All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
12
votes
3
answers
1k
views
Chow Groups of varieties over number fields
I believe that there is a conjecture that for any smooth projective variety $X$ over a number field $K$, its Chow groups $CH^i(X)$ (or at least $CH^i(X)\otimes_{\mathbf Z} \mathbf Q$) are finitely ...
- 2,923
0
votes
0
answers
206
views
Does there exist a homogeneous polynomial $F$ whose partial derivatives satisfy the following inequalities?
Given $n \geq 2$, I would like to find either a homogeneous polynomial $F \in \mathbb{Q}[x_1, \ldots, x_n]$ of degree $d > 1$ with the following properties:
$W = \{ \mathbf{x} \in \mathbb{R}^n : F(...
- 3,625
4
votes
0
answers
211
views
Point counting and the Torelli theorem
Let $E$ be a CM elliptic curve defined over $\mathbb{Q}$ and $p$ be a prime where $E$ has good reduction. Is there always a curve $C$ defined over $\mathbb{Q}$ such that $\# J(\mathbb{F}_p) = \#E(\...
- 521
8
votes
2
answers
403
views
Mazur's Question on Mod $N$ Galois representations
In Rational Isogenies of Prime Degree, Mazur poses:
"the problem of determining all elliptic curves $E'/\mathbb{Q}$ with symplectic $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ isomorphisms $E'[N]\...
- 901
5
votes
1
answer
411
views
Rationality of trace of endomorphism of Iwasawa-thing
Let $n$ be a positive integer, and $p$ a prime number. Let $K_i$ be the cyclotomic field containing exactly the $np^i$th roots of unity. Let $H$ be the inverse limit of $p$-power torsion of the class ...
- 804
2
votes
0
answers
519
views
Good place to learn about arithmetic schemes?
Where is a good place to learn about arithmetic schemes? There is discussion in Eisenbud-Harris's book The Geometry of Schemes (and also Mumford's red book) and I hear that there is discussion in Liu'...
- 21
-1
votes
1
answer
531
views
What do we know about the number $\prod_p \left( 1 - \frac{1}{p}\right)^4\left( 1 + \frac{4}{p} + \frac{1}{p^2} \right) $?
I am looking at a result of Peyre, and he says for a certain variety, the number of rational points of height less than $B$ is:
$$ N(B) \sim \frac{1}{3} \color{#3DB08E}{\prod_p \left( 1 - \frac{1}{p}\...
- 22.8k
5
votes
0
answers
280
views
Proving that a certain function (related to a volume of a region) has a bounded derivative
Let $F$ be a homogeneous form in $n$ variables with integer coefficients.
Let $D$ be a closed box in $\mathbb{R}^n$ (product of closed and bounded intervals). Assume that the partial $\partial F/\...
- 3,625
4
votes
0
answers
413
views
quasi-finite group schemes
The following is what Mazur wrote on page 91 of his paper, Modular curves and the Eisenstein ideal, published in Publ. IHES in 1977, DOI: 10.1007/BF02684339 (freely available at eudml):
Let $m$ be an ...
- 335
25
votes
1
answer
1k
views
How is Borger's approach to $\mathbb{F_{1}}$ related to previous approaches (e.g. Deitmar's)?
The "traditional" approach to the so-called "field with one element" $\mathbb{F}_{1}$ is by using monoids, or, to put it in another way, by forgetting the additive structure of ...
- 3,309
3
votes
0
answers
255
views
What are the easiest counterexamples to Serre-Lang over non-algebraically closed fields?
Let $k$ be a field of characteristic zero. Let $A$ be an abelian variety over $k$. Let $X\to A$ be a finite etale morphism with $X$ a connected (smooth projective) variety over $k$.
Then, choosing a ...
- 161
10
votes
1
answer
376
views
Numbers of solutions equal on every finite commutative ring
Let $X,Y$ be two schemes finite type over $\Bbb Z$, assume $\#X(A)=\#Y(A)$ for every finite commutative ring $A$, then
must these two schemes be isomorphic ?
What invariants of schemes coincide on $...
- 6,622
3
votes
1
answer
209
views
How does the minimal size of a rational solution to a system of polynomial equations depend on parameters?
