All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
5
votes
1
answer
220
views
The generic fiber pullback for $p$-divisible groups in characteristic $p$
Let $R$ be a discrete valuation ring with the field of fractions $K$ and the residue characteristic $p$. If $K$ is of characteristic $0$, then a celebrated theorem of Tate says that the pullback ...
- 2,663
10
votes
1
answer
719
views
what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)?
Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent ...
- 10.7k
3
votes
1
answer
193
views
Pathological behavior of Lie algebra under a map of abelian schemes
I am trying to understand Example 7.5/9 from the book "Neron models". There one has a discrete valuation ring $R'$ that is the localization of $\mathbb{Z}[\zeta_p]$ at $p$, so that the absolute ...
- 1,103
13
votes
1
answer
771
views
Abelian $\ell$-adic representations in $\widehat{\mathrm{SL}(2,\mathbb{Z})}$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Gal{Gal}\newcommand{\Z}{\mathbb{Z}}$In Grothendieck's Esquisse he claims that the action of
$$\Gal(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\...
- 3,799
8
votes
1
answer
451
views
p-adic Stein spaces
The higher cohomology of coherent sheaves vanish on Stein spaces (both complex and p-adic). In the case when the space ($X$) is a curve and we're working in the complex world, this shows that all ...
- 83
8
votes
2
answers
657
views
Adjoining torsion points from abelian varieties
Let $L/\mathbb{Q}$ be the field generated over $\mathbb{Q}$ by all of the (projective) coordinates of all of the torsion points of all abelian varieties defined over $\mathbb{Q}$. Is $L$ algebraically ...
- 11.3k
4
votes
0
answers
166
views
Is this $S$-birational map an open immersion on its domain of definition?
My question is about a claim on the bottom of p. 121 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud, so I will freely use the general terminology recalled in this book, but will ...
- 1,103
48
votes
4
answers
4k
views
Fermat's last theorem over larger fields
Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here $\...
- 11.3k
0
votes
1
answer
285
views
The number of solutions of a Diophantine equation [closed]
Is $\lim_{n \rightarrow \infty} |\{(x,y) \in \mathbb{Q}(\zeta_n)^2 : y^3 = x^3 + x + 1\}| < \infty ?$ where $\zeta_n$ is a primitive $n$-th root of unity.
That is, I am asking whether the number ...
- 11.3k
17
votes
0
answers
1k
views
What is the expected dimension of the Zariski closure of the rational points on the moduli space of curves?
For each genus $g$, there are many curves of genus $g$ defined over $\mathbb Q$. How many? We might study this question by considering the rational points of the Deligne-Mumford moduli space of curves ...
- 149k
5
votes
1
answer
517
views
Disjoint images of polynomials
Are there any $f,g \in \mathbb{Q}[x]$ such that for every root of unity $\zeta$, and every $a,b \in \mathbb{Q}(\zeta)$, $f(a) \neq g(b)?$
- 11.3k
5
votes
0
answers
194
views
How to check whether a scheme of finite type over Spec Z is regular or not [duplicate]
Let $f_1, f_2, \ldots f_k$ be a set of polynomials in $n$ variables, with integer coefficients. These define an affine scheme $X$ of finite type over $Spec \mathbb{Z}$. (We could also consider ...
- 5,637
4
votes
0
answers
196
views
A Diophantine equation revisited
No integer solution of this Diophantine equation $$x^4+y^4+1=z^2$$ is known other than the trivial ones.
While I was reading a paper of Don Zagier, I realized that his idea on the Euler's sum of ...
- 3,337
17
votes
2
answers
1k
views
Images of polynomials
Let $f,g \in \mathbb{Q}[x]$ be polynomials such that $\{f(a) : a \in \mathbb{Q}\} \subseteq \{g(a) : a \in \mathbb{Q} \}$. Must there be some $h \in \mathbb{Q}[x]$ such that $f(x) = g(h(x))$ for all $...
- 11.3k
2
votes
1
answer
162
views
DL-problem on abelian variety
Let $A$ be an abelian variety over $\mathbb{F_q}$ with dimension $n$. Let $q$ be a constant.
Is there polynomial algorithm of finding discrete logarithm in $A$?
UPD: really I don't undestend: can we ...
- 2,576
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
17
votes
1
answer
745
views
Special fiber of $X(p)$ in characteristic $p$
Let $p \geq 5$ be a prime. Let $Y(p)$ be the fine moduli space representing elliptic curves + basis of the $p$-torsion over $\mathbb{Q}_p$ and let $Y_0(p)$ be the fine moduli space representing ...
- 1,109
5
votes
1
answer
354
views
Is there a simple proof that Milnor $K_2$ of a number field is torsion?
