All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
19
votes
0
answers
4k
views
On 'A proof of ABC conjecture after Mochizuki' [closed]
It has been pointed out to me that Go Yamashita has a preprint on his website, A proof of ABC conjecture after Mochizuki, that is not in the RIMS Preprints online archive.
Is that work intended for ...
user75426
3
votes
1
answer
239
views
Explicit endomorphisms of Jacobians of genus $2$ and the Theta divisor
I hope this question is good to be here.
Let $J$ be the Jacobian of a hyperelliptic curve $H$ of genus $2$ given by $y^2 = x^5 + h$.
I was calculating an explicit formula in Mumford coordinates of $[...
3
votes
0
answers
329
views
Etale cohomology of rigidification
Let $K/\mathbb Q_p$ be a discretely valued non-archemedean field, let $X$ be a smooth scheme over $\mathcal O_K$. To $X$ one can associate two rigid-analytic spaces over $K$:
1) the analytification $...
- 790
29
votes
0
answers
2k
views
A modern perspective on the relationship between Drinfeld modules and shtukas
Shtukas were defined by Drinfeld as a generalization of Drinfeld modules. While the relationship between the definitions of Drinfeld modules and shtukas is not obvious, one does have a natural ...
- 148k
7
votes
1
answer
1k
views
Szpiro's conjecture for function fields and Mochizuki's approach to the number field case
Where can I find more details on the proof of Szpiro's conjecture for function fields, as mentioned in Minhyong Kim's answer to this MO question?
I am looking at this in the context of Mochizuki's ...
- 3,309
2
votes
1
answer
270
views
BSD conjecture for abelian schemes and the classical version
I would like to know the relation between the BSD conjecture for abelian schemes (as stated for example in T Keller's thesis) and the clasical BSD conjecture.
In particular, can one state the ...
- 21
24
votes
3
answers
4k
views
How are motives related to anabelian geometry and Galois-Teichmuller theory?
In Recoltes et Semailles, Grothendieck remarks that the theory of motives is related to anabelian geometry and Galois-Teichmuller theory. My understanding of these subjects is not very solid at this ...
- 3,309
1
vote
0
answers
551
views
An infinitely-many-variable formal power series ring ${\Bbb F}_p[[X_1,\ldots]]$
We shall define a infinitely-many-variable formal power series ring ${\Bbb F}_p[[X_1,\ldots]]$ as follows$\colon$
${\Bbb F}_p[[X_1,\ldots]]\colon= \underset{n \geq 1}{\varprojlim}\, {\Bbb F}_p[[X_1,\...
- 563
10
votes
0
answers
854
views
Scholze's infinite to finite type ring theory reductions?
In following essay "The Perfectoid Concept: Test Case for an Absent Theory" by Michael Harris, there is the following sentence I found to be quite striking.
The most virtuosic pages in Scholze's ...
- 119
3
votes
1
answer
213
views
5-Descent or ($\sqrt{5}$-Descent?) on certain genus 2 Jacobians
I would like to know if there is something I can read to compute the following:
Let $H$ be a hyperelliptic curve of genus $2$ given by $y^2=x^5 + 10$ and let $J$ be its Jacobian.
How can I prove ...
4
votes
1
answer
807
views
question regarding Faltings' proof of the Tate conjecture for Abelian varieties over number fields
Why does Faltings in his "Endlichkeitssätze für Abelsche Varietäten über Zahlkörpern" in the proof of Theorem 3/4 assume that $W$ is a maximal isotropic $\pi$-invariant subspace? Tate also assumes ...
user19475
2
votes
0
answers
124
views
torsion sections in abelian schemes
Let $A \to S$ be an abelian scheme of relative dimension $g$ over an algebraically closed field of characteristic $p$. We have the following short exact sequence on $S$
$$0 \to R^i\pi_*(\mathbb{Z}_l) \...
- 21
5
votes
1
answer
369
views
Do Abelian varieties isomorphic to all their conjugates descend?
Suppose $A$ is an abelian variety over $\overline{\mathbb{Q}}$ of dimension $g$, such that $A$ is isomorphic to all of its Galois conjugates. Note that I'm not including any polarization data.
