Questions tagged [arithmetic-geometry]
Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
2,047
questions
8
votes
1
answer
324
views
Alterations and smooth complete intersections
Let $k$ be an algebraically closed field, and $X$ a projective variety over $k$. Let $i : X\subset \mathbf{P}^d_k$ be a closed immersion into a projective space of high enough dimension.
Is there a ...
2
votes
1
answer
165
views
Cohen-Macaulay fiber products
Let $R$ be a regular local ring, $X$ and $Y$ smooth $R$-schemes, $T\to Y$ a regular closed immersion over $R$ with $T$ smooth over $R$, and $f: X\to Y$ an $R$-morphism.
Is the fiber product scheme $...
2
votes
1
answer
238
views
Images of smooth schemes under lci morphisms
Let $S$ be a Noetherian scheme, $f : X\to S$ a smooth quasi-projective morphism, $g : X\to Y$ a morphism of finite type, and $h : Y\to S$ a smooth projective morphism with $h\circ g =f$.
Can we say ...
5
votes
0
answers
290
views
Are there toy models of full poly-isomorphism from Inter-universal Teichmüller theory?
I am learning Inter-universal Teichmüller theory and I am interested in the concept of full poly-isomorphism. According to "INTER-UNIVERSAL TEICHMÜLLER THEORY I: CONSTRUCTION OF HODGE THEATERS&...
3
votes
1
answer
260
views
Leading constant in Batyrev-Tschinkel's refinement of Manin conjecture
Background: Let $X$ be a Fano variety over number field $K$, where its anticanonical bundle $K_X^{-1}$ is ample. Let $i: X \to \mathbb{P}^n$ be the anticanonical embedding, where $K_X^{-m} \cong i_*O(...
0
votes
0
answers
140
views
Norm map of Tate-Shafarevich group $\mathrm{Sha}(E/K)\to \mathrm{Sha}(E/\Bbb{Q})$
Let $K$ be a quadratic number field. Let $\sigma$ be a generator of Galois group of $K/\Bbb{Q}$. Let $E$ be an elliptic curve defined over $\mathbb{Q}$.
Let $\mathrm{Sha}(E/K)$ denote the Tate-...
0
votes
0
answers
253
views
Geometric meaning of cusps/component labels in Katz-Mazur book
In Katz-Mazur book "Arithmetic moduli of elliptic curves" there is a short section (see image below) regarding the cusp-labels and component-labels.
The set of cusps labels intuitively ...
4
votes
1
answer
260
views
Discrepancy in the calculation of $2$-Selmer group by Magma and LMFDB
The result of LMFDB claims (https://www.lmfdb.org/EllipticCurve/Q/1640/c/1 )
that (2-part of) Tate-Shafarevich group $\mathrm{Sha}(E/\Bbb{Q})$ of elliptic curve $y^2=x^3-8747x-314874$ has order $16$. ...
1
vote
0
answers
56
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
176
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
123
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
313
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
422
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
330
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
295
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
145
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
852
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
138
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.
$$\mathit{Sha}(E/\Bbb{Q})\stackrel{\text{def}}{=} \operatorname{Ker}...
5
votes
0
answers
183
views
Image of $\gamma-1$ on etale $(\varphi,\Gamma)$-modules
Let $p\geq 3$ be a prime, $D$ be an etale $(\varphi,\Gamma)$-modules over the classical period ring $A_{\mathbb{Q}_p}=\mathbb{Z}_p[\![T]\!][1/T]^{\widehat{\phantom{xx}}}_p$ and $\gamma$ be a ...
5
votes
1
answer
703
views
Symmetric powers, localisation and Frobenius
I am trying to understand the proof of lemma 2.2.18 in Lucas Mann's thesis. Its statement is surprising for me, because it talks about general rings which are not necessarily characteristic $p$, and ...
0
votes
0
answers
76
views
When can the valuative criterion of universal closure be checked on complete DVRs
Assume that I have a morphism of nice algebraic stacks $f : X \to Y$ that I want to show is universally closed. Suppose I have checked that for every complete DVR $R$ with algebraically closed residue ...
8
votes
1
answer
3k
views
Where to start reading into p-adic non-abelian Hodge theory?
I'm curious about Faltings' "A $p$-adic Simpson correspondence". Do you know more detailed, introductory, expositions, surveys, texts of seminars on that?
Edit: Annette Werner's survey &...
1
vote
0
answers
128
views
Characterization of Selmer group in terms of two descent
This question is about p 337 of Silverman's book ''The arithmetic of elliptic curves'', p 337, http://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf.
Let $E:y^2=x^3+ax^2+bx$ and $E':Y^2=...
4
votes
1
answer
351
views
The notion of morphisms between two moduli problems in Katz-Mazur
I am reading Katz-Mazur Arithmetic Moduli of Elliptic Curves, and have some questions about the notion of morphisms between two moduli problems.
What is the proper definition of morphisms between two ...
6
votes
1
answer
268
views
Definition of modular curve associated to $\Gamma(N)$
For a positive integer $N$, we define $$\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
481
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
379
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
144
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
237
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
114
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
110
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
813
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
165
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
447
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
1
answer
198
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
146
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 $G$-...
3
votes
1
answer
272
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
740
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
157
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
132
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
206
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 $X\times X$ ...