The undecidability of Hilbert's tenth problem implies the following (there is a stronger statement here, Theorem 9):
For any computable function $f$, there is a family of integer polynomials (where ...
- 934
2
votes
1
answer
463
views
Modular parametrization in terms of the moduli of shtukas
The modular parametrization of an elliptic curve over $\mathbb{Q}$ (and maybe over a number field in general?) is well-known; also for an elliptic curve over global function field with some condition (...
- 647
3
votes
0
answers
153
views
Arguing that weakly holomorphic modular forms give rise to Katz modular forms
Let $\Gamma = \Gamma_1(n)\le\text{SL}_2(\mathbb{Z})$ for some $n$ (I don't want to assume that $\Gamma$ is torsion-free)
Let $\mathcal{H}$ be the upper half plane, then on $\mathcal{H}$ we have a ...
- 7,527
20
votes
1
answer
2k
views
"Adelic" Arakelov Geometry
In Soule's Lectures on Arakelov Geometry, he suggests the following "improvement" of Arakelov geometry:
As we said earlier, Arakelov geometry is a static generalization
of infinite descent. For ...
- 3,309
1
vote
1
answer
584
views
Etale cohomology of singular varieties?
In all the literature I have read, etale cohomology is defined for smooth varieties. Suppose $X/\mathbb{Q}$ is a singular variety over $\mathbb{Q}$, does there exist etale cohomology $H_{\text{et}}^*(...
- 2,971
20
votes
0
answers
994
views
Finiteness of etale cohomology for arithmetic schemes
By an arithmetic scheme I mean a finite type flat regular integral scheme over $\mathrm{Spec} \, \mathbb{Z}$.
Let $X$ be an arithmetic scheme. Then is $H_{et}^2(X,\mathbb{Z}/n\mathbb{Z})$ finite ...
- 21.3k
3
votes
1
answer
352
views
When does the module of Katz modular forms contain a basis for the vector space of classical modular forms?
Let $\Gamma\le SL(2,\mathbb{Z})$ be a congruence subgroup of level $N$. Let $R$ be a $\mathbb{Z}[1/N]$-algebra.
Let $\mathcal{Y}(\Gamma)_R$ denote the moduli stack over $R$ of elliptic curves ...
- 7,527
13
votes
1
answer
444
views
Finite generation of module of modular forms
Given a commutative $\mathbb{Z}[\frac1n]$-algebra $R$, we can consider the ring of modular forms $M_*(\Gamma_1(n), R)$. If $R$ is a subring of $\mathbb{C}$, these can be defined as those (holomorphic) ...
- 12.1k
0
votes
1
answer
343
views
Relative Bertini Theorem
Let
$A \colon= {\Bbb C}[S_1,\ldots,S_n]$ with $1 \leq n < \infty$
$B \colon= A[X_1,\ldots,X_d]$ with $2 \leq d < \infty$.
$O \colon= (0,\ldots,0)$ be the origin of ${\mathrm{Spec}}\,B$.
...
- 563
0
votes
1
answer
242
views
Primes of the form $p=3a^2+3ab+b^2$ or $p=27a^2+27ab+7b^2$ and the number of points of $y^2=x^3+2$ modulo $p$
Trying to generalize this answered question based on limited numerical evidence.
Let $E / \mathbb{F}_p : y^2=x^3+2$.
Conjecture 1 Let $p=3a^2+3ab_0+b_0^2$ be prime and $a,b_0$
positive integers.
...
- 25.4k
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
3
votes
0
answers
113
views
Cohomologies of $[V/GL_n]$ in characteristic $p$ for a representation $V$ of $GL_n$
Let $V$ be a representation of $G=GL_n$ (or more generally any reductive group $G$) over an algebraically closed field $\mathbb k$ of characteristic $p$. Let $[V/G]$ be the corresponding quotient ...
- 790
7
votes
3
answers
768
views
Sato-Tate and the angles of split primes
I was reading this blog by Evan Chen about complex multiplication. He's discussing Sato-Tate Conjecture. We can have elliptic curve $E/\mathbb{Q}$ and solve it over finite fields.
\begin{eqnarray*}
...
