All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
5
votes
0
answers
209
views
Reducibility of a cubic over a number field
Given an extension $K = \mathbb{Q}(\alpha)$ by a cubic polynomial $g(x)\in \mathbb{Q}[x]$ (not necessarily Galois extension) is there a criterion for a cubic polynomial $f(x) \in K[x]$ to be reducible ...
- 481
32
votes
0
answers
1k
views
Cubic function $\mathbb{Z}^2 \to \mathbb{Z}$ cannot be injective
It is easy to show, with an explicit construction, that a homogeneous cubic function $f: \mathbb{Z}^2 \to \mathbb{Z}$ is not injective. I am seeking a proof of the same result without the condition ...
- 2,902
6
votes
1
answer
308
views
An explicit equation of the canonical morphism $X_1(N) \to X_0(N)$
I know there are some research about explicit equations for affine models in $\mathbb{A}^2$ of many modular curves over $\mathbb{Q}$, for example of $X_i(N), X(N)$ (where $i = 0, 1, 2$) for small $N$.
...
- 1,364
2
votes
0
answers
109
views
Shimura curves and quaternion orders without elements of norm -1
Let $O$ be an order of a quaternion algebra over $\mathbb{Q}$ such that $O$ does not contain elements of norm $-1$. Such orders exist, but seems less used, in particular these orders are not Eichler.
...
- 91
3
votes
0
answers
87
views
Norm $-1$ elements of quaternion algebras and Shimura curves [duplicate]
Let $Qa$ be an indefinite quaternion algebra over $\mathbb{Q}$.
Let $O$ be an order of $Qa$. The moduli space of abelian surfaces with quaternionic multiplication by $O$ is usually designed as the ...
- 91
4
votes
1
answer
418
views
Explicit natural correspondence between cusps of $X(N)$ and isomorphism classes of level $N$ structures on Tate($q^N$)
In Katz' paper Antwerp III, section 1.4 (Ka-14) one reads (we assume $n \geq 3$ integer):
''The scheme $\overline{M}_n - M_n$" over $\mathbb{Z}[1/n]$ is finite and étale, and over $\mathbb{Z}[1/n,...
- 65
3
votes
0
answers
443
views
What is the definition of smoothness in scheme theory?
I want to ask if there is a somewhat desirable definition of "smoothness".
Definition. Let $k$ be a field and $X$ be a separated finite type scheme over $k$. Then $X$ is smooth if the ...
user196717
4
votes
0
answers
277
views
Explicit computations of the fundamental groups of perfectoid spaces
If $X$ is a perfectoid space then it has the same étale site as its tilt $X^\flat$. This means that the fundamental groups (suitably defined) of $X$ and $X^\flat$ are isomorphic.
Can you give ...
- 41
2
votes
0
answers
242
views
Cartier operator and logarithmic differentials
Let $k$ be an algebraically closed field of characteristic $p$, let $C$ be a curve over $k$ and let $\omega$ be a meromorphic differential form on $C$. If $\omega$ gets mapped to itself by the Cartier ...
- 75
1
vote
0
answers
107
views
Topologically finitely generated non-abelian isomorphic absolute Galois groups
Let $K$ be a field of positive characteristic and $L$ be a field of characteristic zero.
Assume the absolute Galois groups of $K$ and $L$ are topologically finitely generated, non-abelian and ...
- 55
3
votes
1
answer
184
views
Non-abelian isomorphic absolute Galois groups of fields of different characteristic
Let $K$ be a field of positive characteristic and $L$ be a field of characteristic zero.
Assume the absolute Galois groups of $K$ and $L$ are non-abelian and isomorphic as profinite groups.
Can $L$ ...
- 55
16
votes
1
answer
2k
views
Proof of main theorems in étale cohomology theory
(In this question, $p$ can be $0$.)
I'm curious if theorems on étale cohomology can be proved by easier way.
For example, proper base change theorem. This theorem can be stated as the following way.
...
user196717
4
votes
1
answer
411
views
Z2xZ6 elliptic curves with missing generators
By implementing the techniques described in and similar to
A. Dujella, J. C. Peral, Elliptic curves with torsion group Z/8Z or Z/2Z x Z/6Z, arXiv, Number Theory [math.NT] (2013), arXiv:1306.0027v1
A....
