All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
15
votes
0
answers
673
views
Exposition of Drinfeld's proof of function field Langlands for GL(2)
I know, or think I know, the vague outline of the proof: the Galois-to-automorphic direction is "classical," i.e. follows from converse theorems due to Grothendieck et al., and for the ...
- 311
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
8
votes
1
answer
855
views
What is the motivation for excellent rings?
First of all I am not formally educated in mathematics so pardon my ignorance if this is obvious and I am skipping something vital, but I am interested nonetheless in what the original motivation and ...
- 81
-1
votes
1
answer
342
views
Finite or polynomial number integral points clarification on Coppersmith's theorems (possibility of ellipse counter example?)
Coppersmith states if $f(x,y)$ is an irreducible bivariate with total degree $\delta$ then if he can list all roots $(X,Y)$ of the polynomial in $\mathsf{poly}(\log D,\delta)$ time if the roots ...
- 13.9k
14
votes
0
answers
358
views
How do we deduce the Jacquet-Langlands correspondence from Fargues' two towers?
In trying to understand the geometric proof of the local-Langlands and Jacquet-Langlands correspondence which uses Fargues's two tower theorem, I am having trouble finding a nice source on this, and I ...
- 3,539
1
vote
0
answers
158
views
Is this an explicit construction of a Hurwitz space with Galois group Z/p, p distinct branch points, and inertia group Z/(p-1)?
I am desperately confused and would like a sanity check that the following moduli space/stack is a Hurwitz space/stack. I would also appreciate any references on the topic of the explicit construction ...
- 3,539
1
vote
0
answers
108
views
The numerical values of the $L$-function of a weight-5 modular form
Suppose $f$ is a weight-5 modular form that is labeled as 27.5.b.a in LMFDB. Let $\chi_3$ be the Dirichlet character
\begin{equation}
\chi_3(p) =\left(\frac{3}{p} \right).
\end{equation}
I want to ...
- 2,971
5
votes
3
answers
932
views
Automorphy of mixed Tate motives over $\mathbb{Z}$
Deligne, Goncharov and Levine have constructed a Tannakian category of mixed Tate motives, MTM($\mathcal{O}_{K,S}$), over the ring of integers of a number field $K$ unramified outside a finite set of ...
- 1,267
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
9
votes
1
answer
458
views
Prehomogeneous varieties
A prehomogeneous vector space is a pair $(G,V)$ where $V$ is a finite dimensional $\mathbb{C}$-vector space of dimension $n$ and $G$ is a reductive group of complex dimension $n$, such that $G$ admits ...
- 26.9k
5
votes
1
answer
213
views
Can the strongest Hensel lemma over integer rings imply smoothness over $\mathbb Z_p$?
Let $X \rightarrow \mathbb Z_p$ be a flat finite type morphism, with reduced special fiber and smooth generic fiber.
Assume $X(O_K) \rightarrow X(O_K/m_K)$ is surjective for all fintie extension $K$ ...
- 461
65
votes
2
answers
9k
views
Who is the "young student" André Weil is referring to in his letter from the prison?
I am reading a nice booklet (in Italian) containing the exchange of letters that André and Simone Weil had in 1940, when André was in Rouen prison for having refused to accomplish his military duties.
...
- 66.3k
1
vote
0
answers
192
views
Integer solutions of Diophantine equation $y^2= 1+4n^{\underline k} $
I am looking for the integer solutions for the diophantine equation $y^2 =4n(n-1)(n-2)\cdots (n-k+1)+1$ for a given $k$ where $n>k+1>5$.
In other words,
$$y^2=1+4n^{\underline k},\tag{I}$$
where ...
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
4
votes
1
answer
264
views
Reference request: Long exact sequence in profinite Galois cohomology up through $H^2$
I'm looking for a reference of the following statement. Let $G$ be the Galois group of a Galois extension $L/K$, not necessarily finite. Let $A,B,C$ be groups with a continuous $G$-action, and let
$$1\...
- 7,527
2
votes
1
answer
227
views
Global section of vertical differential 1 forms on universal elliptic curve
Let $B$ be a modular curve (of some level) over a number field $K$ (here, we implicitly assume that $K$ is large enough to make sense the phrase "$B$ is a $K$-variety"). Let $E\to B$ the ...
- 1,428
2
votes
0
answers
478
views
Is there bijective correspondence between $P_n$ and $A_n$?
Let $K \supset \mathbb{Q}_p$ be the $p$-adic field and let $O_K$ be its ring of integers and $M_K$ be the maximal ideal with integral closure $\bar{M}_K$. A power series is invertible if its lowest ...
- 930
4
votes
0
answers
155
views
Can we attach (formal) abelian varieties to $p$-adic modular forms?
The Jacobian of the modular curve $X_1(N)$ over $\mathbb Q$ $J_1(N)$ can be decomposed up to isogeny, as a product of abelian subvarieties $A_f$ corresponding to Galois conjugacy classes of Hecke ...
