist
$$t_1,\ldots,t_N\in H^{2*}(X,\mathbf{Q})(*)$$
algebraic classes (...

2
votes

1
answer

165
views

### Cup products and correspondences

Suppose $X$ is a smooth projective complex variety, connected of dimension $n$.
Let $a$ be an algebraic correspondence in $A^n(X\times X)$, the group of cycles modulo homological equivalence in $H^{2n}...

3
votes

0
answers

101
views

### Local global principle over infinite extension of $\Bbb{Q}$ which is not algebraically closed

Let $A$ be an algebraic variety over a field $K$, which is finite extension of $ \Bbb{Q}$.
We say local global principle holds if $A(K_v) \neq \emptyset$ implies $A(K) \neq \emptyset$, where $K_v$ is ...