All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
2
votes
0
answers
214
views
Number of lines on a weak del Pezzo surface
By a line I mean a (-1)-curve. Given a weak del Pezzo surface $X$ of degree $d$, how many lines would $X$ contain?
- 481
10
votes
1
answer
602
views
Example of a $p$-divisible group that is not representable by a formal scheme
Let $R$ be a ring such that $p^nR=0$ for some integer $n$, and $G$ be a $p$-divisible group over $R$.
We think of a $p$-divisible groups as an fppf sheaf $G\colon \mathrm{Alg}^{op}_{R}\to \mathbf{Gps}$...
- 2,923
12
votes
1
answer
534
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
2
votes
1
answer
474
views
Chevalley–Warning theorem for rational field $\mathbb{Q} $
At 1st we consider some weak statement of Chevalley–Warning theorem for any finite field: If $f$ is a homogeneous polynomial of degree $d$ with $n$ independent variables over a finite field $F$. Then ...
- 923
1
vote
0
answers
96
views
Locally symmetric spaces dependence on number field
A special case of locally symmetric spaces is the moduli space of abelian varieties of a given dimension $g$ (over a given base field $k$), lets call it $\mathcal{A}_g$ and is a $k$-scheme. For any ...
- 4,418
28
votes
2
answers
2k
views
When did people start thinking of elliptic curves as groups?
I have been reading some old papers of Cassels and Selmer from around 1950, and they talk about generators of rational solutions to elliptic curves, in the sense of Mordell–Weil, but do not appear to ...
- 6,039
2
votes
0
answers
149
views
Singular fiber of a family of elliptic curves with Kodaira's symbol mI_V
From https://en.wikipedia.org/wiki/Elliptic_surface, one classifies all fibers of a minimal elliptic fibration via Kodaira's symbols:
$I_v$, $mI_v$, $II$, $III$, $IV$, $I_v^*$, $II^*$, $III^*$, $IV^*$
...
- 429
2
votes
2
answers
399
views
What fraction of polynomials with integer coefficients are indecomposable?
It is well-known that "most" integers are composite: the Prime Number Theorem tells us that only about $1/\log(N)$ of the integers in the interval $1 \ldots N$ are prime. For polynomials, ...
- 1,703
2
votes
0
answers
242
views
Solving $x^k+y^k+z^k=w^k$ non-trivially in strictly positive integers
Consider the equation $x^k+y^k+z^k=w^k$ in $x$, $y$, $z$ and $w$ with $k\in\mathbb{N}_{\geq2}$.
If we look for solutions that are strictly positive and non-trivial i.e. $x\neq-y$, $x\neq w$ etc... ...
- 4,862
26
votes
3
answers
3k
views
Crux of Dwork's proof of rationality of the zeta function?
As the question title suggests, what is the crux of Dwork's proof of the rationality of the zeta function? What is the intuition behind the proof, what are the key steps that the proof boils down to?
user97565
55
votes
3
answers
5k
views
What are the higher homotopy groups of Spec Z ?
The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π1 is trivial because Spec Z has no unramified étale covers,...
- 8,004
13
votes
3
answers
1k
views
Faltings height in short exact sequences
Let $K$ be a number field and $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ a short exact sequence of abelian varieties over $K$. Let $h(A)$ denote the logarithmic Faltings height (...
- 1,103
4
votes
2
answers
447
views
Topology of the real points of Shimura varieties
Snowden has studied the topology of the real points of modular curves. Are there analogous results for other Shimura varieties defined over $\mathbb{R}$?
- 81
11
votes
1
answer
1k
views
Generating functions of Collatz iterates?
Let $C(n) = n/2$ if $n$ is even and $3n+1$ otherwise be the Collatz function.
We look at the generating function $f_n(x) = \sum_{k=0}^\infty C^{(k)}(n)x^k$ of the iterates of the Collatz function.
The ...
- 3,064
14
votes
3
answers
3k
views
Quadratic reciprocity and Weil reciprocity theorem
I was told that Weil reciprocity theorem (one has two meromorphic function $f,g$ on a complex curve $C$, so $\prod\limits_{x\in C} g(x)^{ord_xf}=\prod\limits_{x\in C}f(x)^{ord_xg} \ $ where $ord_xf$ ...
- 5,063
34
votes
3
answers
3k
views
irreducibility of discriminant
This must be well-known to everyone but me, but here goes: take a general (monic) polynomial $p(x) = x^d + a_{d-1} x^{d-1} + \dotsc + a_0.$ The discriminant is a polynomial $D(a_0, \dotsc, a_{d-1}).$ ...
- 96.4k
14
votes
3
answers
4k
views
Recent progress toward Birch and Swinnerton-Dyer conjecture
Has there been any progress toward the Birch and Swinnerton-Dyer conjecture after
The current status of the Birch & Swinnerton-Dyer Conjecture
- 141
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
26
votes
1
answer
959
views
What automorphic forms are expected to occur in the zeta function of moduli space of curves?