- 461
3
votes
0
answers
162
views
Cohomology of Siegel modular varieties
$\mathcal{A}_g(N)$ is the moduli space of principally polarized abelian varieties with a level $N$ structure.
Set $C_g=\displaystyle{\lim_{\rightarrow}} H^3(\mathcal{A}_g(N), \mathbb{F}_p)$ where the ...
user163668
16
votes
0
answers
274
views
Why should an abelian variety with few places of bad reduction and a lot of endomorphisms not have many points?
In the paper "Points of Order 13 on Elliptic Curves" by Mazur-Tate, they say in the introduction:
It seemed ... that if such an abelian variety $J$, which has bad reduction at only one ...
- 7,746
4
votes
0
answers
181
views
Sato-Tate over function fields
Suppose we have an elliptic surface $\pi: \mathscr E \to C$ over a curve over a finite field $\mathbb F_q$. We consider only the places on $C$ over which we have good reduction.
Over any point $x\in C$...
- 7,746
9
votes
0
answers
194
views
Methods to compute the Kodaira dimension of moduli spaces
It is known that the moduli space $\bar{M_g}$ of genus $g$ stable curves over $\mathbb C$ is of general type for $g \geq 24$ with Kodaira dimension $3g-3=\dim \bar{M_g}$.
The idea is that one can ...
- 461
6
votes
1
answer
465
views
Etale fundamental group of an order in a number field
Let $\mathcal{O}$ be an order in a number field $K$, that is a subring of $K$ with rank as abelian group equal to $[K:\mathbb{Q}]$. What is known about the SGA3-étale fundamental group of $X=\mathrm{...
- 617
13
votes
0
answers
663
views
On a kind of Hilbert irreducibility theorem
Let us work over a number field $k$. Let $C$ be a non-empty open subscheme of $\mathbb{P}^{1}_{k}$, and $X\to C$ a family of smooth, projective hyperbolic curves such that $X(k)\to C(k)$ is surjective....
- 1,000
3
votes
1
answer
220
views
Reference request: relationship between discriminant and smoothness of a conic over arbitrary fields
I'm looking for an (ideally modern) reference of the relationship between the discriminant and smoothness of projective conics over arbitrary fields (including those of characteristic 2). Let $k$ be a ...
- 7,527
8
votes
0
answers
174
views
Geometry of moduli problem in practice: how to check it is connected / irreducible / normal / reduced / locally complete interesection...?
Moduli spaces are very common and useful in the world of algebraic geometry. From the point view of functors, one can already check many geoemtric properties of it. I like examples, and you can assume ...
- 461
9
votes
1
answer
860
views
Complex manifold defined over $\mathbb{Q}$
If we consider complex projective varieties, to be defined over $\mathbb{Q}$ means that there is a projective embedding whose image is the vanishing locus of a polynomial system with coefficients in $\...
user164740
0
votes
0
answers
110
views
Common integer roots of polynomials
I have two polynomials of form
$$f_1(w,x)=M_1$$
$$f_2(y,z)=M_2$$
and I have two polynomials of form
$$g_1(w,x,y,z)=M_3$$
$$g_2(w,x,y,z)=M_4$$
where $f_1,f_2,g_1,g_2\in\mathbb Z[w,x,y,z]$ and $M_1,M_2,...
- 13.9k
1
vote
0
answers
174
views
What are the irreps in this canonical action of $\operatorname{PGL}_2(F_q)$?
Consider the permutation action of $\operatorname{PGL}_2(\mathbb F_q)$ on $\mathbb P^1(\mathbb F_q)$ by fractional linear transformations. We can consider the associated (complex) representation of ...
- 7,746
6
votes
1
answer
298
views
Weil cohomologies with given field of definition and coefficient field
Fix a perfect field $k$. Fix a field $K$ of characteristic $0$.
A Weil cohomology induces a functor from the category of smooth projective geometrically connected $k$-schemes to the category of $\...
- 61
2
votes
0
answers
165
views
Is the cohomology of rigid varieties semisimple?
Let $X$ be a smooth projective geometrically connected scheme over $\mathbb{Q}_p$. Assume that $H^1(X, T_{X/\mathbb{Q}_p})=0$.
Is the Galois representation $H^*(X_{\overline{\mathbb{Q}_p}}, \mathbb{Q}...
- 21
19
votes
1
answer
1k
views
Deligne's letter to Bhargava from March 2004
I am quite interested in moduli spaces for Rings and Ideals, a letter from Deligne to Bhargava is cited in Melanie Wood's thesis Moduli spaces for Rings and Ideals (pdf), studying the minimal free ...
