All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
9
votes
0
answers
478
views
Abelian (possibly ramified covers) of elliptic curves
I'd like to understand the fields of definition of abelian covers of elliptic curves In particular, let $E$ be an elliptic curve over a field $K$ (say, a number field).
I'd like to understand the ...
- 7,527
6
votes
2
answers
576
views
Explicit formulas Belyi maps for a rational dessin d'enfants
What are the best references for finding explicit formulas for Belyi maps for rational dessin d'enfants?
I am most interested in a formula for the Belyi map that corresponds to a specific rational ...
3
votes
0
answers
260
views
Hilbert's irreducilibity theorem over function fields
I have a very naive question concerning Hilbert's irreducilibity theorem over function fields.
Let $f: V \to \mathbb{A}^2_\mathbb{Q}$ be a finite morphism, with $V$ an irreducible variety over $\...
- 21.3k
7
votes
2
answers
467
views
multiple zeta values and knots invariants
I have heard several times that MZV appear in the context of knot invariants and deformation quantisation. Could anyone explain how and give some references?
- 71
7
votes
0
answers
483
views
independence of $\ell$ for $p$-adic cohomology of varieties over finite fields
Let $X/k$ be a smooth projective geometrically integral variety ($X = A$ an Abelian variety suffices) over $k = \mathbf{F}_q$ with absolute Galois group $\Gamma$, $\bar{X} = X \times_k \bar{k}$, $q = ...
user19475
16
votes
0
answers
591
views
Are there any examples of hyperbolic curves over finite fields such that the action of frobenius on its prime-to-$p$ fundamental group is known?
Let $X$ smooth curve over a finite field $\mathbb{F}_q$ of type $(g,n)$ - that is, $X$ is an open subscheme of its genus $g$ compactification obtained by removing $n$ points.
Any such curve ...
- 10.7k
-3
votes
1
answer
964
views
On the maximal ideal m of the formal power series ring [closed]
Let $A \colon= K[[X_1,X_2,\ldots,X_{\infty}]]$ be the formal power series ring with infinitely many variables over a field $K$. We can represent it also by the following manner$\colon$
\begin{equation*...
- 781
2
votes
0
answers
281
views
Galois cohomology of cyclotomic extension
Let $K$ be a complete discrete valuation ring with algebraically closed residue field $F$ of characteristic $p > 0$. Suppose ${\Bbb Q}_p \subset K$ and the absolute ramification index v$_{\pi_K}(p) ...
- 781
17
votes
1
answer
5k
views
Algebraic Geometry in Number Theory
It appears to me that there are two main ways by which algebraic geometry is applied to number theory. The first is by studying polynomials over fields of number-theoretic interest (which does not ...
- 3,309
4
votes
0
answers
392
views
On nearby cycle sheaves and a 2-fibered product of topoi
In SGA7 Exposé XIII, Deligne introduces an algebraic theory of nearby cycle sheaves and vanishing cycle sheaves on schemes over the $S$, where $S$ is the spectrum of a Henselian discrete valuation ...
- 181
4
votes
1
answer
649
views
On Ramanujan's beautiful cubic identity
Let $a_i, b_i, c_i$ be defined by the following$\colon$
$\frac{1 + 53X + 9X^2}{1 - 82X - 82X^2 + X^3} = a_0 + a_1X + \ldots$.
$\frac{2 - 26X - 12X^2}{1 - 82X - 82X^2 + X^3} = b_0 + b_1X + \ldots$.
...
- 781
4
votes
0
answers
279
views
On De Shalit's Lemma in Wiles' proof of R=T
In Wiles' proof of ``$R = {\Bbb T}$", if we associate the extra primes the set of which is denoted $D$, there is a famous result in the following$\colon$
Theorem(De Shalit's Lemma). Let ${\Bbb T}_N$ ...
- 781
11
votes
1
answer
478
views
Uniform Faltings [duplicate]
Suppose I give you positive integers $g\geq 2$ and $N.$ Is it always possible to find an absolutely irreducible curve of genus $g$ over $\mathbb{Q}$ which has at least $N$ rational points? For that ...