Assume $g \geq 1$ and $n \geq 0$, the moduli stack ${\mathcal {M}}_{g,n}$ classifies families of smooth projective curves of genus $g$ with $n$ marked points , together with their isomorphisms. It has ...
- 6,622
15
votes
2
answers
2k
views
sum of three cubes and parametric solutions
The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$, i.e., there are finite many polynomial triples $(x(t),y(t),z(...
- 3,337
5
votes
0
answers
206
views
$\operatorname{GL}_2$-type abelian varieties with full level-$p$ structure
For a prime $p$, does there exist a $\operatorname{GL}_2$-type abelian variety $A$ over $\mathbb{Q}$ having full level-$p$ structure? That is, $A[p]=(\mathbb{Z}/p)^d \oplus \mu_{p}^{d}$ as a $G_{\...
- 51
5
votes
0
answers
278
views
Torsion points of an elliptic curve over number fields. Another proof of Silverman AEC theorem
I am studying the following theorem from Silverman's AEC:
I am wondering whether there exists another proof that doesn't make use of formal groups and is still valid for a number field $K$. Could you ...
- 233
21
votes
0
answers
2k
views
Recent developments in the proof of Fermat's last theorem
I posted on Mathematics Stack Exchange, but was encouraged to post on MathOverFlow instead.
It has been 20 years since Fermat's last theorem was proved by Andrew Wiles.
Has there been any ...
- 311
2
votes
0
answers
134
views
Differential equation of Van Gorder type for zeta of global fields, or: Does the zeta function of a global field satisfy a differential equation?
Let
\begin{equation*}
\zeta(s):=\prod_{p\text{ prime}}\frac{1}{1-p^{-s}}
\end{equation*}
be the Riemann zeta function. Van Gorder has shown that $\zeta$ satisfies a differential equation
\begin{...
- 1,805
14
votes
1
answer
1k
views
If it quacks like an abelian variety over a finite field
Consider smooth projective varieties over a finite field. If a curve "looks like" an elliptic curve (i.e. has genus $1$) then it can be made into an elliptic curve.
Is there something ...
- 117
4
votes
1
answer
363
views
The numbers of isomorphism classes of abelian variety over finite fields
It is known that there are only finitely many isomorphism classes of abelian variety over a finite field. I am curious about the exact number of these isomorphism classes.
Explicitly, fix $g$, let $\...
- 547
20
votes
5
answers
2k
views
Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$
As a generalisation to the equation of Fermat, one can ask for rational solutions of $X^n+Y^n+Z^n=1$ (or almost equivalently integer solutions of $X^n+Y^n+Z^n=T^n$).
Contrary to the case of Fermat, ...
- 7,722
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
3
votes
1
answer
395
views
Finding $K$-rational points on $X_0(35)$
Let $K=\mathbb{Q}(\sqrt{-2})$. How can I compute the $K$-rational points on the modular curve $X_0(35)$?
Recall that $X_0(35)$ is a hyperelliptic curve of genus 3 and has the simplified affine model:
\...
- 31
0
votes
0
answers
125
views
Does an isogeny between tori induce an isomorphism of the Lie algebras of their lft Néron models?
Let $f:T_1 \to T_2$ be an isogeny of tori over a number field $K$. Does $f$ induce an isomorphism of the Lie algebras of the lft Néron models of $T_1$ and $T_2$ ? Are there some interesting properties ...
- 617
35
votes
4
answers
8k
views
What would a "moral" proof of the Weil Conjectures require?
At the very end of this 2006 interview (rm), Kontsevich says
"...many great theorems are originally proven but I think the proofs are not, kind of, "morally right." There should be better proofs......
- 1,764
1
vote
0
answers
261
views
Integer points on genus 1 curves using CAS
How can I practically find integer points on genus 1 curves with small coefficients using computer algebra systems (CAS), like Mathematica, Maple, SageMath, Magma, etc.?
As a specific example, do ...
- 7,182
4
votes
0
answers
219
views
Generate periods only by smooth varieties
Like explained in this passage that a period is a complex number whose real and imaginary parts are integrations of rational functions over $\mathbb{Q}$ on some $\mathbb{Q}$-semi-algebra set in $\...
- 275
6
votes
3
answers
605
views
Natural number solutions for equations of the form $\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}$
Consider the equation $$\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}.$$
Of course, there are solutions to this like $(a,b,c) = (9,8,6)$.
Is there any known approximation for the ...
- 749
9
votes
0
answers
380
views
How can I "see" that a map is birational?
This came up with the Euler brick.