- 913
3
votes
1
answer
330
views
Generalization of torsion points on Jacobian of genus 2 over finite fields (with respect to the theta divisor)
Let $J(C)$ be the jacobian of a hyperelliptic curve $C$ of genus 2 defined over finite field $\mathbb{F}_q$. Let $\Theta$ be the image of the curve on the Jacobian under the embedding $P \mapsto P - \...
- 125
1
vote
0
answers
151
views
Rational points on towers of surfaces
Take infinitely many 2-variable polynomials $p_k(X,Y)\in \mathbb{Q}[X]$ ($k\in \mathbb{N}$) and let $S_n$ be the surface given by $p_1(X,Y)=Z_1^2,\dots, p_n(X,Y)=Z_n^2$
Assume that no $p_k$ equals the ...
- 2,959
4
votes
1
answer
571
views
Relation between stacky curves and "M-curves"
A tame stacky curve over a field $k$ is a geometrically connected proper smooth DM stack of dimension 1 which has a dense open substack which is a scheme, and whose automorphism group of each ...
- 1,364
0
votes
0
answers
177
views
Passing over $O_K \otimes_{\mathbb{Z}} A$ from $O_K$, how it affects the rank of a module?
This question was asked in MSE as well.
Let $K$ be a finite extension of the rationals $\mathbb{Q}$ with $O_K$ its the ring of integers.
Consider a $\mathbb{Z}$-algebra $A$ such that $|A|<\infty$.
...
- 930
3
votes
0
answers
186
views
Maximum value of newform from Galois representation
One can attach $\ell$-adic Galois representations to holomorphic cuspidal newforms of weight $2$ on the upper half-plane.
If a newform is $L^2$-normalized, can one extract its maximum value from the ...
- 39
3
votes
0
answers
174
views
Reference Request: CM Motives over Function Fields
Inspired by Schutt and Shioda's lovely "An interesting elliptic surface over an elliptic curve", I have been investigating the following surface:
$$
\mathcal{E} : y^2 = x^3 - 27ux - 54v \...
12
votes
1
answer
532
views
Open immersion of affinoid adic spaces
If $R$ and $S$ are complete Huber rings with $\varphi: R \to S$ a continuous map, then is it true in general that if $\mathrm{Spa}(S, S^\circ) \to \mathrm{Spa}(R, R^\circ)$ is an open immersion of ...
- 503
5
votes
0
answers
328
views
Ramification behavior of field given by adjoining $p$-torsion point on formal group of abelian variety
Setup. Let $p > 2$ be a prime, let $K$ be the completion of the maximal unramified extension of $\mathbb{Q}_p$, and fix an algebraic closure $\overline{K}$ of $K$. Let $A/K$ be an abelian variety ...
- 998
3
votes
1
answer
514
views
Field generated by Kontsevich-Zagier periods
What is the (transcendence) degree of the field $\mathbb{Q}^*$ generated by Kontsevich-Zagier periods over $\overline{\mathbb{Q}}$? What is its algebraic closure, $\overline{\mathbb{Q}^*}$?
What are ...
- 31
3
votes
1
answer
244
views
A local-to global principle for splitting of Azumaya algebras
Let $S$ be a finitely generated domain with the field of fractions $F.$ Let X be a smooth,
geometrically connected affine variety over $S.$ Let $A$ be an Azumaya algebra over $X.$
Assume that for all ...
- 33
9
votes
2
answers
518
views
Chevalley-Warning-Ax for double covers
Let $f(x_1,\ldots,x_n)$ be a polynomial of degree $d$ with coefficients in the finite field $\mathbb F_q$ and let $V(f)\subseteq\mathbb F_q^n$ be its set of zeroes. Assume $d<n$. Then Chevalley ...
- 15.5k
39
votes
6
answers
6k
views
Using algebraic geometry to understand class field theory
In Algebraic Number Theory, S. Lang says "[a geometrical approach] allows one to have a much clearer insight into the whole class field theory, since the existence theorem and
the reciprocity law ...