This is a theorem of Garland. I had a look at the original paper which looks pretty complicated. I was wondering if the proof has been simplified over the years or if a different approach is nowadays ...
- 71
3
votes
1
answer
664
views
Question on effective Mordell conjecture
Suppose $F(x,y,z)$ is a homogeneous polynomial over $\mathbb{Q}$, where $C:F(x,y,z)=0$ is a curve of genus $g\geq 2$.
Question: Faltings proved that $C$ has finite many rational points. Suppose that ...
- 3,337
13
votes
3
answers
3k
views
$j$-invariants of elliptic curves over finite fields
Let $K$ be a finite field, and $\overline{K}$ its algebraic closure. It is well known that two curves are isomorphic over $\overline{K}$ if and only if they have the same $j$-invariant. If two such ...
- 647
3
votes
0
answers
168
views
Invariant Theory over finite adeles
Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$.
I am ...
- 2,384
1
vote
0
answers
148
views
Clarifications on twisted forms
Suppose $F = F(\bar{k})$ is a finite algebraic group over a number field $k$. The absolute Galois group $\Gamma_k$ of $k$ acts on $F$ by group automorphisms via a homomorphism $\rho: \Gamma_k \to S_n$,...
- 11
1
vote
2
answers
247
views
Equidistribution of rational points on an algebraic variety
Suppose that we have a variety $X \subset \mathbb{P}^{n}$ defined over $\mathbb{Q}$. Suppose we have $S$ many rational points on $X$ inside the box defined by $|x_i| \leq B_i$ for $i = 0, \cdots, n$, ...
- 26.9k
5
votes
0
answers
278
views
Tate's conjecture and symmetry of Hodge-Tate weights
I'm reading Bellaiche's notes on the Block-Kato conjecture (Hawaii summer school). Here is the link http://people.brandeis.edu/~jbellaic/BKHawaii5.pdf
At page 10 he claims that an indirect ...
- 231
1
vote
1
answer
398
views
Is the Cassels-Tate pairing defined for elliptic curves over function fields?
The Cassels-Tate pairing is typically defined for elliptic curves (or abelian varieties) over number fields, but it seems like it should be defined for elliptic curves over function fields as well. ...
user60591
5
votes
0
answers
522
views
Moduli interpretation of Hecke operators on Shimura curves
In his book on Automorphic Forms, Shimura gives (chapter 9) definitions of the the Hecke operators for Shimura curves.
One can give definitions of the Hecke and Atkin-Lehner operators in terms of the ...
- 3,555
12
votes
0
answers
624
views
How tight is the Weil bound for this exponential sum?
Let $f$ be a nonconstant polynomial of degree $d$. Let $\psi$ be a character of the additive group of $\mathbb F_p$. By Weil, we have:
$$ \left| \sum_{x\in \mathbb F_p} \psi ( f(x)) \right| \leq (d-1)...
- 149k
7
votes
1
answer
471
views
Intermediate Jacobians of intersections of two quadrics
Let
$$X: \quad Q_1(x)=Q_2(x) = 0 \quad \subset \mathbb{P}^{2n+1},$$
be a smooth complete intersection of two quadrics of odd dimension over a field $k$, not of characteristic $2$. Let $J(X)$ denote ...
- 21.3k
4
votes
1
answer
254
views
Rational mapping related to cubic surfaces
A theorem in Manin's book on cubic forms leads to the conclusion that the least degree of a rational mapping from $\mathbb{P}^2$ to general cubic surfaces(e.g. Picard rank=1) over $\mathbb{Q}$ is 6.
...
- 3,337
17
votes
1
answer
2k
views
Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?
I am wondering if there is a multi-dimensional analog of the
Birch/Swinnerton-Dyer (BSD) conjecture.
The recent famous result inching toward resolution of that conjecture is:
Bhargava, Manjul, and ...
- 151k
2
votes
1
answer
308
views
convergence of L-functions of curves
Let $C$ be a smooth projective curve over $\mathbb{Q}$. Its associated L-function is defined by
$$
L(C, s)=\prod_{p \text{ prime}} L_p(C, s),
$$
where, if $p$ is a prime of good reduction, $L_p(C,...
- 21
2
votes
0
answers
179
views
Questions about transformation or integral transformation
I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...
- 3,094
1
vote
1
answer
193
views
L-function of twist
I'd like to ask the following easy question, since I can't find a reference.
Let $X/\mathbb{Q}$ a smooth projective variety. How does one express the $L$-function of the twist, $L(H^i(X)(r), s)$ in ...
- 3,555
5
votes
1
answer
247
views
transcendence of beta values
(1) Can anybody suggest a readable reference for Schneider's theorem that the number
$$
\beta(a, b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
$$ is transcendental for $a, b \in \mathbb{Q}$ such that none ...
