All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
6
votes
2
answers
397
views
Hyperelliptic Jacobians with (or without) CM
Let $C$ be a hyperelliptic curve $y^2 = f(x)
$ defined over $\mathbb{Q}$, where $f(x) \in \mathbb{Q} [x]$ is a polynomial of degree $n=5$ or $6$, and $J = Jac(C)$ its Jacobian.
I know Zarhin's result [...
- 95
14
votes
8
answers
2k
views
Applications of the idea of deformation in algebraic geometry and other areas?
The idea of proving something by deforming the general case to some special cases is very powerful. For example, one can prove certain equalities by regarding both sides as functions/sheaves, and show ...
0
votes
1
answer
149
views
Power series rings and the formal generic fibre
Let $S = K[[S_1,\ldots,S_n]]$ and consider $d$ elements
\begin{equation*}
f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]]
\end{equation*}
and the prime ideal ${\frak P} \colon\!= (f_1,\ldots,f_d)$ generated ...
- 563
1
vote
0
answers
102
views
Maximum number of bounded primitive integer points in a zero-dimensional system
Given a set of $n$ many degree $2$ algebraically independent and thus zero-dimensional system of homogeneous polynomials in $\mathbb Z[x_1,\dots,x_n]$ with absolute value of coefficients bound by $2^{...
- 1,826
3
votes
1
answer
471
views
Extending section of étale morphism of adic spaces
This question is related to Lifting points via étale morphism of adic spaces.
Fix a complete non-archimedean field $k$. Let $(A,A^+)$ be a complete strongly noetherian Huber pair over $(k,k^\...
- 83
3
votes
0
answers
166
views
Automorphy Factor from Vector Bundles on Compact Dual
So I'm coming from an algebraic geometry perspective and I'm trying to carefully piece together the story of interpreting automorphic forms as sections of vector bundles on Shimura varieties. I think ...
- 1,701
1
vote
0
answers
278
views
Level structures in deformation spaces of $p$-divisible groups
I am reading (parts of) the paper "Moduli of $p$-divisible groups" by Scholze and Weinstein, and I am stuck at understanding the definition of level structures in Rapoport-Zink spaces (cf. Definition ...
- 83
30
votes
4
answers
3k
views
Motivation for zeta function of an algebraic variety
If $p$ is a prime then the zeta function for an algebraic curve $V$ over $\mathbb{F}_p$ is defined to be
$$\zeta_{V,p}(s) := \exp\left(\sum_{m\geq 1} \frac{N_m}{m}(p^{-s})^m\right). $$
where $N_m$ is ...
- 901
2
votes
1
answer
251
views
Étale fibration for $K[[X_1,...,X_n]]$
Let us consider a formal power series ring $A_n \colon= K[[X_1,\ldots,X_n]]$ with $0 \ll n < \infty$ and we shall consider a prime ideal ${\frak P}$ of $A_n$ such that $1 < {\mathrm{ht}}({\frak ...
- 563
0
votes
0
answers
116
views
Gauss lemma for a complete Noetherian domain
Suppose that $R$ is a Noetherian complete domain over a field $K$.
Suppose that a monic polynomial $f(X) \in R[X]$ (i.e., the highest degree $X^e$ in $f$ has the coefficient $1$), satisfies the ...
- 563
4
votes
1
answer
386
views
Smooth proper variety over a number field with prescribed bad reductions
Given a number field $K$, and a finite set $S$ containing finite places of $K$. When can we find a smooth proper geomerically connected variety $X$ over $K$ such that $X$ has good reduction outside $S$...
- 6,622
5
votes
2
answers
250
views
What integer value can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$?
Fix a positive integer $g$. What positive integer $N$ can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$ ?
For example, as there is no abelian varieties over $\mathbb Z$, $N$ ...
- 6,622
6
votes
0
answers
342
views
How to decide whether the isogeny between Neron models is etale?
Let there be an isogeny $f:A_1 \rightarrow A_2$ between two abelian varieties over a $p$-adic field $F$ and assume $f$ has degree $p^n$. By the universal property we get a moprhism $f_0: \mathcal{A}_1 ...
