Questions tagged [arithmetic-geometry]
Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
2,047
questions
4
votes
2
answers
329
views
$p$-adic series bounded if and only if it has finitely many zeros
Let $L\subseteq\mathbb{C}_p$ be a finite extension of $\mathbb{Q}_p$, $r$ be a positive real number, and $f$ be a series $\sum_{n\in \mathbb{Z}} a_nz^n$ convergent in $D= \{x\in \mathbb{C}_p|0<v(x)\...
- 40.2k
0
votes
1
answer
240
views
Fundamental group of a smooth projective curve of char $0$
In this note of Akhil MATHEW, when he proves the fundamental group of a smooth projective curve over a algebraic closed field $k$ of characteristic $0$ admits $2g$ topological generators, there are ...
CommunityBot
- 1
2
votes
0
answers
245
views
Rigid analytic geometry and Tate curve
I am stuck in the proof of theorem 5.1.4 in the book rigid analytic geometry and its applications on page 126. The authurs define $\Gamma:=G^{an}_{m,k}/<q\gt$ where $k$ is a complete non-...
CommunityBot
- 1
2
votes
0
answers
278
views
Fiber of normalisation morphism
Let $X$ be an integral, excellent scheme and $\eta: \widetilde{X} \to X$ be its normalization. If $x \in X$ is a closed point there is the following powerful tool (from EGA, Ch IV, 7.8.3, vii) to find ...
- 617
4
votes
0
answers
261
views
Generalized Tate Conjecture
I have seen a statement of the generalized Tate conjecture over finite fields (see for example page 4 of Milne's "The Tate conjecture over finite field" (https://jmilne.org/math/articles/2007e.pdf)).
...
- 726
7
votes
1
answer
211
views
Subfields of Hilbertian fields
This question is about the Remark on the top of page 22 of Serre's Topics in Galois Theory, available here :
http://www.ms.uky.edu/~sohum/ma561/notes/workspace/books/serre_galois_theory.pdf
My ...
- 19.3k
5
votes
0
answers
457
views
A functor on Abelian varieties corresponding to this operation on Weil numbers
Let $A/\mathbb F_q$ be an abelian variety over a finite field with Weil numbers $q^{1/2}\alpha_1,\dots,q^{1/2}\alpha_n$.
Consider the numbers $q^{d/2}\alpha_1,\dots,q^{d/2}\alpha_n$. These are still ...
- 7,646
7
votes
2
answers
2k
views
Faltings height of a CM abelian variety
Let A be a CM abelian variety, say simple of dimension g, with $End(A) = O_K$, where $K$ is a CM field
of degree $2g$.
Is there an upper bound for the Faltings height $h(A)$ in terms of the ...
- 6,141
3
votes
0
answers
312
views
Integral points on affine varieties
Consider Siegel's theorem. It says that for a smooth affine algebraic curve $C$ over $\mathbb{Q}$ such that $g(C)>0$ any model $\mathcal{C}$ of $C$ over $\mathbb{Z}$ has finitely many $\mathbb{Z}$-...
CommunityBot
- 1
0
votes
0
answers
201
views
Endomorphism rings of flat group schemes
Let $R$ be a commutative ring and $X$ be a flat $R$-group scheme.
We call $\text{End}_R(X)$ the ring of endomorphisms of the $R$-group scheme $X$, defined over $R$.
Let $R\to S$ be a ring map ...
CommunityBot
- 1
3
votes
0
answers
262
views
A complete Tate Huber ring is Banachizable (maybe not)?
I have questions of technical nature.
A complete Tate Huber ring is a complete topological (commutative) ring $A$ admitting an open subring $A_0$ whose topology is the $\varpi A_0$-adic topology, for ...
- 5,492
2
votes
0
answers
248
views
Which fields and schemes "have enough finite residue fields"?
I am looking for assumptions on the spectrum $S$ of a field $K$ that ensure the following: there exists an excellent noetherian finite dimensional (integral) scheme $S'$ such that $S$ is its generic ...
- 16.5k
5
votes
0
answers
306
views
motivations of classifying $p$-divisible groups
Let $k$ be a perfect field of characteristic $p>0$ and $W:=W(k)$ is the witt ring. Let $K$ be a totally ramified extension of $K_0:=W(\frac{1}{p})$ and $\Lambda:=W[[u]]$ is the formal series ring ...
CommunityBot
- 1
3
votes
0
answers
85
views
Self-contained reference for projective embedding of moduli of polarized abelian varieties via modular forms
I've been working on reading and understanding Arakelov's '71 paper and he uses the fact that the moduli space of complex abelian varieties of dimension $g$ with polarization of degree $d$ admits an ...
