Questions tagged [arithmetic-geometry]

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

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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 ...
Larry Smith's user avatar
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 $p>0$ ...
Matt's user avatar
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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 $\...
XL _At_Here_There's user avatar
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 ...
Coherent Sheaf's user avatar
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 ...
Desunkid's user avatar
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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?
LaGra's user avatar
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2 votes
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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}...
BrauerManinobstruction's user avatar
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 ...
Yijun Yuan's user avatar
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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 ...
C.D.'s user avatar
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1 answer
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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 ...
Bogdan Grechuk's user avatar
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 ...
Bruno Kahn's user avatar
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 ...
TopGatLu's user avatar
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35 votes
1 answer
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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'...
Jan's user avatar
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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 ...
user1000039's user avatar
1 vote
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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=...
BrauerManinobstruction's user avatar
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{...
Coherent Sheaf's user avatar
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 ...
BrauerManinobstruction's user avatar
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 ...
Dimitri Koshelev's user avatar
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 ...
Coherent Sheaf's user avatar
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 ...
Matt's user avatar
  • 165
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 ...
Matt's user avatar
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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 $E$ ...
BrauerManinobstruction's user avatar
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 $\...
Matt's user avatar
  • 165
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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 ...
BrauerManinobstruction's user avatar
-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 $a=2,b=17$, ...
BrauerManinobstruction's user avatar
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....
Maarten Derickx's user avatar
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} ...
stupid_question_bot's user avatar
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)=\...
BrauerManinobstruction's user avatar
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 $d=1$, ...
Bogdan Grechuk's user avatar
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 ...
Bogdan Grechuk's user avatar
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 ...
BlaCa's user avatar
  • 6,770
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}...
BrauerManinobstruction's user avatar
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^*_{\...
OOOOOO's user avatar
  • 187
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 $G$-...
Doron Grossman-Naples's user avatar
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: $abc$-...
Stanley Yao Xiao's user avatar
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" ...
Roxana's user avatar
  • 519
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....
k.j.'s user avatar
  • 1,444
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 \...
BrauerManinobstruction's user avatar
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 (...
Matt's user avatar
  • 165
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 ...
Asvin's user avatar
  • 7,302
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 $X\times X$ ...
Jan's user avatar
  • 537
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 ...
Fernando Peña Vázquez's user avatar
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 ...
Hetong Xu's user avatar
  • 435
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 \...
Stanley Yao Xiao's user avatar
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 ...
Jan's user avatar
  • 537
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 = ...
Mor's user avatar
  • 443
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 ...
BrauerManinobstruction's user avatar
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 (...
user avatar
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}...
user avatar
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 ...
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