- 6,622
9
votes
0
answers
216
views
Kronecker's theorem in higher dimension
Recall the following classical theorem of Kronecker: if $P(x) \in \mathbb{Z}[x]$ is a monic irreducible polynomial with all roots on the unit circle $S^1$, then $P(x)$ is a cyclotomic polynomial (and ...
- 20.8k
1
vote
0
answers
119
views
On the exponent of a certain matrix $A$ in characteristic $p > 0$
Let $A$ be a square matrix in characteristic $p > 0$ with both column and row having length $(1 + p^0 + p + \cdots + p^i)$, where $i \geq 0$.
Suppose that further the $(m,n)$-component $a_{m,n}$ ...
- 563
5
votes
1
answer
653
views
rank of Jacobian of Fermat curve and Chabauty-Coleman method
Consider the fermat curve $F(p)$ over $\mathbb Q$ which is the projective closure of $X^p+Y^p=1$ inside projective plane, where $p$ is a prime number and without loss of generality we assume $p>2$. ...
- 6,622
6
votes
1
answer
821
views
Does there exist a discrete valuation subring $R$ of $K((t))$ ($K$ a number field) of residue characteristic $p$ with $\mathrm{Frac}(R) = K((t))$?
Let $K$ be a number field, and let $K((t))$ be the field of formal Laurent series. Let $p > 0$ be a prime.
I have two questions:
Does there exist a discrete valuation subring $R$ of $K((t))$ of ...
- 10.7k
12
votes
0
answers
265
views
Mixed characteristic analogue of algebraicity of the diagonal of two-variable power series?
Let $f=\sum_{n,m \geq 0}^{\infty}[a_{nm}]p^ny^m \in \mathbb Z_p[[y]]$, where $a_{nm} \in \mathbb F_p$ and $[\cdot]$ means the Teichmüller lifting. Define $I(f)=\sum_{n \geq 0}[a_{nn}]p^nt^n \in \...
- 6,622
2
votes
0
answers
247
views
field of definition of CM abelian varieties
When $A$ is a CM abelian variety of dimension $1$ (i.e., an elliptic curve), then we have a result that if it has CM by a maximal order then it has a model over a number field $F$ where $F$ is the ...
- 443
4
votes
0
answers
195
views
lemma II.2.4 in Harris-Taylor (about drinfeld-katz-mazur level structure on 1-dimensional $p$-divisible groups)
Lemma II.2.4 on page 82 in Harris and Taylor's "The Geometry and Cohomology of Some Simple Shimura Varieties" (or lemma 3.2 here), says that given a Drinfeld(-Katz-Mazur) level structure $\alpha:(p^{-...
- 371
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
6
votes
1
answer
524
views
How to compute Galois representations from etale cohomology groups of a generalized flag variety?
Let $G$ be a connected reductive group over a number field $K$, $P$ be a parabolic subgroup of $G$ defined over $K$, $X=G/P$ be the generalized flag variety which is a smooth projective variety over $...
- 6,622
11
votes
0
answers
252
views
Can the orbit with respect to a rotation on the torus hit an algebraic variety infinitely often?
Question: Does there exist a $d$-tuple $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$ (with $1,\alpha_1, \dots,\alpha_d$ linearly independent over $\mathbb{Q}$) and an algebraic variety $V \...
- 1,914
3
votes
0
answers
148
views
Maximum number of integral roots in degree $d$ polynomial?
Given $f(x_1,\dots,x_n)\in\mathbb Z[x_1,\dots,x_n]$ such that
Each coefficient is bound in absolute value by $B$
Degree of each variable in any monomial is bound by $d$
Total degree is $d'$
$f(x_1,\...
- 13.9k
0
votes
0
answers
132
views
Final step in Coppersmith?
In the final step in Coppersmith technique we have $n$ polynomials (possibly non-homogeneous) in $\mathbb Z[x_1,\dots,x_m]$ where $m\leq n$ and using elimination theory we extract the common integer ...