- 96.4k
12
votes
1
answer
563
views
reference request: rational points on the unit sphere
I wonder what would be a good/early reference for the fact:
rational points on the unit sphere (centered at the origin) are dense.
Stereographic projection (from a rational point in the sphere) ...
- 10.7k
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
3
votes
0
answers
297
views
Flatness of R[X]/I over R
In the famous paper Simple flat extensions, Journal of Algebra Volume 16, Issue 1, September 1970, p. 105-107, Wolmer V. Vasconcelos proves the following
Theorem (Vasconcelos). For a noetherian ...
- 781
75
votes
5
answers
3k
views
When the automorphism group of an object determines the object
Let me start with three examples to illustrate my question (probably vague; I apologize in advance).
$\mathbf{Man}$, the category of closed (compact without boundary) topological manifold. For any $M,...
2
votes
0
answers
98
views
Value distribution of polynomials
Take a polynomial $p \in \mathbb{Z}[x_1, \dotsc, x_k],$ and consider the values of $p(a_1, \dotsc, a_k)$ as $\mathbf{a}$ (the vector of the $a_i$) ranges over, say, the integer lattice points in the ...
- 96.4k
4
votes
1
answer
282
views
Serre tensor construction on finite flat group schemes
Let $K/\mathbb{Q}_p$ be a finite field extension and $\mathcal{O}_K\subseteq K$ be its ring of integral elements. Let also $G/\mathcal{O}_K$ be a finite flat $p$-group scheme that is also an $\mathcal{...
- 443
55
votes
3
answers
11k
views
What is precisely still missing in Connes' approach to RH?
I have read Connes' survey article http://www.alainconnes.org/docs/rhfinal.pdf
and I am somewhat familiar with his classic paper on the trace formula: http://www.alainconnes.org/docs/selecta.ps
Very ...
- 1,338
4
votes
2
answers
696
views
Rank of certain elliptic curves
I need to calculate rank of the some
elliptic curves,(espicially getting generators or finding a rational point
on the elliptic curves) but I cannot do this by my computer.
I am interested in ...
31
votes
2
answers
1k
views
The Sylvester-Gallai theorem over $p$-adic fields
The famous Sylvester-Gallai theorem states that for any finite set $X$ of points in the plane $\mathbf{R}^2$, not all on a line, there is a line passing through exactly two points of $X$.
What ...
- 20.8k
4
votes
1
answer
621
views
finitness of syntomic/fppf cohomology with coefficients in a finite flat group scheme
Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathscr{A}/X$ be an Abelian scheme.
Is then $H^1_\mathrm{SYN}(X,\mathscr{A}[p]) = H^1_\mathrm{fppf}(X,\mathscr{...
user19475
5
votes
1
answer
824
views
Understanding Siegel's Theorem on integral points
Siegel's theorem states the following:
Let $C$ be a smooth projective curve over a number field $K$. Let $\tilde C\subset C$ be an open affine subvariety, and $i:\tilde C\hookrightarrow \mathbb{A}^...
- 2,071
5
votes
1
answer
541
views
Surjectivity of map between Néron models $\mathcal{E} \to \mathcal{E}'$
My question is related to a previous question on the Mordell-Weil rank of the elliptic curve $E/\mathbf{Q} : y^2 = x^3- 2$ asked here. More precisely, I want to understand the following. Let $E'/\...
- 2,607
2
votes
0
answers
364
views
Find all rational points on the hyperelliptic curve $y^2 = x^5 - 6x^3 + \frac{2}{9} x^2 -3x$
I'm trying to classify all of the rational points on the hyperelliptic curve $y^2 = x^5 - 6x^3 + \frac{2}{9} x^2 -3x$.
This is a genus 2 curves and MAGMA gives a RankBound on this curve's Jacobian of ...
- 21
6
votes
0
answers
429
views
Euler Bricks in High Dimensions
It is a well-known and open problem to determine whether there exists a rectangular cuboid where the distance from any corner to any other corner is an integer. Such a beast, if it exists, is called ...