- 115
12
votes
1
answer
594
views
Is it easy to prove that $\sum_n |X(\mathbb{F}_{q^n})| t^n$ is rational?
Background: Let $X$ be an algebraic variety over a finite field $\mathbb{F}_q$. One of the successes of Etale cohomology - previously achieved by Dwork- was proving the rationality of the Zeta ...
- 2,824
0
votes
1
answer
153
views
Hasse principle and twists of $\mathbb{P}^n$ [closed]
Let $X$ be a twist of the $n$-th projective space, seen as a $K$-variety for some number field $K$. For $n = 1$, the Hasse principle holds for $X$.
My question is: for which $n >1$ does the ...
- 11
4
votes
1
answer
198
views
odd degree $0$-cycles and rational points on a quadric hypersurface
Is it true that a smooth quadric hypersurface has a rational point if and only if it has an odd degree $0$-cycle?
I think this is true. If so, can someone give a (geometric) proof?
- 43
5
votes
2
answers
674
views
Why is the supersingular locus the zero locus of a modular form?
This question is related to my other question here: Examples of subspaces singled out by modular forms.
Here I am wondering if there is a philosophical explanation about why the supersingular locus ...
- 231
8
votes
1
answer
617
views
Rational distance from vertices of an equilateral triangle
A colleague in my department posed the following question...
Let $A=(0,0)$, $B=(1,0)$, and $C=(1/2,\sqrt{3}/2)$. Then $\Delta ABC$ is an equilateral triangle with sides of length 1. Let $B_{\epsilon}(...
- 1,197
2
votes
2
answers
267
views
Openness of finite index subgroups of $\mathrm{GL}_n(\prod O_v)$
Let $K$ be a global field and set $O := \prod_{v\nmid \infty} O_v$ where $v$ runs over the finite places of $K$. Equip $\mathrm{GL}_n(O) = \prod_v \mathrm{GL}_n(O_v)$ with the product of the $v$-adic ...
- 1,103
75
votes
4
answers
5k
views
What are reasons to believe that e is not a period?
In their 2001 paper defining periods, Kontsevich and Zagier (pdf) without further comment state that $e$ is conjecturally not a period while many other numbers showing up naturally (conjecturally) are....
- 2,493
5
votes
1
answer
251
views
Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q)
I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form
$
\left(
\begin{...
- 641
7
votes
0
answers
267
views
Invariant obstructions to gluing Galois representations on elliptic curves
Let $E$ be an elliptic curve over $\overline{\mathbb F_p}$, or another separably closed field of characteristic $p$. Let $K$ be the function field of $E$, and let $K_p$ be the local field at a point.
...
- 149k
17
votes
1
answer
3k
views
Why is the section conjecture important?
As in the title, I want to know the reason for importance of the section conjecture. Of course, the statement of conjecture is important as itself, even I cannot fully grasp the soul of it. However, ...
- 601
0
votes
0
answers
79
views
Stable analytic manifold under simple action
For an integer $m > 1$, let us define the action
$$
f: X_i \to (1+X_i)^{m} - 1
$$
on $C[[X_1,...,X_N]]$, where $C$ is the complex number field. Consider the analytic manifold $V(I)$ defined by the ...
- 1
13
votes
1
answer
2k
views
What is the arithmetic Nullstellensatz?
The only precise statement (coming from a reliable source) of the "arithmetic Nullstellensatz" I can find is in Gowers's book, stating that two polynomials with integral coefficients have the same ...
- 239
9
votes
2
answers
791
views
Rational points techniques on curves not using their Jacobian
Let $C/K$ be a curve of genus > 2 over a number field $K$ and suppose there exists a $p \in C(K)$. Then a recurring theme in studying $C(K)$ is using the map $C \to J(C)$ normalized by sending $p$ to ...
- 2,224
9
votes
0
answers
720
views
Non-hyperelliptic families of curves with trivial Ceresa class (or Gross-Schoen class)
Suppose $X/K$ is a curve over a field $K$, which we want to think of as non-algebraically closed, and let $x$ be a point of $X(K)$. The Ceresa cycle is defined as follows; you can embed $X$ in $Jac(X)$...
- 19.2k
13
votes
1
answer
1k
views
Applications of anabelian geometry to Galois representations?
One aspect of anabelian geometry is the study of the action of the absolute Galois group of a field $K$ on the etale fundamental group $\pi_1(X_\overline K)$, where $X$ is a (anabelian) variety and $...
- 3,799
0
votes
1
answer
630
views
Obstruction and 1st order infinitesimal deformations of Generalized Elliptic Curves (Deligne-Rapoport)
We consider the deformation theory of a generalized elliptic curve $(C_0,+)$ over a field $k$. Let $D$ be the deformation functor.
And now we only consider the case that $C_0$ is irreducible as in D-R ...
- 997