- 13.9k
1
vote
0
answers
127
views
Universal elliptic curve over anticanonical tower
While I'm reading Scholze's paper "On torsion in the cohomology of locally symmetric varieties" he constructs the anticanonical tower passing through the construction of an integral model $X_{\infty}$ ...
- 445
13
votes
1
answer
1k
views
Is there a substitution that relates every Fermat curve to an elliptic curve?
I asked this question on MSE but didn't get any response, so I'm asking here. I apologize in advance if this question is not research level.
A Fermat Curve of degree $n$ is the set of solutions to $x^...
- 236
11
votes
1
answer
660
views
Finiteness or infiniteness for Galois representations with unusual Hodge numbers
Say a representation $\operatorname{Gal}(\mathbb Q) \to GL_n(\overline{\mathbb Q}_\ell)$ has big monodromy if the Zariski closure of the image of $\operatorname{Gal}(\mathbb Q) $ contains $SO_n$ or $...
- 148k
27
votes
3
answers
3k
views
Where's the best place for an algebraic geometer to learn some algebraic number theory?
There are lots of introductions to number theory out there, but typically they are streamlined to assume as little prerequisite knowledge as possible. I'm looking for a text which does the opposite -- ...
- 64k
10
votes
1
answer
562
views
Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$?
Do there exist infinitely many real quadratic fields $F$ such that there is an abelian surface $A$ over $\mathbb Q$ whose ring of endomorphisms, tensored with $\mathbb Q$, is $F$?
Do there exist ...
- 148k
11
votes
1
answer
284
views
Does every section of the map Gal$(\overline{k(\!(t)\!)}/k(\!(t)\!))\rightarrow$ Gal$(\overline{k}/k)$ stabilize a compatible system of roots of $t$?
There may be some technical issues with the question, but hopefully what I mean is clear...
Let $k$ be a number field (or maybe any finitely generated field over $\mathbb{Q}$ of characteristic 0)
...
- 7,527
13
votes
0
answers
503
views
Hensel lemma and rational points in complete noetherian local ring
Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal.
If we have several polynomials $f_i \in A[X_1, \dots, X_m]$ which have a common zero $x_n$ in $A/\mathfrak{m}^n$ ...
- 6,622
2
votes
0
answers
127
views
Generalizing characterizing crystalline representations of dimension 2 to certain special classes of crystalline representations of higher dimension
Let $A$ be an abelian variety defined over a number field $K$, and let $v$ be a prime of $K$ such that $A$ has good reduction modulo $v$. Let $\rho$ be the representation of $G_K = \text{Gal}(\...
- 26.9k
20
votes
1
answer
1k
views
Curves over number fields with everywhere good reduction
My question is the following:$\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
5
votes
0
answers
156
views
Reference Request: Cohomology and limits of coherent topoi. (Non-abelian case)
SGA 4 VI Discusses finiteness conditions one can impose on topoi to make limits behave correctly. I am not that familiar with SGA but it is my impression that this expose only discusses abelian ...
- 435
4
votes
1
answer
418
views
Subsets $E$ of $\mathbb{F}_{p^k}$ with vanishing polynomial subset sums
The following question arose in some discussions recently as a misunderstanding of another problem.
Question: Which subsets $E\subset \mathbb{F}_{p^k}$ satisfy the property that $ \sum\limits_{x\in E}...
- 3,209
1
vote
0
answers
128
views
Point Counts on $G$-torsors over Finite Fields
Let's assume we have a $G$-torsor $X_{1} \to X_{2}$, where $G$ is a finite abelian group, and both $X_{1}$ and $X_{2}$ are defined over $\text{Spec}(\mathbb{Z})$. Is there an easy way to compute $\#...
- 1,701
10
votes
2
answers
1k
views
Are the ideles literally a Picard group?
I understand that in the number field / function field analogy, the ideles $\mathbb I_K$ of a number field $K$ are supposed to be analogous to the Picard group of a function field.
Question: Is this ...
