All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
7
votes
0
answers
234
views
Reconstitution from reduction and tropicalization for $p$-adic varieties
For a variety $X$ over a $p$-adic field $k$, its reduction to a variety $X_p$ over residue field $k_p$ of $k$ and its tropicalization to $X_{tr}$ seem to me to be somewhat orthogonal processes.
Taken ...
user122285
7
votes
1
answer
333
views
$p$-adic lifts of tropical varieties
What is currently known about lifts of tropical varieties to varieties over $\mathbb{Q}_p$ or its extensions? Starting with an appropriate rational polyhedral cone complex what are the obstructions ...
user122285
4
votes
0
answers
262
views
$p$-adic lifts of varieties over finite fields
Fix a finite field $k_p$ of characteristic $p$, as well as a $p$-adic lift of it, $k_0$. In other words $k_0$ is a $p$-adic field and $k_p$ its residue field.
Let $X_p$ be a non-singular variety over ...
user122285
3
votes
0
answers
333
views
Philosophical question on the role of motivic cohomology
As a physicist who has read almost nothing on the beautiful theories of motives and motivic cohomology, I have some philosophical questions on motivic cohomology. I apologize first for my naive ...
- 2,971
4
votes
0
answers
273
views
Ring of Witt vectors and Fontaine's deRham period ring
The construction I am interested in is the following:
Let $V$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with algebraically closed residue field $k$. Let $K=Frac(V)$ and ...
- 455
0
votes
0
answers
184
views
What pure motives have Hodge realisations isomorphic to $\mathbb{Q}(0)$
Suppose $k$ is a number field and $\sigma:k \hookrightarrow \mathbb{C}$ is an embedding. If $\sim$ is an adequate equivalence relation, we can construct the category of pure motives with rational ...
- 2,971
5
votes
1
answer
1k
views
Misunderstanding of Hodge conjecture
I ask a question on math stack exchange about Hodge conjecture and have not got any reply or comment.
https://math.stackexchange.com/questions/2704988/clarify-hodge-conjecture
so I decide to post ...
- 2,971
10
votes
1
answer
502
views
Arithmetic representation stability and Galois action
I am thinking about representation stability phenomena (as considered by Church, Farb, Ellenberg and others) in arithmetic settings where Galois action enters the picture, and am curious about their ...
user122285
4
votes
0
answers
195
views
Geometric fundamental group and algebraically closed residue field
my questions relates to the following talk of Tsuji:
https://www.youtube.com/watch?v=2brDj26phP0
At around 10:30 of the video, Tsuji is interrupted by a man stating that his construction does not ...
- 455
4
votes
0
answers
338
views
$L$-function of induced Galois representation
Suppose $L \subset K$ are number fields and let's choose and fix an algebraic closure $\overline{L}$ of $L$ such that $L \subset K \subset \overline{L}$, hence $H:=\text{Gal}(\overline{L}/K)$ is a ...
- 2,971
7
votes
0
answers
430
views
quasi-finite group schemes and associated Galois modules
Let $p$ be an odd prime.
Let $A$ be an abelian variety over $\mathbb{Q}$ and suppose that it has semistable reduction at $p$. Let $\mathcal{A}$ denote the Neron model of $A$ over $\mathbb{Z}_p$. Now ...
- 335
4
votes
0
answers
172
views
Abelian variety associated to Hilbert modular forms
Is there any construction of Abelian variety associated to Hilbert modular forms with arbitrary nebentypus?
I am aware of the Sho-Wu Zhang's 2001 Annals paper for $\Gamma_0(N)$ under Jacquet-...
- 248
2
votes
0
answers
147
views
Genus Zero Diophantine Equations and Infinite Valuations
I'm interested in an explicit upper bound for the integral solutions of a certain genus zero curve $F(X,Y)=0$. I found some papers that address this problem:
[1] Solving genus zero diophantine ...
- 211
4
votes
0
answers
235
views
kernel of multiplication in the Neron model of an abelian variety
Let $p$ be a fixed prime.