Can I ...
- 2,824
2
votes
1
answer
212
views
When will the value of automorphic function $f(x)$ satisify an algebraic equation?
When will the value of automorphic function $f(x)$ satisfy an algebraic equation? Or what is the value of $x$ such that the value of automorphic function $f(x)$ is algebraic?
If the question is too ...
- 3,094
3
votes
1
answer
237
views
Selmer $p$-Groups
I searched so many articles about Bloch-Kato $p$-selmer Groups defined by $p$-adic representation but it seems that $p$ is not necessary to be odd.
Hence, I am wondering if it is worth considering ...
- 41
3
votes
0
answers
148
views
Given an embedding of $X$ into $\mathbb{P}^n_K$, do you get an induced embedding of any twist of it into $\mathbb{P}^n_K$?
Let $X$ be a projective algebraic curve over some number field $K$, and let $\varphi:X\hookrightarrow \mathbb{P}^n_K$ be an embedding of it (defined over $K$) into some projective space.
Now let $X'$ ...
- 460
5
votes
1
answer
385
views
A Naive Question about Nekovar's Paper on Beilinson's Conjecture
I post this naive question on Math stackexchange, but have got no reply, so I decide to bother the mathoverflow community.
https://math.stackexchange.com/questions/2350436/rational-structures-of-...
- 2,971
11
votes
0
answers
324
views
Why is the CM-type preserved after base changing from char 0 to char p?
There is a transition in the theory of complex multiplication which seems to be glossed over in all expositions I can find. I would like to explicitly find a theorem that allows me to do this.
...
- 3,539
7
votes
1
answer
321
views
optimal estimate for generalized Kloosterman sum
Let $p$ be an odd prime. Denote $e(x):=e^{2\pi i\frac{x}{p}}$. Let $n\ge 2$ be an integer. Consider the exponential sum
$$
S(f,g)=\sum_{g(x_1,\dots,x_n)=0, x_i\in\mathbb{F}_p}e(f(x_1,\dots,x_n)),
$$
...
- 103
5
votes
0
answers
294
views
On toroidal compactifications of Hilbert Kuga-Sato varieties
Let $F$ be a totally real field of degree d. There are Hilbert modular varieties over $\mathbb{Q}$ that paramatrize abelian varieties of dimension d with an action of $\mathcal{O}_F$ the ring of ...
- 845
3
votes
0
answers
139
views
Cartan decomposition for $G[z]$
Let $G$ be a reductive group over complex numbers. Fix some maximal torus $T$. Let $\Lambda^{+}$ be the monoid of dominant coweights. It is known that one has a Cartan decomposition $$G((z))=\coprod\...
- 31
7
votes
0
answers
642
views
Automorphisms of semistable $G$-bundles
Let $X$ be a projective smooth curve over an algebraically closed field $\mathbb k$ of any characteristic. Let also $G$ be a reductive group over $\mathbb k$. (Probably) following Ramanathan, a $G$-...
- 790
11
votes
1
answer
526
views
Oesterlé's unpublished bound on Uniform Boundedness
The bound in Merel's solution to the Uniform Boundedness conjecture is not explicit, as it relies on Falting's work on the Mordell conjecture. I think this still is the case.
But there are known ...
- 17.6k
4
votes
2
answers
333
views
estimate for a sum of products of Weil's sum
Let $p$ be a prime and consider the field $\mathbb{F}_p$. Fix $f\in\mathbb{F}_p[X]$ a polynomial of degree $d\ge 2$. Define
$$
K(x,y)=\frac{1}{\sqrt{p}}\sum_{z\in\mathbb{F}_p}e_p(xz+yf(z)),
$$
where $...
- 463
1
vote
0
answers
201
views
Galois Cohomology and $\sqrt{k} \notin \mathbb{Q}$
I know it seems excessive, but I have been trying to understand the relationship between two concepts:
Galois cohomology
Fermat Descent
The first one is very abstract and I know very little about it....
- 22.8k
7
votes
1
answer
482
views
Geometry of Hecke Operators on Jacobi Forms?