- 18.7k
7
votes
1
answer
794
views
Special fiber of the Néron model of the generalized Jacobian of a singular curve
Let $C$ be a curve over $\mathbf{Q}_p$ (or a finite extension) whose minimal regular model $\mathcal{C}$ over $\mathbf{Z}_p$ has a "nice" special fiber (maybe singular, with at most ordinary double ...
- 1,109
4
votes
1
answer
579
views
Find all rational solutions of $x^2(x+1)(x^2+1)(x-1)=2(y+1)(y-1)$
Find all rational solutions of
$$x^2(x+1)(x^2+1)(x-1)=2(y+1)(y-1).$$
Clearly the following six solutions hold:
$$(x,y)=(1,1),(-1,-1),(-1,1),(1,-1),(0,1),(0,-1)$$
But how to find all rational ...
- 4,280
14
votes
1
answer
1k
views
A quantitative version of Hensel's Lemma
I've been reading some papers on Igusa zeta functions, and they seem to be implicitly using a "quantitative version" of Hensel's Lemma, which also asserts the number of lifts of a $\mathbb{Z}/p\mathbb{...
- 21.3k
5
votes
2
answers
469
views
Enumerate rational points on the unit sphere with bounded height
This might be an easy one. Motivated by this answer. I would like rational pointts on the unit sphere of bounded height:
$$ \{ (x,y,z)\in \mathbb{Q}^3: x^2 + y^2 + z^2 = 1 \}$$
This seems ...
- 22.8k
3
votes
1
answer
1k
views
Is the Tate-Shafarevich group of a rational elliptic curve finite?
It seems that Lan Nguyen proved in a preprint on arxiv of 2013 that the Tate-Shafarevich group of a rational elliptic curve is finite. However, I couldn't find any published version thereof. So is it ...
- 6,984
11
votes
1
answer
385
views
Is there a presentation to the kernel of the prime-to-$p$ fundamental short exact sequence of curves over finite fields?
Let $X$ be $\mathbb{P}^1_{\mathbb{F}_q}\smallsetminus \{a_1,...,a_r\}$, where $a_1,...,a_r$ are some $\mathbb{F}_q$-rational points. Let $\bar X :=X_{\bar{\mathbb{F}}_q}$. There is a short exact ...
- 2,071
5
votes
0
answers
676
views
Basic question on p-adic Hodge theory
I am starting to study the rudiments of p-adic Hodge theory and I have the following basic question. Let $\chi$ be the unramified quadratic character of $G_p = \mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{...
- 111
18
votes
1
answer
1k
views
On Tate's "Endomorphisms of Abelian Varieties over Finite Fields", sketch of proof of main result?
Let $k$ be a field, $\overline{k}$ its algebraic closure, and $A$ an abelian variety defined over $k$, of dimension $g$. For each integer $m \ge 1$, let $A_m$ denote the group of elements $a \in A(\...
user97565
14
votes
1
answer
2k
views
Reduction mod $p$ map is injective?
Let $E$ be an elliptic curve over $\mathbb{Q}$ with good or multiplicative reduction at $p$. How do I see that the reduction mod $p$ map from the endomorphisms of $E$ over $\overline{\mathbb{Q}}$ to ...
- 141
6
votes
1
answer
362
views
Infinitely many primes $p \in \mathbb{Z}$ where reduced curve $E/\mathbb{F}_p$ has Hasse invariant $1$?
Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and fix a Weierstrass equation for $E$ having coefficients in $\mathbb{Z}$. Are there infinitely many primes $p \in \mathbb{Z}$ such that the ...
- 211
3
votes
0
answers
128
views
On Abhyankar's results cited in a paper of Manin titled "Correspondences, Motifs and Monoidal Transformations"
Consider the following from this paper "Correspondences, Motifs and Monoidal Transformations" of Manin here.
Theorem. Nonsingular three-dimensional projective unirational varieties $V$ over ...