Let $T=(p,q,r)$ be a Randall triple, i.e. $$(p^2-1)(q^2-1)(r^2-1)=8pqr\ \qquad\text{[eq.1]}.$$ There are tons of maps that map a triple $T$ to another $T'=(p',q',r')$...
- 4,829
2
votes
0
answers
278
views
Rational points on surfaces
Let $k$ be a field of characteristic zero. In the affine space $\mathbb{A}_{x,y,t}^3$ consider a surface $S$ of the form
$$
S = \{a_0(t)x^2+a_1(t)xy+a_2(t)x+a_3(t)y^2+a_4(t)y+a_5(t) = 0\}
$$
where $...
user168611
2
votes
0
answers
124
views
About finite dimensionality of Chow groups of zero cycles
Let $S$ be a connected smooth complex projective surface.
Let $Sym^{d}(S)$, $d\in \mathbb{Z}^+_0$, be the $d$-th symmetric product of $S$ parametrizing $0$-cycles of degree $d$.
Let $Sym^{d,d}(S)=...
- 519
19
votes
3
answers
2k
views
Bhargava's work on the BSD conjecture
How much would Bhargava's results on BSD improve if finiteness of the Tate-Shafarevich group, or at least its $\ell$-primary torsion for every $\ell$, was known? Would they improve to the point of ...
user92332
6
votes
0
answers
196
views
Weyl group and Galois action on cubic surfaces
Let $X$ be a smooth cubic surface over a field $k$. Denote by $\bar{k}$ the separable closure of $k$ and $\bar{X}:=X\times_{k}\bar{k}$. Then it is well know that there exists a homomorphism
$$
\phi:\...
- 481
7
votes
0
answers
307
views
Number of rational points over finite fields mod $q$ is birational invariant
I heard that if $\mathbf F_q$ is a finite field, $X, Y$ are birational smooth proper variety over $\mathbf F_q$, then $\#(X(\mathbf F_q)) \equiv \#(Y(\mathbf F_q)) \pmod q$, and I heard that the proof ...
- 988
19
votes
3
answers
1k
views
Points of elliptic curves over cyclotomic extensions
Let $E$ be an elliptic curve over $\mathbb Q$. Let's look at the group of points of this elliptic curve over $\mathbb Q(1^{1/\infty})$ which we get after adding all roots of unity to $\mathbb Q$. It ...
- 2,305
0
votes
1
answer
131
views
Primes of the form $p=u^2+1$ and number of points on the elliptic curve $x^3+a x z^2=y^2 z$
Let $p$ be prime of the form $p=u^2+1$. For $a \in \mathbb{F}_p,a \ne 0$,
define
$E_a : x^3+a x z^2=y^2 z$
Let $B= \lfloor 2 \sqrt{p}\rfloor$
Must we have $(\#E_a(\mathbb{F}_p) -p - 1) \in \{2,-2,B,-B\...
- 25.4k
12
votes
0
answers
676
views
Kihara-like Z/6Z elliptic curve families
Shoichi Kihara constructed a family of elliptic curves with Mordell–Weil group $\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}^3$ (generic rank at least 3) in 2006. Kihara's family produces a number of rank 8 ...
- 913
8
votes
2
answers
387
views
Milnor lattice and Du Val singularity
I am reading this paper: https://arxiv.org/abs/0810.2687 by A. J. de Jong and Robert Friedman. In the proof of Theorem 4.10, a singularity of the following type shows up $$y^2=x^3+z^{6d-1}.$$ When $d=...
- 409
14
votes
0
answers
821
views
What goes wrong with this alternate proof of Dirichlet's Theorem?
I had an idea for an alternate proof of Dirichlet's theorem, but something goes wrong. Dirichlet's theorem on primes in arithmetic progression says that for $ m,a \in \mathbb{N} $ which are ...
- 782
3
votes
0
answers
530
views
The cycle class map with values in crystalline cohomology
Let $ k = \mathbb{F}_q $ be a finite field of characteristic $ p > 0 $.
Let $ X $ be a smooth proper scheme of dimension $ d $ over $ k $.
Consider the associated $ K $ - linear cycle class map ...
- 595
28
votes
1
answer
4k
views
Arithmetic Morse theory?
Is there any analogue of Morse theory in Number theory? Naive idea arising in my head is that defining a Morse function on scheme and find etale cohomology using that function. Since I'm not an expert ...
- 2,215
2
votes
1
answer
215
views
The growth of the number of Fano complete intersection families
I recently calculated the number (possible multidegrees) of Fano complete intersections of dimension $n$ , because I wanted to make the remark that it grows "very rapidly" as $n \rightarrow \...
- 6,995
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
10
votes
1
answer
459
views
Why are root data a natural candidate for classifying connected reductive groups?
For the purpose of this question, you may assume that we are working over the complex numbers.
Given a connected reductive group $G$, one can choose a maximal torus $T$, and then let $T$ act on the ...
- 557