In something I've been thinking about recently, the following object appears:
$$\mathcal{F}_{g} = \sum_{n=0}^{\infty} Q^{n} T_{n}\big( \phi_{2g-2}(\tau, z) \big)$$
where $T_{n}$ is the $n$-th Hecke ...
- 1,701
5
votes
0
answers
327
views
Is there a concrete way to show the existence of canonical model for non-modular Shimura curves?
I am trying to read Carayol's article on the construction of Galois representations associated to Hilbert modular forms (http://archive.numdam.org/article/CM_1986__59_2_151_0.pdf). The main geometric ...
- 2,842
4
votes
1
answer
311
views
Are there "primitive" modular functions for $\Gamma(n)$ with Fourier coefficients in $\mathbb{Q}$?
The modular curve $Y(n)$ which over $\mathbb{C}$ is $\mathcal{H}/\Gamma(n)$ is often viewed as a curve defined over $\mathbb{Q}(\zeta_n)$. However, if one twists the moduli problem to be given by ...
- 10.7k
20
votes
2
answers
2k
views
Tate's definition of residues
In http://www.numdam.org/article/ASENS_1968_4_1_1_149_0.pdf, Tate defines residues on a curve over an arbitrary field as a trace of some commutator. What is the intuition for the definition? If I knew ...
- 431
14
votes
3
answers
4k
views
Recent progress toward Birch and Swinnerton-Dyer conjecture
Has there been any progress toward the Birch and Swinnerton-Dyer conjecture after
The current status of the Birch & Swinnerton-Dyer Conjecture
- 141
7
votes
0
answers
401
views
Integrality of the mirror map -- non-GKZ examples? Counterexamples?
The mirror map in mirror symmetry is the change-of-variables between the natural coordinatizations on the two mirror sides and is typically a highly-complicated transcendental function (indeed, should ...
- 553
6
votes
2
answers
460
views
Torsion points on twists of elliptic curves and products of fine modular curves over $\mathcal{M}_{1,1}$ vs over the $j$-line
Let $Y_1(n)$ (for $n\ge 4$) be the fine moduli scheme over $\mathbb{Q}$ parametrizing elliptic curves with a rational point of order $n$. Let $\mathbb{A}^1_j$ be the $j$-line over $\mathbb{Q}$, the ...
- 10.7k
4
votes
0
answers
384
views
Parity in degrees of determinantal varieties
Let $M_{m,n}(\Bbb{C})$ be the space of $m\times n$ matrices with entries in $\Bbb{C}$, and let $U_{k,m.n}(\Bbb{C})\subset M_{m,n}(\Bbb{C})$ be the variety of matrices of rank $\leq k\leq\min(m,n)$. ...
- 43.2k
6
votes
2
answers
519
views
Seeking for a meaning: a curious symmetry
Suppose $\Phi(m,n)=(2m)!^n\prod_{k=1}^n\binom{2m+2k+x}{2k+x}$.
Then, algebraically, it is trivial to see that
$$(2m)!^n\prod_{k=1}^n\binom{2m+2k+x}{2k+x}=(2n)!^m\prod_{k=1}^m\binom{2n+2k+x}{2k+x}.$$
...
- 43.2k
11
votes
3
answers
1k
views
Why linearization leads to arithmetization?
Sorry for this question, but I think it is really important the intuition here.
Motives can be seen as the 'best' way of linearizing the study of schemes, des-composing them into "cohomological atoms"...
- 1,720
4
votes
1
answer
698
views
Reference for the proof of Langlands conjecture for $GL_n$ over function fields
Is there any reference written in English for the proof of Langlands conjecture for $GL_n$ over function fields?
- 781
6
votes
1
answer
728
views
Relation - Anabelian geometry and Tate conjecture
A lot has been said about the relation between Anabelian algebraic geometry and Mordell conjecture.
I would like to know what is the relation between Anabelian algebraic geometry and Tate ...
- 1,720
7
votes
0
answers
666
views
High dimensional analogue of Ramanujan's pi formula
The question below comes to my mind when I am trying to explore something related to the formulas found by Jesus Guillera:
a)Generalized hypergeometric function
$${}_3 F_2\left(\begin{matrix}1/4&...