- 64k
11
votes
0
answers
604
views
Fourier transforms and nontrivial vector bundles
We know that in arithmetic, geometry and analysis, Fourier transforms of various forms show up. For example, we have the classical Fourier transform, Fourier-Mukai transforms in the setting of ...
- 281
4
votes
0
answers
156
views
Traces of Frobenius Endomorphism on Etale Cohomology and $G$-torsors
I have a smooth, projective, and rigid Calabi-Yau threefold $X$ defined over $\mathbb{Q}$. Such spaces always have integral models. Let's assume we have an action on $X$ by a finite abelian group $G$...
- 1,701
9
votes
1
answer
751
views
Nontrivial p-divisible groups over $\mathbb Z$ for general prime $p$
In Tate's famous paper about $p$-divisible groups, for a prime number $p$ he asks whether there exists a $p$-divisible group $G$ over $\mathbb Z$ such that $G$ is not a direct sum of $\mu_{p^\infty}$ ...
- 6,622
8
votes
0
answers
377
views
Comparison between Faltings height and Modular Height
Motivation/Context: In Faltings’ proof of the Mordell conjecture, there is a theorem that establishes a finiteness of abelian varieties with respect to the Faltings height under certain conditions. ...
- 993
1
vote
1
answer
157
views
Homogeneous polynomials which ramify completely on a hypersurface
Let $F \in \mathbb{Z}[x_0, \cdots, x_n]$ be a homogeneous polynomial. Let $V \subset \mathbb{P}^n(\mathbb{C})$ be a hypersurface (defined over $\mathbb{Q}$ say), given by a homogeneous polynomial $G(...
- 26.9k
12
votes
0
answers
285
views
Statistics for rational points on curves of genus $g$ over $\mathbb{F}_q$, $g\gg q$
Consider the distribution of the number of $\mathbb{F}_q$ points as I range over smooth projective curves of genus $g$ (defined over $\mathbb{F}_q$). If $q\gg g,$ the Hasse-Weil bounds give me a lot ...
- 5,958
8
votes
1
answer
251
views
Can discriminant polynomials become perfect powers on hyperplanes?
Let
$$\displaystyle f(x) = a_d x^d + a_{d-1} x^{d-1} + \cdots + a_0.$$
Consider the discriminant of $f$, denoted by $\Delta(f)$, defined as
$$\displaystyle \Delta(f) = a_d^{2d-2} \prod_{i < j} (\...
- 26.9k
5
votes
1
answer
415
views
What is $\mathrm{O}_q/\mathrm{SO}_q$ if $q$ is a quadratic $\mathbb{Z}$-form which is degenerate?
Any binary quadratic $\mathbb{Z}$-form $q$ induces a symmetric bilinear form
$$ B_q(u,v) = q(u+v) - q(u) -q(v) \ \ \forall u,v \ \in\mathbb{Z}^2 $$
and it is considered non-degenerate (over $\mathbb{...
- 359
5
votes
1
answer
412
views
Hasse principle and its failure for a special class of plane cubics
Let $X$ be a variety defined over a number field $K$. The Hasse principle, or the local-to-global principle, asserts that $X(K) \ne \emptyset$ if and only if for each completion $K_v$ of $K$, we have $...
- 26.9k
35
votes
0
answers
1k
views
Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function?
The adèles $\mathbb A$ arise naturally when considering the Berkovich space $\mathcal M(\mathbb Z)$ of the integers. Namely, they are the stalk $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
- 64k
2
votes
1
answer
290
views
Dimension formulae for Jacobi forms
I'm interested in locating dimension formulae for (more general) Jacobi forms associated with a lattice $L$ (where the Jacobi forms of Eichler-Zagier correspond to $L=A_1$).
Unfortunately, the ...
- 61
4
votes
0
answers
232
views
holomorphic continuation of motivic $L$-functions
The question is rather easy to formulate: when is the $L$-function of a pure motive over $\mathbb{Q}$ expected to have a holomorphic (as opposed to simply meromorphic) continuation to the complex ...
- 41