Let $A$ be an abelian variety of $\mathbb{Q}_p$ and let $\mathcal{A}$ denote the Neron model of $A$ over $\mathbb{Z}_p$. Now, the multiplication by $p$ map in $\text{End}(A)...
- 335
19
votes
3
answers
2k
views
Bhargava's work on the BSD conjecture
How much would Bhargava's results on BSD improve if finiteness of the Tate-Shafarevich group, or at least its $\ell$-primary torsion for every $\ell$, was known? Would they improve to the point of ...
user92332
28
votes
2
answers
2k
views
When did people start thinking of elliptic curves as groups?
I have been reading some old papers of Cassels and Selmer from around 1950, and they talk about generators of rational solutions to elliptic curves, in the sense of Mordell–Weil, but do not appear to ...
- 6,039
2
votes
1
answer
84
views
reduction of torsion modules
Let $G$ be a profinite group.
Let $K(G,\mathbb Z_\ell)$ be the Grothendieck group of the derived category of finitely generated $\mathbb Z_\ell$-modules with continuous $G$-action.
Let $K(G,\mathbb ...
- 135
2
votes
0
answers
110
views
Dual sheaf of universal pointed unipotent connection and the canonical de Rham torsor
I'm trying to understand a particular point in the paper The Unipotent Albanese Map and Selmer Varieties on Curves by Kim.
We fix a basepoint $b \in X(L)$ on a curve $X$ over $L$ a characteristic 0 ...
- 121
16
votes
3
answers
1k
views
Number of solutions to polynomial congruences
Suppose I have $R$ homogeneous polynomials $F_1, ..., F_R$ with integer coefficients. Let $V$ be the affine variety defined by these polynomials over $\mathbb{C}$. I was wondering if some bound that ...
- 3,625
3
votes
0
answers
289
views
Prerequisites to read Katz's Gauss Sums, Kloosterman Sums, and Monodromy Groups
What is the minimal background one must have to read Katz's Gauss Sums, Kloosterman Sums, and Monodromy Groups?
- 1,890
2
votes
0
answers
103
views
Bound for the number of solutions to a system of congruence relations
Suppose I have $n$ polynomials in $n$ variables $G_j(x_1, \ldots, x_n)$ with integer coefficients. Let $u_j$ be some fixed $p$-adic integers.
Consider the system of congruences
$$
G_j(\mathbf{x}) \...
- 3,625
12
votes
3
answers
790
views
$K[[X_1,...]]$ is a UFD (Nishimura's Theorem)
Let us define the infinitely-many-variable formal power series ring
$$
K[[X_1,\ldots]] \colon= \underset{m \geq 1}{\varprojlim}\,K[[X_1,\ldots,X_m]].
$$
$K[[X_1,\ldots]]$ is known to be a UFD by a ...
- 563
2
votes
2
answers
509
views
Question about Zeta Function of Singular Plane Curve
I am working on a project which involves learning about the zeta function (weil zeta function) for plane curves, but I do not know much algebraic geometry. (I do not know anything about schemes).
I ...
- 53
6
votes
0
answers
200
views
Units in $B\otimes_{A}B$, where $B/A$ is a finite Galois extension of number rings
I need information or directions to the literature regarding the structure of the group of units of $B\otimes_{A}B$, where $B/A$ is a finite Galois extension of rings of integers, say associated to a ...
8
votes
0
answers
400
views
The definition of Drinfeld modules
I have an embarrassingly basic confusion about the definition of Drinfeld modules. I think that the definition of a Drinfeld module over $S$ should be "a $\mathbb{G}_a$-torsor over $S$ and ...".
...
- 2,809
-1
votes
1
answer
919
views
coprime and strictly coprime ideals
in the book "etale cohomolgy" milne is using two notions: coprime ideals and strictly coprime ideals. It seems to me that both the notions are same.
because (f(t))+(g(t))=(f(t),g(t)).
What am i doing ...
user111251
18
votes
1
answer
1k
views
How does Saito's treatment of the conductor and discriminant reconcile with an elliptic curve?