- 3,337
1
vote
0
answers
183
views
Can we choose $h(\mathcal{O})$ elliptic curves such that they have same trace and endomorphism ring $\mathcal{O}$ but distinct j-invariants?
Let $\mathbb{F}_q$ be a finite field of characteristic $p$ with $q=p^a$ elements. Let $E$ be an ordinary elliptic curve defined over $\mathbb{F}_q$. We know that $\text{End}(E)\otimes\mathbb{Q}$ is ...
- 139
10
votes
1
answer
505
views
Can a index 2 subgroup of $\pm\Gamma(n)\le \text{SL}_2(\mathbb{Z})$ be noncongruence?
One way of interpreting the question might be: Is the property of being congruence a topological property? Ie, is it detected at the level of Riemann surfaces $\mathcal{H}/\Gamma$?
My motivation is ...
- 10.7k
3
votes
0
answers
329
views
Rational cohomology of formal multiplicative group
Let $\hat{\mathbb G}$ be a formal group over a field $k$, and let $V$ be a finite dimensional algebraic representation of $\hat{\mathbb G}$ (meaning we have fixed a homomorphism of algebraic groups $\...
- 18.7k
25
votes
2
answers
1k
views
Why it is difficult to define cohomology groups in Arakelov theory?
I have been puzzled by the following Faltings' remark in his paper Calculus on arithemetic surfaces (page 394) for a few months. He says:
If $D$ is a divisor on $X$, we would like to define a ...
- 6,259
7
votes
0
answers
731
views
The role of Honda-Tate theory in (Scholze's refinement of) the Langlands-Kottwitz method?
I was wondering if somebody could give me a quick primer on Honda-Tate theory, specifically the number of isogeny classes of elliptic curves, and how specifically it's utilized in (Scholze's ...
4
votes
2
answers
443
views
Counting Points on an Elliptic Curve over Finite Field $X^2Y+Y^2Z+Z^2X=kXYZ$
Let $p$ be a prime number(maybe generic large odd prime). Let $k$ be integer with $(k,p)=1$.
The equation in question is $$X^2Y+Y^2+X=kXY.$$
In homogeneous form, it is alternating
$$X^2Y+Y^2Z+Z^2X=...
- 585
1
vote
0
answers
104
views
On dimension of Segre embedding of lattice translations
Consider three lattices $L_1$, $L_2$ in $\Bbb Z^{n+1}$ and $L$ in $\Bbb Z^{2n+1}$.
Let $L_1+v_1$, $L_2+v_2$ and $L+v$ be their respective translationsfor some $v_1,v_2\in\Bbb Z^{n+1}\backslash\{(0,\...
- 13.9k
1
vote
1
answer
239
views
Local parameters and etale coverings of of elliptic curves
I found some absurd observation which I could not fix by myself. For an elliptic curve $E$ over $\mathbb{Q}$, let $\overline{E}=E\otimes\overline{\mathbb{Q}}$. Every multiplication-by-$n$ map $\...
- 21
4
votes
0
answers
267
views
Dimension of the moduli space of abelian varieties with a prescribed endomorphism algebra
Let $D$ be a division algebra over a number field $K$, and consider abelian varieties $A$ over the complex numbers, of dimension $g$ with an action of (an order of) $D$. Is it known when this set is ...
- 2,824
5
votes
0
answers
188
views
Zeta function of $\Delta[\text{det},m]$
In Geometric complexity theory the following variety $\Delta[\text{det},m]$ is crucial.
Let $X=(x_1,\ldots,x_r)$ be a tuple of $r=m^2$ variables, so that $X$ can be thought of as an $m\times m$ ...
- 2,576
28
votes
1
answer
4k
views
Arithmetic Morse theory?
Is there any analogue of Morse theory in Number theory? Naive idea arising in my head is that defining a Morse function on scheme and find etale cohomology using that function. Since I'm not an expert ...
- 2,215
2
votes
0
answers
100
views
Quasi-algebraically closed fields reference request
I am looking for a road to understanding what quasi-algebraically closed fields are with the ultimate goal of understanding the paper by Lang 1952.
My current background is the first 6 chapters from ...
- 87