- 22.8k
1
vote
1
answer
127
views
Can we drop smoothness in uniformity conjecture if we don't count singular points?
The uniformity conjecture
basically states that the number of rational points on a smooth curve of genus $g >1$ over number field is bounded.
If we drop smoothness, there is counterexample coming ...
- 25.4k
14
votes
3
answers
666
views
Patterns in solutions to $a^2 + b^2 + c^2 = n$
I have plotted solutions $(a,b,c)$ to $a^2 + b^2 + c^2 = n$ for $12000 \leq n < 12100$, rescaled to $S^2$ and projected onto the first two coordinates. (these are read from the lower left, across ...
- 22.8k
14
votes
2
answers
545
views
Distribution relation in the Euler system of Heegner points
I am trying to understand the details behind the so-called "distribution relations" between Heegner points on the modular curve $X_0(N)$, as given (for instance) in Gross's paper Kolyvagin's work on ...
- 329
7
votes
1
answer
858
views
Teichmuller groupoids in Grothendieck's esquisse d'un programme
Grothendieck in his Esquisse d'un programme mentioned without any precise definition and construction that the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on the whole "tower" ...
- 21.8k
1
vote
2
answers
505
views
Base change of a finite morphism
Let $X$, $Y$, $Z$ be integral schemes of finite type over a field $K$ (i.e., locally affine opens are finitely-generated algebras over $K$). Suppose we have the following condition$\colon$
$f \colon ...
- 563
3
votes
0
answers
344
views
Dirichlet Character, Galois Representation and Motives
Suppose $\chi$ is a real Drichlet character of modulus $N$,
\begin{equation}
\chi: (\mathbb{Z}/N\mathbb{Z})^* \rightarrow \mathbb{C}
\end{equation}
which induces an Artin representation
\begin{...
- 2,971
9
votes
0
answers
374
views
Clarification on relationship between Grothendieck-Messing and Honda systems
It's well-known, due to work of Fontaine (see e.g. this), that if $k$ is a perfect field of characteristic $p$, then one can classify $p$-divisible groups over $W(k)$ by Honda systems. Namely, these ...
- 843
7
votes
1
answer
270
views
Is $\Gamma(p) := \text{Ker}(SL_2(\mathbb{Z}_p)\rightarrow SL_2(\mathbb{F}_p)$ a "standard" subgroup?
Let $\Gamma(p) := \text{ker}(SL_2(\mathbb{Z}_p)\rightarrow SL_2(\mathbb{Z}_p/p))$.
Viewing $SL_2(\mathbb{Z}_p)$ as an analytic group, is there a formal group law $F$ in three variables, defined over $...
- 7,527
3
votes
0
answers
187
views
Lifting the Cartier operator to the p-adics
The Cartier operator is defined for smooth varieties over finite fields. If $C$ is an algebraic curver over $\mathbb{Z}_p$. Let $C_p$ the reduction of $C$ to positive characteristic. My question is if ...
- 527
10
votes
2
answers
1k
views
periodic cyclic homology and tilting in the sense of Scholze
Suppose $R$ is a perfectoid ring in mixed characteristic, and $R'$ its characteristic-$p$ tilt. Scholze's results on tilting say that the étale theories over $R$ and $R'$ are equivalent in an almost ...
- 7,952
0
votes
1
answer
353
views
Strange subscheme in ${\mathrm{Spec}} R \times {\Bbb A}^1_{\Bbb C}$
Let ${\Bbb C}[X_1,\ldots,X_n]$ be a $n$-variable polynomial ring over a complex number field ${\Bbb C}$. For its maximal ideal $(X_1,\ldots,X_n)$, we define the geometric regular local ring as
$R \...
- 563
3
votes
1
answer
776
views
On the coherence of a Néron-ring
Let $A:= \underset{\lambda \in \Lambda}{\varinjlim} \,A_{\lambda}$ be an inductive limit of geometric regular local ring $(A_{\lambda}, {\frak m}_{\lambda})$, whose transition map $\phi_{\mu\lambda} \...