Saito (1988) gives a proof that
$$\textrm{Art}(M/R) = \nu(\Delta)$$
Here, $M$ is the minimal regular projective model of a projective smooth and geometrically connected curve $C$ of positive genus ...
- 457
4
votes
0
answers
313
views
Action of the Picard Scheme of an Elliptic Fibration
Suppose that we have a surface $X$ defined over a field $k$ (I am interested in $k$ being a number field) and an elliptic fibration $f: X \rightarrow \mathbb{P}^1$, i.e. $f$ is proper and almost all ...
- 241
5
votes
0
answers
158
views
Maass-Saito-Kurokawa Lift of Weak Jacobi Forms
Given a holomorphic Jacobi form $\varphi_{k,1} \in \mathbb{J}_{k,1}$ of weight $k$ and index 1, we know we can use the Hecke operators $V_{m}$ to lift $\varphi_{k,1}$ to a Siegel modular form
$$\...
- 1,701
6
votes
1
answer
337
views
Computing Tamagawa number of torus in Quaternion algebra
Consider a rational Quaternion algebra $M$ over $\mathbb{Q}$ that does not split at $\infty$. For example take the rational Hamilton quaternions $M=\mathbb{Q}(-1,-1)$.
For the adele ring $\mathbb{A}$ ...
- 63
18
votes
0
answers
740
views
Infinite extensions such that every elliptic curve has finite rank
The comments to this answer seem to make the following claim.
Claim. Let $K$ be the maximal abelian extension of $\mathbf Q$ that is unramified away from $p$ (more generally, away from a finite set $S$...
- 28.7k
6
votes
0
answers
369
views
What happens to Neron-Ogg-Shfarevich when characteristic of the residue field equals the prime at which Tate module is considered?
Neron-Ogg-Shafarevich criterion states that an elliptic curve $E$ over a local field $K$ has a good reduction if and only if the Tate module $T_{\ell}(E)$ is unramified for some prime $\ell$ which ...
- 3,625
3
votes
1
answer
480
views
Index one weak Jacobi forms and weakly holomorphic modular forms?
At the beginning of Eichler and Zagier's book on Jacobi Forms, there is the following diagram, summarizing part of the special role played by Jacobi forms of index 1.
One of the things confusing me ...
- 1,701
13
votes
1
answer
559
views
Minimal cardinality of a field where a polynomial has a root
Let $P(n)$ be the set of all monic polynomials of degree $n$ with integer coefficients, such that all coefficients have absolute value at most $2^n$.
Given a positive integer $n$ let us define $A(n)$ ...
- 3,897
2
votes
0
answers
249
views
An exponential sum like the Kloosterman sums
I encounter a tricky sum like the Kloosterman sum
$${\sum_{x \mod P}}^\ast e\left(\frac{ax+\overline{x}}{P}+\frac{lx^2}{P^2}\right),$$
where $l$ is a positive integer co-prime with $P$ and here $P$ ...
- 353
7
votes
1
answer
1k
views
Fields of Definition of Elliptic Curves
I am currently studying the theory of complex multiplication and I find myself confused by the language in a lot of the literature.
In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves,...
- 901
32
votes
1
answer
8k
views
$p$-adic Hodge Theory for rigid spaces, after P. Scholze
I was going over P. Scholze's paper on $p$-adic Hodge Theory for rigid analytic varieties.
This question is around the "Poincaré Lemma" in the paper.
Throughout, let $X$ be a proper smooth rigid ...
user113453
2
votes
1
answer
217
views
Diagonalising a symmetric matrix with polynomial entries
Suppose I have a symmetric $2$ by $2$ matrix $M$ whose $(i,j)$-th entry $F_{i,j}(\mathbf{x})$ belongs to $\mathbb{R}[x_1, \ldots, x_n]$ for each $i,j$. I know that for each $\mathbf{x} \in \mathbb{R}^...