- 711
1
vote
4
answers
717
views
Given an integer $N$, find solutions to $X^3 + Y^3 + Z^3 - 3XYZ \equiv 1 \pmod{N}$
Given an integer $N > 0$ with unknown factorization, I would like to find nontrivial solutions $(X, Y, Z)$ to the congruence $X^3 + Y^3 + Z^3 - 3XYZ \equiv 1 \pmod{N}$. Is there any algorithmic way ...
- 1,703
4
votes
0
answers
289
views
Formal integration (?) in Chabauty’s method
In Mccallum, Poonen’s paper “The method of Chabauty and Coleman”,
the authors define, for the Jacobian $J$ of a geometrically connected smooth proper curve over the rational field and for $\omega \in ...
- 1,364
3
votes
1
answer
174
views
Historical context of rationality problem for algebraic torus
I have found that a lot of research has been done in rationality problem for algebraic tori. (For example, https://arxiv.org/abs/1210.4525). So I got to wonder what historical context or elementary ...
- 483
4
votes
1
answer
415
views
3-, 6-, 12-descent for Z2xZ6 elliptic curves
We are trying to write a snippet of Magma code to clarify the steps in the simplified procedure of applying $3$-, $6$-, $12$-descent and hopefully resolve the missing generator of the following $\...
- 913
2
votes
1
answer
303
views
Is the reduction of an absolutely irreducible plane curve still irreducible except for the finite number of cases?
Help me please.
Let $k$ be an algebraically closed field (I am mainly interested in $k = \overline{\mathbb{Q}}, \overline{\mathbb{F}_q}$). Consider a plane curve $C \subset \mathbb{A}^2$ of degree $d$ ...
- 1,833
7
votes
1
answer
421
views
Integral refinements of rigid cohomology
Disclaimer: I know absolutely nothing about p-adic cohomology, so it is possible that even the premises of this question are incorrect. But it turns out that I need to apply the theory of rigid ...
- 5,958
4
votes
0
answers
155
views
Is the unipotent section map of hyperbolic curve over local field injective?
Let $X/\mathbb{Z}_p$ be a smooth hyperbolic curve and $\pi^{un}_1(X_{\overline{\mathbb{Q}}_p},b)$ denotes the pro-unipotent completion (over $\mathbb{Q}_p$) of the etale fundamental group of $X$ base ...
- 123
5
votes
0
answers
264
views
Can arithmetic geometry accelerate the search for rational points in high dimensions?
There are several ideas in arithmetic geometry that can help in proving the absence of rational points as well exhibiting rational points on algebraic curves.
I am aware there are some obstructions (e....
11
votes
2
answers
679
views
Z/8Z elliptic curve with a missing generator
We are searching for the rank $6$ elliptic curves with the torsion subgroup $\mathbb{Z}/8\mathbb{Z}$ using the families similar to Allan MacLeod's as described in
A. J. MacLeod, A Simple Method for ...
- 913
1
vote
3
answers
383
views
What heuristic suggest for the number of solutions of $x^n+y^n=A$?
For integers $x,y,n,A$ with $n>1$ and $A>0$ we are interested
how many solutions $x^n+y^n=A$ has for fixed $n$ and infinitely many
$A$.
What is unconditionally known $n=2$ or $n=3$ the number of ...
- 25.4k
3
votes
1
answer
356
views
Understanding the implementation of the $p$-adic(?) sigma function in SageMath
I'm trying to understand the (pretty undocumented) method .sigma() method for formal groups of elliptic curves, as listed here. The source code looks like this:
<...
- 2,044
3
votes
1
answer
214
views
Distribution of the rank of $y^2=x^4+x+b^2$
For positive integer $b$ define the curve $C_b : y^2=x^4+x+b^2$.
$C_b$ is genus one and has the rational points: $(0,\pm b),(-1,\pm b)$
and one more point from the reciprocal of the polynomial y=0
...
- 25.4k
0
votes
0
answers
222
views
To show equivalence and full faithfulness of a functor PRESERVED under an action of a finite flat algebra
I have explained the two questions and then showed my effort on question $(1)$ as follows (Please at least check my effort below and suggest to make it perfect):
Let $R, S,T$ be three commutative ...