- 113
0
votes
0
answers
197
views
'Adelic torus' not arising from a rational torus
Let $G$ be a reductive group over a global field $F$, and $\gamma$ a strongly regular semi-simple element of $G(F)$. Then the centralizer $G_\gamma$ is defines an $F$-torus $T$, and hence by base ...
- 3,799
0
votes
2
answers
211
views
Can one reconstruct a diagonalisable endomorphism from its action on the exterior algebra?
Let $K$ be a field. Let $V$ be a finite-dimensional $K$-vector space; and let $f$ be an automorphism of $V$. Assume that $f$ is diagonalisable over some extension of $K$. Form the exterior algebra $\...
6
votes
0
answers
369
views
Cohomology and Riemann-Roch in Number Theory (Neukirch Chapter 3)
In chapter 3 of Neukirch's Algebraic Number Theory, an analogue of the classical Riemann-Roch theorem is developed for number fields. To achieve this, notation suggestive of cohomology with sheaves of ...
- 3,309
5
votes
1
answer
797
views
Euler characteristic of local system depends only on rank?
Let $X$ be a proper variety over a finite field $k$ of characteristic $p>0$, and let $\mathcal F$ be a finite rank $\mathbb F_\ell$ local system on (the etale site of) $X$. Is it true (and, if so, ...
- 18.7k
6
votes
2
answers
336
views
Modular parametrization abelian varieties
Let $N$ be a positive integer, and let $f$ be a newform for $S_2(\Gamma_0(N))$. Then by Shimura's construction, the variety $J_0(N)$ has a quotient $A_f$ which is an abelian variety attaced to $f$.
$$\...
- 619
0
votes
1
answer
147
views
Closure of non-closed subset in Ring theory
We say that the ring $R$ is topological, when we are given neighbourhoods $\{O_{\lambda};\lambda \in \Lambda\}$ of $0_R$.
We say ${\cal I}$ is a closed ideal of $R$ if and only if for any sub-index ...
- 781
10
votes
1
answer
845
views
How to compute the formal group law of a Shimura variety (using its invariant differentials)?
I have a 3 dimensional abelian variety whose formal group law breaks into a formal summand where one of the pieces is one-dimensional.
I am desperately wondering how to compute the $p$-series of ...
- 3,539
2
votes
2
answers
615
views
twists of elliptic curves over finite fields
Let $p\ge 5$ be prime. For every $j\in\mathbb{F}_p$, there are at most 6 twists of any elliptic curve over $\mathbb{F}_p$ with $j$-invariant $j$, and in general only two twists.
Is there a formula ...
- 7,527
4
votes
1
answer
322
views
Albanese of Siegel modular variety $\mathcal{A}_2$
Jacobian varieties of Shimura curves are very interesting objects. For one thing they provide a geometric relation between elliptic curves and modular forms of weight 2 (say we are over $\mathbb{Q}$). ...
- 845
6
votes
2
answers
2k
views
Sketch of Weil's proof of the Riemann hypothesis for curves
I was wondering if anybody could provide a sketch of Weil's proof of the Riemann hypothesis for curves that uses the Jacobian $\text{Pic}^0(X)$ and a bit of the intersection theory on $X \times X$ and ...
user97565
2
votes
1
answer
466
views
Do $PGL_n$-torsors induce elements of the Brauer group
Let $K$ be a field and let $n\geq 2$. If $n=2$, then the set of $K$-isomorphism classes of $PGL_n$-torsors is in bijection with the $n$-torsion of the Brauer group of $K$.
Is this only for $n=2$?
Is ...
- 23
7
votes
1
answer
428
views
Reference result: proof of theorem of Kazhdan-Margulis on monodromy group of a Lefschetz pencil of odd fiber dimenion is "as big as possible"
In Deligne's paper on his first proof of the Weil conjectures, we have the following result.
Theorem 5.10 (Kazhdan-Margulis). L'image de $\rho: \pi_1(U, u) \to \text{Sp}(E/(E \cap E^\perp), \psi)$ ...
user97565