- 461
6
votes
0
answers
357
views
Moduli interpretation of Hirzebruch-Zagier divisors
In their famous 1976 paper, Hirzebruch and Zagier define certain divisors $T_N$ on the Hilbert modular surface corresponding to the group $\text{SL}_2(\mathcal{O}_F)$ for $F=\mathbb{Q}(\sqrt{p})$. ...
- 2,044
4
votes
0
answers
304
views
How to apply Newton polygon to discuss about the roots of a multi-variate p-adic power series?
How to apply Newton polygon to discuss about the roots of a multi-variate p-adic power series ?
We know that if $f(x)=\sum_{i=0}^{\infty} a_ix^i$ be a power series over $p$-adic field, then the Newton ...
- 930
1
vote
0
answers
109
views
Euler systems over abelian number fields [duplicate]
Im confused with the following statement:
Coleman’s conjecture concerning circular distribution imply that Euler systems over abelian number fields arise in “an elementary” way from the theory of ...
- 99
6
votes
0
answers
243
views
Computing Hodge numbers by point counting
In the lecture note of Bhatt from Arizona winter school 2017, there is an exercise which claims if X is a proper smooth scheme defined over $\mathbb{Z}[1/N]$ and if there is a polynomial $P$ such that ...
- 1,093
10
votes
1
answer
425
views
When is a twisted form coming from a torsor trivial?
Consider a sheaf of groups $G$, equipped with a left torsor $P$ and another left action $G$ on some $X$. Form the contracted product $P \times^G X := (P \times X)/\sim$ where $\sim$ is the ...
- 1,084
18
votes
4
answers
621
views
What are immediate applications of the classification of connected reductive groups?
After years of putting it off, I finally sat down, read, and understood the classification of connected reductive groups via root data.
That's a non-trivial theory! I'm hoping that now that I am done ...
- 557
10
votes
1
answer
458
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
12
votes
0
answers
552
views
When does Matiyasevich's theorem "kick in"?
Hilbert's 10th problem was famously resolved by the Matiyasevich–Robinson–Davis–Putnam theorem: the theorem implies that there is no algorithm which decides whether a given polynomial equation with ...
- 26.9k
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
0
votes
1
answer
325
views
On the elliptic curve $y^2 = x^3 + z^{4k}$
Are there any rational numbers $x, y, z$ with $xyz \neq 0$ such that $y^2 = x^3 + z^{4k}$ for some $k \in \mathbb{Z}_{>1}$ ?
- 1,019
7
votes
0
answers
104
views
Line bundles on pointless plane cubics
Let $k$ be a field, let $C\subset\mathbb{P}^2_k$ be a smooth plane cubic.
Suppose $C$ does not admit $k$-rational points, and for every degree-$3$ closed point $P$ in $C$, the Galois closure of $k(P)/...
user39380
-3
votes
2
answers
608
views
Rational points on the elliptic curve $y^2 = x^{3} - t^{2}z^3$
What are the rational points on the elliptic curve $y^2 = x^3 - t^{2}z^3$ ? I seem not to find any besides the trivial ones whereby $txyz=0$ or $x= \pm z$.
ADDENDUM 1. I have just noticed that if $z^3 ...
- 1,019
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
16
votes
0
answers
400
views
Quadratic non-residues in elliptic divisibility sequences
Let $E: y^2 = x^3 + ax + b$ be an elliptic curve over $\mathbb{Q}$ with $a,b \in \mathbb{Z}$. Recall that any rational point $P = (x,y)$ can be written uniquely as $P = (u/d^2, v/d^3)$ with $u,v,d \in ...
- 21.3k
4
votes
0
answers
552
views
Modern example of a reciprocity law and intuition behind it
I'm very new to the Langlands program and I was going through the Gauss reciprocity law, Hilbert's 9th problem, Artin's reciprocity law which allowed him to identify the Artin's L-functions with the ...
- 3,804
1
vote
0
answers
89
views
Sifted sets can't accumulate on a curve
Let $f,g,h$ be elements in $\mathbb{Z}[x,y]$, each geometrically integral and at least two of them are distinct. Without loss of generality, suppose that $f$ is not proportional to $g$ over $\mathbb{C}...
- 26.9k
0
votes
0
answers
165
views
Counting points in elliptic curves
Given an elliptic curve over $\mathbb Z_n$
Is it $\#P$ hard to compute $\# E(\mathbb Z_n)$?
Is it $PP$-hard to compute $\# E(\mathbb Z_n)\leq\frac n2$?
Is it $\oplus P$ hard to compute $\# E(\...
- 1,826
3
votes
0
answers
660
views
While solving the 1988 IMO problem 6, I have questions about new solutions without using Vieta Jumping [closed]
I think most of you may know the well-known problem:
"Let $x$ and $y$ be positive integers such that $xy + 1$ divides $x^{2} + y^{2}$. Show that $\frac {x^{2} + y^{2}}{xy + 1}$ is the perfect ...
- 39