- 930
12
votes
0
answers
676
views
Kihara-like Z/6Z elliptic curve families
Shoichi Kihara constructed a family of elliptic curves with Mordell–Weil group $\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}^3$ (generic rank at least 3) in 2006. Kihara's family produces a number of rank 8 ...
- 913
2
votes
1
answer
187
views
Is every sufficiently general monic quartic rational square infinitely often?
Let $f(x)=x^4+b_3 x^3+ b_2 x^2+b_1 x + b_0$.
Let $g(x)=x^4 f(1/x)$. Let $C : g(x)=y^2$.
$C$ is birationally equivalent to $f(x)=y^2$.
The constant coefficient of $g(x)$ is $1$ since $f$ is monic
and $(...
- 25.4k
8
votes
2
answers
387
views
Milnor lattice and Du Val singularity
I am reading this paper: https://arxiv.org/abs/0810.2687 by A. J. de Jong and Robert Friedman. In the proof of Theorem 4.10, a singularity of the following type shows up $$y^2=x^3+z^{6d-1}.$$ When $d=...
- 409
2
votes
2
answers
541
views
A new simple formula is needed
The following question is related to the families of high rank elliptic curves with torsion subgroup $\mathbb{Z}/6\mathbb{Z}$.
The SageMath/Python code below produces a list of small fractions $a$ for ...
- 913
8
votes
6
answers
2k
views
How many solutions are there to the equation $x^2 + 3y^2 \equiv 1 \pmod{p}$?
Let $p$ be a prime. How many solutions $(x, y)$ are there to the equation $x^2 + 3y^2 \equiv 1 \pmod{p}$? Here $x, y \in \{0, 1, \ldots p-1\}$. This paper (https://arxiv.org/abs/1404.4214) seems like ...
- 1,703
0
votes
0
answers
77
views
On four non-cocyclic integral points on ellipse
Let $(x_i,y_i)\in\mathbb Z^2$ at $i\in\{1,2,3,4\}$ be four (not five and I assume an unique curve exists because of Diophantine constraints and not geometric constraints) non-cocyclic integral points ...
- 13.9k
19
votes
0
answers
1k
views
Mumford-Tate conjecture for mixed Tate motives
Let $X$ be a (not necessarily smooth or proper) variety over a number field $k$. Suppose we are given
A subquotient $V_{dR}$ of the algebraic de Rham cohomology $H_{dR}^i(X)$ (defined in the non-...
- 23k
9
votes
0
answers
607
views
Geometric meaning of twist
It is sometimes the case that a Galois representation or a motive acquires a desirable property only after a twist by a character, usually a Tate twist. The latest example of this I have come across ...
- 1,267
0
votes
0
answers
148
views
The scheme of intersection points of algebraic plane curves defined over a number field
This question concerns two related but different notions of "plane curves", where we consider both curves in the projective plane $\mathbb{P}^2$ and the affine plane $\mathbb{A}^2$.
Given ...
- 26.9k
14
votes
2
answers
571
views
Number of d-Calabi-Yau partitions
This problem arises from algebraic geometry/representation theory, see https://arxiv.org/pdf/1409.0668.pdf (chapter 2).
We call a partition $p=[p_1,...,p_n]$ with $2 \leq p_1 \leq p_2 \leq ... \leq ...
- 26.5k
5
votes
3
answers
448
views
Variation of centraliser in $\operatorname{GL}(n,\mathbb{Z})$
$\DeclareMathOperator\GL{GL}$Let $n$ be a positive integer $\geq 2$. The setting is that $K \in \GL(n,\mathbb{Z})$, and people are interested in understanding the centralizer:
$$
C(K)=\{ B \in \GL(n,\...
- 145
1
vote
0
answers
62
views
Polynomial sized arithmetic map from circle to ellipse preserving integral points
Let $n$ be a square free integer and a product of $O(m/\log m)$ number of primes $1\bmod 4$ where $m$ is $\log_2n$.
Take the circle around origin of radius $n^2$. It has ${\exp}(m/\log m)$ number of ...
- 13.9k