All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
2
votes
0
answers
186
views
Is the group of rational points of an anisotropic absolutely quasi-simple algebraic group over a non-archimedean local field known to be perfect?
Suppose that $G$ is an algebraic group defined over a non-archimedean local field $k$ which is absolutely quasi-simple and anisotropic over $k$. Is it known whether the group $G(k)$ is necessarily ...
- 2,125
4
votes
1
answer
611
views
Fermat surface known to have very few rational integer solutions
The motivation for this question is the Selmer curve, given by
$$\displaystyle 3x^3 + 4y^3 + 5z^3 = 0.$$
One can show that this curve has no rational integer solutions, despite having a solution ...
- 26.9k
8
votes
1
answer
753
views
Hasse principle and Brauer-Manin obstruction for forms of large degree
The Hasse principle is perhaps an at-first naive generalization of the Chinese remainder theorem; that if a linear equation can be solved modulo $p$ for any prime $p$, then it can be solved in the ...
- 26.9k
1
vote
0
answers
103
views
degree of isogenies between Jacobians and Abelian Varieties
Let $K$ be a local field of characteristic zero and positive residual characteristic. Let $A$ be a simple abelian variety and assume we have an isogeny $f:Jac_C\rightarrow A$ with $C$ a smooth curve ...
- 11
12
votes
2
answers
424
views
Existence of local sections
I would like to know when the property of "having a section" for a morphism of varieties in characteristic $0$ can be detected by spreading out to characteristic $p$.
Take a number field $K$, and let ...
- 121
1
vote
1
answer
244
views
Arithmetic property of a surface of general type
In my previous post I asked about the hyperbolicity of the affine surface $S'=\{zw \neq o\}$ in the projective surface $z^2 = P(x) Q(y)$ in $\mathbb{P}^3$, where $P$ and $Q$ are two general ...
- 39
0
votes
1
answer
721
views
Kernel of a 3-isogeny between two elliptic curves
Suppose $E_1$ and $E_2$ are two elliptic curves defined over $\mathbb{Q}$ and there exists a 3-isogeny $\varphi$: $E_1 \longrightarrow E_2$. If $E_1$ has no $\mathbb{Q}$-rational point of order 3, ...
- 303
3
votes
1
answer
360
views
What is the relation between KC and height of rational number?
Roughly speaking,Kolmogorov Complexity of a bits string or a description is the minimal length of programs outputing a bits string,and height of rational number is logarithm of the largest numerator ...
- 3,094
3
votes
1
answer
734
views
Curves of high genus with many rational points
The seminal theorem of Faltings confirms Mordell's conjecture: that is, curves of genus at least 2 have at most finitely many rational points. The proof of Faltings' theorem is not effective, meaning ...
- 26.9k
3
votes
0
answers
143
views
Question related to the number of rational curves on a hypersurface
The following is from a PhD thesis "Algebraic Circle Method" by Thibaut Pugin, which I am currently reading.
Let $k$ be a finite field of order $q$.
Let $\mathfrak{X} \subseteq \mathbb{P}^n_k$ be ...
- 579
4
votes
0
answers
202
views
Continuity of the Hilbert pairing
I would like to know if the Kummer pairing (or the analogue of the Hilbert Symbol) for a one dimensional group defined over the ring of integers of a higher-dimensional local field is continuous (with ...
- 41
3
votes
0
answers
197
views
Number of rational curves on varieties over finite fields
Let $k$ be a finite field. Let $P_r$ be set of degree $r$ binary forms
over $k$.
We define
$$
\mathcal{M}_r = \{ (x_0, ..., x_n) \in P_r^{n+1} : x_0^d + ... + x_n^d = 0 \text{ and the }x_i \text{'s ...
user48211
10
votes
2
answers
2k
views
Main conjecture for elliptic curves
Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ \lambda_{E}^...
- 1,209
0
votes
0
answers
117
views
Consistency of the u-invariant under field extension
A algebraic field extension L/k induces of homomorphism between the Wittrings. We get
$\phi: W(k) -> W(L)$. If every anisotropic isometry class of $W(k)$ stays anisotropic, the kernel of $\phi$ ...
- 193
8
votes
2
answers
902
views
Forms of algebraic varieties
Let $X$ be an algebraic variety (say, projective, irreducible and smooth), defined over a field $K$, and let $L$ be a Galois extension. I am interested in algebraic varieties $Y$, defined over $K$, ...
- 7,722
11
votes
1
answer
775
views
Degeneration of the Hodge spectral sequence for scheme over truncated Witt ring?
Let $f:X\to S$ be a smooth proper morphism of schemes. If $S$ is of characteristic zero (i.e., $S$ is a $\mathbb{Q}$-scheme), then Deligne has shown:
The Hodge-De Rham spectral sequence $E^{a,b}_1=...
- 699
16
votes
1
answer
1k
views
Examples of elliptic curves over $\mathbb{Q}$
I need examples of two non-isogenous elliptic curves $E_{1}, E_{2}$ over $\mathbb{Q}$ having the following 2 properties -
1) $E_{1}, E_{2}$ have no rational torsion points.