- 3,625
2
votes
1
answer
196
views
Circuit Reduction on Dual Graph of an Algebraic curve
I want to compute the resistance function r(p,q) between any two vertices of a fairly complicated graph. This resistance function is the one in Admissible pairing on a curve by Shouwu Zhang, section 3 ...
- 727
12
votes
2
answers
823
views
GRH and the rank of elliptic curves
I have been using the Magma calculator recently, and while calculating ranks of elliptic curves with very big coefficients, there is a possibility to assume GRH is true, which signaficantly speeds up ...
- 3,738
0
votes
1
answer
278
views
Proof of the coherence of ${\Bbb F}_q[[X_1,\dots,X_{\infty}]]$
We shall define the infinitely-many-variable formal power series ring $A = {\Bbb F}_q[[X_1,\ldots,X_{\infty}]]$ over a finite field ${\Bbb F}_q$ as the following$\colon$
$A \colon= \underset{n \geq ...
- 563
9
votes
2
answers
2k
views
Any simple concrete proof of Faltings theorem?
Are there simple proofs of some concrete special cases of Faltings's theorem? Any help would be appreciated.
- 3,094
6
votes
0
answers
437
views
Are there infinitely many integer solutions to $a^4+b^4-c^4=N$?
Is there non-zero integer $N$ such that
$$ a^4+b^4-c^4=N \qquad (1)$$
has infinitely many integer solutions $(a,b,c)$ with $a,b \ne \pm c$?
(1) is a surface, so possible approach is to find genus 0 ...
- 25.4k
7
votes
3
answers
926
views
Lefschetz fixed-point theorem for the Frobenius map
Where can one find a proof of Lefschetz fixed-point theorem for the Frobenius map on elliptic curves over algebraic closures of $F_{p}$ ?
This could immediately follow if their coholomogies (for the ...
- 305
26
votes
1
answer
4k
views
Underlying structure behind the infamous IMO 1988 Problem 6
This is the infamous Problem 6 from the 1988 IMO which has recently been popularised by the YouTube channel Numberphile:
Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^{2} + b^{...
user117786
4
votes
0
answers
369
views
Weierstrass model of an elliptic curve: a line bundle over the base
Let $S$ be a Weierstrass model of an elliptic surface (for me it works better to understand it as an elliptic fibration), that is a map $\pi : S \to C$ where $C$ is a compact Riemann surface.
...
- 587
5
votes
0
answers
323
views
Existence of infinitely many Heegner points that are divisible by $p^{n}$ in $K_{\lambda}$
Let $E$ be an elliptic curve over $\mathbb{Q}$, $p>2$ a prime and $n,s\in\mathbb{N}$.
For $j\in\{1,...,s\}$ let $n_{j}\in\mathbb{N}$ be a natural number which may or may not be coprime to $p$.
Let $...
- 1,805
4
votes
2
answers
571
views
What are positive divisors of degree 2 on Elliptic Curve $y^2=x^3-x-1$ over $\mathbb{F}_3$?
Let (E) be an elliptic curve $y^2=x^3-x-1$ over $\mathbb{F}_3$, $a_0=1$, $a_n$ is the number of positive divisor of degree $n\geq 1$. $a_1$ in this case is the number of points of E, i.e., $a_1=1$ ...
- 41
5
votes
2
answers
252
views
Does a $K_{\upsilon}$-point of a variety $V$ give a point of $V$ in $K_{(\upsilon)}=K^{sep}\cap K_{\upsilon}$ for a global field $K$?
Let $K$ be a global field and $\upsilon$ a place of $K$. Let $K_{\upsilon}$ denote the completion of $K$ at $\upsilon$ and $K_{(\upsilon)}:=K^{sep}\cap K_{\upsilon}$ the henselization (which is the ...
- 111
12
votes
0
answers
811
views
Number field analog of Artin-Tate $\Rightarrow$ BSD?
What is the difference between the alternating product of the Hasse-Weil $L$-functions of the generic fiber of an arithmetic scheme $X\to\text{Spec}(\mathbf{Z})$ and the zeta function of $X$? (each ...
user113452