- 842
3
votes
0
answers
199
views
Endomorphisms of elliptic curves, resp formal groups
Let
$E$ be an elliptic curve over a number field $K$,
$\mathcal{E}^w$ a fixed Weierstrass model for $E$ over $R := \mathbf{Z}[a_1,\ldots, a_6]$,
$\mathcal{E}$ the Néron model of $\mathcal{E}$ over ...
CommunityBot
- 1
1
vote
0
answers
88
views
Algebraic definition of the "pseudo complement" of algebraic curve
Not sure if this makes sense.
Let $K$ be field and $C : f(x,y)=0$ algebraic curve curve over $K$.
Define the "pseudo complement" $\hat{C}$ to be the rational surface
$z f(x,y) - 1=0$ with ...
- 24.2k
26
votes
4
answers
1k
views
Variety acquiring rational point over any quadratic extension
Does there exist a variety $X$ over $\mathbb{Q}$ (or a number field) such that it has no rational points over $\mathbb{Q}$ but acquires points over any quadratic extension $\mathbb{Q}(\sqrt{d})$?
If ...
- 44.9k
2
votes
1
answer
394
views
Explicit semi-stable theorem for elliptic curves over $p$-adic fields
In this paper of Maja Volkov, the authur metions a number called "défaut de semi-stabilité" on page 9. It is defined as $\text{dst}(E)=\frac{12}{\text{pgcd} (12,v_p(\Delta_E))}$ where $E$ is ...
CommunityBot
- 1
4
votes
1
answer
382
views
Motivations of families of modular forms, elliptic curves and Galois representations?
I want to know some reference, why do some number theorists study the families of the elliptic curves, modular forms or Galois representations? As far as I know, I always consider the Galois ...
CommunityBot
- 1
6
votes
2
answers
452
views
Points on hyperelliptic curves: $y^2=5(x^2-3)(x^2+2)(x^2-11/5)$
González-Jiménez and Xarles studied a problem in Diophantine number theory and they obtained several nice results via elliptic curve Chabauty's method over quadratic number fields. At page 73 in paper ...
- 11.4k
48
votes
4
answers
4k
views
Fermat's last theorem over larger fields
Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here $\...
- 1
4
votes
1
answer
202
views
Commensurator of a subgroup of matrices
Let $k$ be a totally real number field and let $\mathcal{O}_k$ denote its ring of integers. If $H$ is a subgroup of $\text{GL}(n, \mathbb{R})$ let denote with $H(k)$ and $H(\mathcal{O}_k)$ the ...
- 3,349
8
votes
1
answer
444
views
The product of two supersingular elliptic curves is independent of which ones we pick
In a comment on this MO question, Qing Liu says "In positive characteristic p, if you take two supersingular elliptic curves $E_1,E_2$, then $E_i×E_j$ is isomorphic to $E^2_1$ for any pair $i,j$."
...
- 4,745
5
votes
1
answer
400
views
Fourier coefficients of Siegel Eisenstein series
I am looking for reference about Fourier coefficients of Eisenstein series. Currently I am mainly interested Eisenstein series given by Siegel parabolic subgroup of $SP_{2n}$ and $U(n,n)$. Let's ...
CommunityBot
- 1
6
votes
1
answer
466
views
Strengthened supercongruences for Ramanujan-type formulas for $1/\pi^k$
The question below is again a follow-up of an old question.
Motivation: Zhi-Wei Sun listed a number of supercongruences attached to Ramanujan-type $1/\pi$ formulas in the arXiv paper which can be ...
- 704
2
votes
0
answers
267
views
Étale group scheme exact sequence
Consider the exact sequence of finite flat group schemes over the $2$-adic integers ring $\mathbb{Z}_2$:
$$0\longrightarrow\mathbb{Z}/2\mathbb{Z}\longrightarrow A\longrightarrow\mathbb{Z}/2\mathbb{Z}\...
CommunityBot
- 1
16
votes
3
answers
1k
views
Is Multilinear Hilbert's tenth problem version undecidable?
A multilinear polynomial $f\in\mathbb Z[x_1,\dots,x_t]$ has terms only of form $$b\prod_{i=1}^tx_i^{a_i}$$ where $a_i\in\{0,1\}$ and $b\in\mathbb Z$.
Is there no general purpose algorithm for ...
- 6,159
2
votes
0
answers
233
views
Standard application of Oort-Tate classification theorem
$\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\tors}{tors}$In Mazur's paper “Modular curves and the Eisenstein ideals”, on the bottom of page 159, it says that if $T$ is a open subscheme of $...
CommunityBot
- 1
3
votes
1
answer
416
views
References for the early history of Fontaine's tilting construction
Scholze attributes the tilting construction for perfectoid rings to Fontaine, who calls it "a classical construction in $p$-adic Hodge theory".
Would anyone happen to know an early reference where ...