2) $E_1[9] \cong E_2[9]$ ...
- 1,209
4
votes
1
answer
265
views
What is the interpretation of this galois cohomology set?
Let $K$ be a field of characteristic zero. Let $G_K:=Gal(\bar{K}/K)$
The nontrivial elements of the set $H^1(G_K,PGL_2)$ correspond to $\bar{K}/K$-forms of $\mathbb{P}^1$; i.e. curves that are ...
- 605
1
vote
0
answers
70
views
Unbranched cover of a curve of CM type
Let $C$ be a curve of genus $g\geq 2$ over complex number. Assume that $C$ has complex multiplication (CM).
Does there exist such a curve $C$ such that $C'$ is also of CM type for any unbranched ...
- 288
3
votes
1
answer
229
views
On Universal Abelian surfaces over a Shimura curve.
Let ${\cal O}, {\cal O}'$ be two order in ${\mathrm M}_2({\Bbb R})$ that are sets of all $2 \times 2$ matrices over real number ${\Bbb R}$. Assume that we have the relation ${\cal O}' = a{\cal O}a^{-1}...
- 181
7
votes
0
answers
462
views
Looking for a paper of Hartshorne
In a famous paper
Hartshorne - Varieties of small codimension,
Hartshorne formulates a conjecture, which roughly says that varieties of small codimension in projective space are complete ...
- 21.3k
5
votes
1
answer
249
views
Conductor CM abelian variety
This is probably well known but I am not an expert in the subject.
Given an abelian variety $A$ of dimension $g$ with CM by $O_K$ where $K$
is a CM field of degree $2g$, let $N_A$ be the norm of the ...
- 547
2
votes
0
answers
429
views
Zariski density of Q-bar points
Let $X \subset \mathbb{C}^n$ be an affine variety which is defined over $\mathbb{Q}$ (i.e. the zero set of some finite collection of polynomials with coefficients in $\mathbb{Q}$). Define $X'$ to be ...
- 31
4
votes
1
answer
517
views
How to obtain the Period matrix from the Igusa Invariants of a genus two curve?
I am looking for an algorithm to obtain the period matrix tau= (tau1 & tau12\ tau12 & tau2) from the Igusa invariants of a genus two curve. More precisely:
I consider a family of genus two ...
- 41
7
votes
2
answers
2k
views
Basics on anabelian geometry and Grothendieck's section conjecture
Even I can find similar questions and some answers on that questions, most of them are not quite unsatisfactory to me. Maybe this is a very stupid question, but there is no other place that I can ask ...
- 601
7
votes
2
answers
417
views
Is there a largest prime p such that J_0(p) completely splits into elliptic curves
The question in the title is related to a more general question. Namely does there exist an integer $N$ such that for all curves $C/\mathbb C$ of genus $> N$ one has that not all simple isogeny ...
- 2,224
1
vote
2
answers
304
views
how to see CM types as functions on the Galois group?
Let $K$ be a CM field, that is, an imaginary quadratic extension of a totally real number field. Its degree $[K: \mathbb{Q}]$ is a en even number $2n$.
(1) For me a CM type is a subset $\Phi \subset ...
- 11
1
vote
0
answers
169
views
another question on the Manin-Drinfeld theorem
A few days ago I asked a question about possible higher dimensional generalizations of the Manin-Drinfeld theorem. Let me come back to the classical statement.
It says that a divisor on the modular ...
- 123
2
votes
1
answer
218
views
A problem in intersection theorem
I'm reading the paper:
SGA 7 II, Intersections sur les surfaces regulieres.
In Papge 6 , I cannot understand why there is sign $-1$ in the formula (1.10.4):
Let $S$ be a trait, for any $\mathcal O_S$...
- 135
9
votes
1
answer
2k
views
Torsors and the fpqc topology
Fix a scheme $S$, a group scheme $G/S$ (let us say smooth, maybe even affine with some finiteness conditions if you like), and suppose I have some other $S$-scheme $P$ with a right $G$-action. We want ...
- 806
15
votes
3
answers
1k
views
Why study CM abelian varieties?
I know that abelian varieties of CM type have central importance in algebraic geomtry and number theory. There are many conjectures and concepts related to them like Andre-Oort, Coleman conjecture, ...
- 2,221
2
votes
1
answer
332
views
Pencil with desired Jet in Algebraic geometry(new!)
Let $k$ be an algebraic closed field.
Let $n$ be a positive integer.
Let $X$ be an irreducible, proper and smooth scheme over $k$ with an immersion $i:X\hookrightarrow E:=\mathbb P^N_k$ with $N$ ...
- 153
1
vote
1
answer
307
views
Weierstrass points on modular curves
What is knowns about Weierstrass points on modular curves? Are there any explicit formulas of them, or any information about Weierstrass gaps? I am interested in (compactifications of) the quotients ...
