All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
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
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
3
votes
0
answers
151
views
Computing the group structure of $J(\mathbb{F}_q)$
Let $k$ be a finite field, $X/k$ a smooth curve, $f$ a polynomial of 2 variables which gives an affine model of $X$ and $J$ its Jacobian.
Then how can I compute $J(k)$?
If $X$ is a hyperelliptic curve,...
- 1,364
21
votes
3
answers
1k
views
Does X(13) have potentially good reduction at 13?
The complete level modular curve $X(p)$ does not have potentially good reduction at $p$ for any $p \neq 2,3,5,7,13$ because then there are cusp forms on $X_0(p)$ showing up in the cohomology of $X(p)$,...
- 148k
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....
37
votes
3
answers
5k
views
Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?
If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves ...
- 45.7k
6
votes
3
answers
2k
views
Is there any theorem like implicit function theorem in $\mathbb{Q}$ ?
My qeustion is that,
is there any theorem like implicit function theorem in $\mathbb{Q}$ ?
More precisely, let $p(\bar{x},\bar{y})$ be in $\mathbb{Z}[\bar{x},\bar{y}]$ such that in $\mathbb{Q}$, for ...
- 69
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
27
votes
3
answers
4k
views
Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,...)
Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many ...
- 32.2k
13
votes
2
answers
2k
views
What is the best reference for motives?
I want to learn about homotopy theory on number fields, and I heard that the theory of motives made it possible, so I want to know what is a good textbook for motive theory.
To be honest, I don’t ...
- 153
13
votes
0
answers
663
views
On a kind of Hilbert irreducibility theorem
Let us work over a number field $k$. Let $C$ be a non-empty open subscheme of $\mathbb{P}^{1}_{k}$, and $X\to C$ a family of smooth, projective hyperbolic curves such that $X(k)\to C(k)$ is surjective....
- 1,000
2
votes
0
answers
478
views
Is there bijective correspondence between $P_n$ and $A_n$?
Let $K \supset \mathbb{Q}_p$ be the $p$-adic field and let $O_K$ be its ring of integers and $M_K$ be the maximal ideal with integral closure $\bar{M}_K$. A power series is invertible if its lowest ...
- 930
7
votes
3
answers
594
views
Hyperelliptic modular curves in characteristic p
Ogg characterized the finitely many N such that $X_0(N)_{\mathbb{Q}}$ is hyperelliptic, and Poonen proved in "Gonality of modular curves in characteristic p" that for large enough N, $X_0(N)_{\mathbb{...
- 10.5k
12
votes
5
answers
2k
views
Clarification on the weak BSD conjecture
It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}
$$
as $x$ tends to ...
user85435
29
votes
3
answers
2k
views
$\zeta(n)$ as a mixed Tate motive
I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that
$M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$
and $\zeta(n)$, ...
- 291
14
votes
0
answers
358
views
How do we deduce the Jacquet-Langlands correspondence from Fargues' two towers?
In trying to understand the geometric proof of the local-Langlands and Jacquet-Langlands correspondence which uses Fargues's two tower theorem, I am having trouble finding a nice source on this, and I ...
- 3,539
18
votes
1
answer
2k
views
Conjecture: The number of points modulo $p$ of certain elliptic curve is $p$ or $p+2$ for $p$ of form $p=27a^2+27a+7$
Numerical evidence suggests a conjecture that the number of
points of certain elliptic curve over $\mathbb{F}_p$ is
either $p$ or $p+2$ for $p$ of certain form.
Let $p$ be prime of the form $p=27a^2+...
- 25.4k
2
votes
1
answer
270
views
Local to global principle for a pair of bilinear equations?
Let $A_{i, j}, B_{i, j}, C, D \in \mathbb{Q}$, and consider the following pair of equations
$$
A_{1, 1} x_1 y_1 + A_{1, 2} x_1 y_2 + A_{2, 1} x_2 y_1 + A_{2, 2} x_2 y_2 = C
$$
$$
B_{1, 1} x_1 y_1 + B_{...
- 3,625
8
votes
1
answer
519
views
Do $p$-adic topological modular forms exist?
Are there $p$-adic topological modular forms? What is the analogue of finite slope and overconvergent?
user139985
1
vote
0
answers
167
views
The existence of two $p$-isogenies implies the existence of one $p^2$-cyclic isogeny
Let $E$ be an elliptic curve over $\mathbb{Q}$.
(or over a number field $K$.)
If $E$ has two $p$-isogenies over $\mathbb{Q}$, then $E$ has $p^2$ cyclic isogeny over $\mathbb{Q}$.
I want to show it ...
- 185
1
vote
2
answers
471
views
Prime ideal of $A[X_1,...,X_d]$
Let $A$ be a UFD domain, i.e. integral and for any height one prime
${\frak p}$ of $A$, we have ${\frak p} = (u_{\frak p})$ for some $u_{\frak p} \in A$.
Once and for all, we fix the algebraic ...
- 781
8
votes
0
answers
400
views
The definition of Drinfeld modules
I have an embarrassingly basic confusion about the definition of Drinfeld modules. I think that the definition of a Drinfeld module over $S$ should be "a $\mathbb{G}_a$-torsor over $S$ and ...".
...
- 2,809
16
votes
3
answers
1k
views
Number of solutions to polynomial congruences
Suppose I have $R$ homogeneous polynomials $F_1, ..., F_R$ with integer coefficients. Let $V$ be the affine variety defined by these polynomials over $\mathbb{C}$. I was wondering if some bound that ...
- 3,625
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
9
votes
1
answer
353
views
The $S$-unit equation for functions on curves
Let $X$ be a smooth projective connected curve over a number field $k$, and let $S \neq \emptyset$ be a finite set of closed points of $X$. The curve $Y = X \setminus S$ is affine, and we denote by $R$...