- 563
16
votes
0
answers
532
views
Are there smooth and proper schemes over $\mathbb Z$ whose cohomology is not of Tate type
Is there an example of smooth and proper scheme $X \to \mathrm{Spec}(\mathbb Z)$, and an integer $i$ such that $H^i(X, \mathbb Q)$ is not a Hodge structure of Tate type?
Alternatively: such that $H^...
- 171
5
votes
1
answer
327
views
Application of Abhyankhar's lemma
I am confused by an application of Abhyankhar's lemma in the proof of Theorem 3.4 of Deligne-Rapoport.
Here is the question with only the relevant parts of the text:
Let $X$ and $Y$ be two curves ...
15
votes
0
answers
2k
views
Inter-Universal Teichmuller Theory and the Field with One Element
The idea of the "field with one element", or $\mathbb{F}_{1}$, is supposed to allow us to do for number fields what we can do for function fields. Hence this idea often comes up regarding problems ...
- 3,309
23
votes
1
answer
3k
views
Geometric intuition for Fontaine-Wintenberger?
I asked my advisor the question in the title. He told me it was a stupid question and that I should focus on my research. Thus we're asking here.
The statement of Fontaine-Winterberger, per their ...
- 273
0
votes
0
answers
100
views
Solutions of the linear equation from K[[X_1,X_2,X_3]] to K[[X_1,X_2]]
Let $A_3 := K[[X_1,X_2,X_3]]$ be a three-variable formal power series ring over a field $K$. We consider a linear equation
$(\sharp) \phantom{aa} a_1(X_1,X_2,X_3)Y_1 + \ldots + a_n(X_1,X_2,X_3)Y_n = ...
- 563
4
votes
0
answers
159
views
identifying components of points on elliptic curves with Kodaira symbol $I_{2n}^{*}$
Let $K$ be a local field that is complete with respect to a discrete valuation.
When an elliptic curve, $E/K$, has reduction type represented by the Kodaira symbol $I_{2n}^{*}$, its component group ...
- 41
43
votes
1
answer
4k
views
A mysterious connection between Ramanujan-type formulas for $1/\pi^k$ and hypergeometric motives
The question below is the follow-up of this question on MathOverflow.
Motivation: As is stated in the former question, those identities(formula (35)-(44)) of $1/\pi$ attributed to Ramanujan are ...
- 3,337
5
votes
0
answers
194
views
Relation between potential good reduction of curves and belyi maps
Let $X$ be a smooth proper curve over a number field $K$, and let $f : X\rightarrow\mathbb{P}^1_K$ a degree $n$ map unramified outside $\{0,1,\infty\}$.
I've read in various places that something of ...
- 10.7k
6
votes
1
answer
464
views
Two definitions of the narrow Mordell-Weil group
Let:
$K = k(C)$, where $C/k$ is a projective non-singular curve,
$E/K$ - an elliptic curve,
$\mathcal{E} \to C$ - the minimal elliptic surface associated to $E$.
Consider the "narrow Mordell-Weil ...
- 421
4
votes
0
answers
237
views
Periodicity in Galois Cohomology
Let $k$ be a field which is not of characteristic $\ell$ and $\nu \geq 1$. Suppose for a moment that $\ell$ is odd. Even without $\ell$-th roots of $1$, then we have isomorphisms $H^*_{et}(k, \mu_{\...
- 2,327
10
votes
4
answers
1k
views
Possible groups of K-rational points for elliptic curves over arbitrary fields
It is known that the group $C(\Bbb R)$ has at most two connected components, and the connected component of the identity is isomorphic to $U(1)$ as a topological group (trivially) and $C(\Bbb Q)$ is ...
- 3,738
4
votes
0
answers
259
views
Galois cohomology of the Serre group in the proof of the fundamental theorem of CM
I am working through J.S. Milne's note on the fundamental theorem of complex multiplication over $\mathbb{Q}$. Let $E$ be a CM-field Galois over $\mathbb{Q}$, and $S^E$ the Serre group corresponding ...
- 570
4
votes
0
answers
233
views
"Lifting" of Jacobi forms of weight zero vs. index one?
In this question I'll try to avoid using the words "Borcherd's Lift" only because I'm not sure in what setting it applies properly. What I will be asking about is sometimes called "second quantized ...
- 1,701