- 5,186
4
votes
1
answer
273
views
A problem on Jets in algebraic geometry
Let $k$ be a perfect field, let $n$ and $m$ be two positive integers.
Consider $X=\mathbb P_k^n\times \mathbb P_k^m$. Let $x_0=(1,0,\cdots,0;1,0,\cdots,0)\in X$ be fixed.
For any pair of integers $(...
- 153
3
votes
0
answers
122
views
Curves on hypersurfaces generated by diagonal sums
This is related to an earlier question of mine ((Non-)Existence of curves of low degree on affine and projective varieties). It seems that the question is too difficult for specific surfaces, although ...
- 26.9k
2
votes
2
answers
1k
views
L-functions and algebraic geometry
Robert Langlands commented in a letter to Deligne that perhaps some of the deepest problems of algebraic geometry lie in L-functions. I want to understand the general philosophy and the connection ...
- 2,106
12
votes
3
answers
411
views
(Non-)Existence of curves of low degree on affine and projective varieties
I am interested in papers that investigates the existence or non-existence of curves of low degree (relative to the degree of the ambient variety). The starting example is that of surfaces and ...
- 26.9k
17
votes
0
answers
1k
views
Katz--Mazur for abelian varieties
Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.
Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac 1N]...
- 18.7k
3
votes
1
answer
420
views
Automorphisms of complete discrete valuation ring
Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb F}...
- 87
3
votes
3
answers
719
views
A neat monodromy group of a family of Kaehler manifolds
Let $X\rightarrow B$ be a family of Kaehler manifolds with possibly singular fibers. Let $G$ be the monodromy group on $H^n(X_b,\mathbb{Z})$, where $n=\dim X_b$ with the smooth fiber $X_b$ over some $...
- 31
9
votes
1
answer
763
views
What is the motivation for defining the conductor of an abelian variety?
Let $K$ be a $p$-adic field, and let $A$ be an abelian variety over $K$. The conductor of the abelian variety is often defined as $2u+t+\delta$, where $u$, $t$ and $\delta$ are invariants related to ...
- 6,348
3
votes
1
answer
325
views
Mordell-Weil and finiteness of rational points
Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order, that is, $\mathrm{End}_K(E)\cong \mathcal{O}_K$. Suppose the class number of $K$ is equal to $1$. Let $\...
- 601
3
votes
1
answer
385
views
Is Mazur's deformation ring R integral?
Consider the absolutely irreducible Galois representation
$\overline{\rho} \colon G_{\Bbb Q} \to {\mathrm{GL}}_2({\Bbb F}_p)$. We apply the Mazur's deformation theory on the lift ${\rho} \colon G_{\...
- 111
0
votes
1
answer
455
views
Iwasawa theory for Mazur's deformation ring R
The ideal class group $\mathrm{Cl}({\cal O}_K)$ and Mazur's deformation ring $R(\overline{\rho})$ for a number field $K$ are said to be similar to each other.
Let ${\Bbb Q}_{\infty}$ be the unique ...
- 87
11
votes
1
answer
831
views
The Sato-Tate conjecture for hypersurfaces?
The Sato-Tate conjecture for elliptic curves $E$ predicts the distribution of the eigenvalues of Frobenius at $p$ on the Tate module of $E$ as $p$ varies in terms of the distribution of the ...
- 118k
12
votes
2
answers
1k
views
An integrality question about expressing an integer as a product of numbers below $n$
Let $n\ge 2$ be a natural number. Suppose that $N$ is a natural number, composed only of primes below $n$, and that can be expressed as
$$
N= \prod_{j=1}^{n} j^{x_j}
$$
where $x_1$, $\ldots$, $x_n$...
- 43.7k
6
votes
2
answers
480
views
Rational points and torsion points of CM elliptic curve
Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order i.e., $\mathrm{End}_K(E)\cong \mathcal{O}=\mathcal{O}_K$. Let $\mathfrak{p}$ be a prime of $K$ such that ...
- 601
2
votes
0
answers
224
views
Abel-Jacobi map isomorphism galois representations
Let $X/\mathbb{Q}$ be an irreducible smooth projective curve with a $\mathbb{Q}$-rational point $p$. Then there is a map $\phi: X \rightarrow \textrm{Pic}^{0}(X)$ with the property that $q \mapsto [q]-...
- 3,555
12
votes
1
answer
895
views
Is the universal elliptic curve $\overline M_{1,2}$ a toric stack?
It is well-known that the compactification $\overline M_{1,1}$ of the moduli space of elliptic curves over $\mathbb C$ is a weighted projective line with
weights $4$ and $6$. As far as I can tell, ...
- 5,186
2
votes
0
answers
146
views
Odds of projections of a point not on the hyperplane
Let $\mathcal{L}=\{\Bbb x\in\Bbb R^n:x_1+x_2+\dots+x_n=0\}$ be a specific hyperplane.
Let a projection of $c\in\Bbb R^n$ be $p(c)=[p_1,p_2,\dots,p_n]$ where $p_i\neq c_i\implies p_i=0$.
Let $\...
- 13.9k