- 20.8k
12
votes
0
answers
552
views
When does Matiyasevich's theorem "kick in"?
Hilbert's 10th problem was famously resolved by the Matiyasevich–Robinson–Davis–Putnam theorem: the theorem implies that there is no algorithm which decides whether a given polynomial equation with ...
- 26.9k
3
votes
2
answers
338
views
Isomorphism between finite algebras over ${\Bbb Z}_p$
Let $\pi \colon R \twoheadrightarrow {\Bbb T}$ be a surjective ring homomorphism between finite algebras over ${\Bbb Z}_p$. Further, we suppose the following three conditions$\colon$
$R$ is a ...
- 563
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
1
vote
0
answers
67
views
Curves covering elliptic curves with polynomially bounded genus
Let $S$ be the set of isomorphism classes of elliptic curves over $\mathbb{Q}$. Consider the following claim.
There is a map $f:S\to \mathbb{N}$ such that $|f^{-1}(n)|$ is finite for all $n\in \...
- 183
3
votes
2
answers
751
views
Witt vectors addition confusing
I raise this confusing because I try to understand the witt vectors for characteristic not equal to p.
Let us assume p=2. The Witt Polynomials is explicitly given by
$$
S_0=X_0+Y_0
$$
$$
S_1=X_1+...
- 397
3
votes
1
answer
220
views
Reference request: relationship between discriminant and smoothness of a conic over arbitrary fields
I'm looking for an (ideally modern) reference of the relationship between the discriminant and smoothness of projective conics over arbitrary fields (including those of characteristic 2). Let $k$ be a ...
- 7,527
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 \...
8
votes
3
answers
1k
views
Sufficient conditions for a polynomial to be reducible over the integers
There are several well-known criteria for a polynomial with integer coefficients to be irreducible over $\mathbb{Z}$, e.g., Eisenstein's criterion. I'm looking for the opposite: other than ...
- 1,703
2
votes
1
answer
206
views
Density of integral points on affine cubic surfaces of a certain type
Let $Q(x,y,z)$ be a cubic polynomial with integer coefficients, such that the terms $x^3, y^3, z^3$ do not appear. That is, it is at most quadratic in each of the variables $x,y,z$.
Is there a general ...
- 26.9k
16
votes
2
answers
4k
views
Elliptic Curves over Rings?
So an elliptic curve $E$ over a field $K$ is a smooth projective nonsingular curve of genus $1$ together with a point $O \in E$.
I was reading Silverman's "Arithmetic of Elliptic Curves" and it ...
- 1,458
11
votes
3
answers
1k
views
Why linearization leads to arithmetization?
Sorry for this question, but I think it is really important the intuition here.
Motives can be seen as the 'best' way of linearizing the study of schemes, des-composing them into "cohomological atoms"...
- 1,720
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
-3
votes
2
answers
608
views
Rational points on the elliptic curve $y^2 = x^{3} - t^{2}z^3$
What are the rational points on the elliptic curve $y^2 = x^3 - t^{2}z^3$ ? I seem not to find any besides the trivial ones whereby $txyz=0$ or $x= \pm z$.
ADDENDUM 1. I have just noticed that if $z^3 ...
- 1,019
4
votes
0
answers
553
views
Modern example of a reciprocity law and intuition behind it
I'm very new to the Langlands program and I was going through the Gauss reciprocity law, Hilbert's 9th problem, Artin's reciprocity law which allowed him to identify the Artin's L-functions with the ...
- 3,804
7
votes
1
answer
217
views
Subfields of Hilbertian fields
This question is about the Remark on the top of page 22 of Serre's Topics in Galois Theory, available here :
http://www.ms.uky.edu/~sohum/ma561/notes/workspace/books/serre_galois_theory.pdf
My ...
- 353
3
votes
0
answers
660
views
While solving the 1988 IMO problem 6, I have questions about new solutions without using Vieta Jumping [closed]
I think most of you may know the well-known problem:
"Let $x$ and $y$ be positive integers such that $xy + 1$ divides $x^{2} + y^{2}$. Show that $\frac {x^{2} + y^{2}}{xy + 1}$ is the perfect ...
- 39
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
43
votes
1
answer
19k
views
What is inter-universal geometry?
I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such ...
- 10.8k
20
votes
1
answer
1k
views
Curves over number fields with everywhere good reduction
My question is the following:$\newcommand{\Q}{\Bbb Q}
\newcommand{\Z}{\Bbb Z}$
What is known about number fields $K$ fulfilling the condition
$C_{g,K}$ "there is a smooth projective curve of ...
- 1,742
6
votes
2
answers
1k
views
Motivation for Hirzebruch-Jung Modified Euclidean Algorithm
Let $a,b \in \mathbb{N} \ \ s.t. \ \ a > b$ have $\gcd(a,b) =1$. We can define the Hirzebruch-Jung modified euclidean algorithm as follows:
Let $e_i \in \mathbb{N} >2$, and $ r_k \in \mathbb{N}$...
30
votes
1
answer
2k
views
Enriques surfaces over $\mathbb Z$
Does there exist a smooth proper morphism $E \to \operatorname{Spec} \mathbb Z$ whose fibers are Enriques surfaces?
By a theorem of, independently, Fontaine and Abrashkin, combined with the Enriques-...
- 148k
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
15
votes
2
answers
1k
views
Modular forms from counting points on algebraic varieties over a finite field
Suppose we are given some polynomial with integer coefficients, which we regard as carving out an affine variety $E$, for example:
$$ 3x^2y - 12 x^3y^5 + 27y^9 - 2 = 0 \tag{$*$} $$
(We might ...
- 1,821
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