- 10.3k
2
votes
0
answers
207
views
Lift the relative Frobenius automorphism to zero characteristic
Let $X$ be a algebraic variety of finite type over $\mathbb{Z}$. Let $\mathcal{F}$ be a foliation in codimension one over $X$. Let $X_p$ and $\mathcal{F}_p$ be the reductions modulo $p$ of $X$ and $\...
- 527
6
votes
0
answers
187
views
Gromov-Witten theory over any field
In this question, VA. said the stack $\overline{\mathcal{M}}_{g,n}(X,\beta)$ has been constructed over $\mathbb{Z}$. I can not find anything about it but I am quite interested in it. Is there any ...
- 27.4k
1
vote
0
answers
276
views
Smooth absolutely irreducible (?) genus 1 plane pointless curve over $\mathbb{F}_{13}$
We got a family of genus 1 plane curves that may violate a bound
in a paper.
Explicitly: Let $F(x,y)$ be the degree 39 polynomial with integer coefficients:
...
4
votes
0
answers
229
views
Why does $\theta: \mathbb{B}^+_{dr} \rightarrow \mathbb{C}_p$ have no continuous or equivariant section?
Fix a $p$-adic field $K$ with perfect residue field $k.$ Let $\mathbb{C}_K$ be the completion of the algebraic closure of $K,$ and let $$R = \varprojlim \mathbb{C}_K/p,$$ where the transition maps in ...
- 197
4
votes
0
answers
942
views
Next step in studying arithmetic geometry
This relates to this post.
I want to study arithmetic, such as Fermat's last theorem, Faltings' theorem, Mazur's torsion points theorem, Weil conjecture and so on.
For understanding these theorems (...
- 1,352
3
votes
0
answers
156
views
Field of definition for sheaves
What follows could be formulated for more general extensions than $\mathbb{R}\rightarrow\mathbb{C}$ but I'll stick to this particular case for now. Further, I am somewhat new to this language and I'm ...
CommunityBot
- 1
1
vote
0
answers
102
views
An irrational complete intersection surface with good reduction everywhere
Do there exist 3 absolutely irreducible homogeneous polynomials in $\mathbb{Z}[a, b, c, d, e, f]$ such that
each one defines a hypersurface in $\mathbb{P}^5_{\mathbb{Z}}$ smooth over $Spec(\mathbb{Z})...
- 11
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.
2
votes
0
answers
497
views
Fontaine - Wintenberger field of norms and imperfect case
Let $K$ be a complete discrete valued field whose residue field $k_K$ has characteristic $p$ and has the property that $[k_K:k_K^p]=p^d$ for some $d$. Let $t_{\alpha}, 1 \leq \alpha \leq d$ be a set ...
- 313
3
votes
1
answer
274
views
Bounds on the size of isogeny classes (over number fields)
I am reading B. Mazur's seminal paper "Rational isogenies of prime degree" (Invent. Math. 44 (1978), 129-162), and Theorem 5 of this paper caught my attention; it states that there exists an absolute ...
- 2,481
14
votes
1
answer
691
views
What numbers are not represented by $5xy+2x+2y$?
What numbers are not represented by $5xy+2x+2y$? Do they have a positive density?
This came up for me while investigating some cases here. Here's what I've found:
All evens are represented with $x=0$...
- 77.5k
5
votes
1
answer
346
views
Curves with isogenous Jacobians
Suppose that $C_1, C_2$ are two curves of genus $g \geq 2$ defined over a number field $K$. Let $J_1, J_2$ respectively be their Jacobians. Suppose that $J_1, J_2$ are isogenous over $K$ and $C_1(K), ...
- 4,467
2
votes
0
answers
156
views
Weighted projective lines and elliptic curves
The modular curves of low level can sometimes be describes as weighted projective lines. For example, over $\mathbb{Z}[1/2]$ the compactified stack of elliptic curves with full level 2 structure is ...
CommunityBot
- 1
6
votes
4
answers
718
views
Texts on moduli of elliptic curves
I want to study FLT (Fermat's Last Theorem), and now I'm studying moduli of elliptic curves.
I've heard that Deligne-Rapoport, Katz-Mazur, Mazur's "Modular curves...", and Katz's "p-adic..." are very ...
- 8,341
2
votes
1
answer
389
views
What is the relationship between the sheaf-function dictionary and cohomology of moduli spaces of shtukas?
I'm a newcomer to the geometric Langlands setting, and have mostly consulted surveys like Laumon's overview of L. Lafforgue's proof or Frenkel's recent advances survey, so apologies if this is ...
- 41
3
votes
1
answer
335
views
Computing Mordell-Weil Groups without Rational Torsion
Summary: How does one compute the Mordell-Weil group of an elliptic curve $E / \mathbb{Q}$, in the case where the torsion points are only defined over larger fields?
More detail: I've been reading ...
- 8,341
2
votes
0
answers
264
views
Which endomorphisms of the Tate module of an abelian variety are "algebraic"?
For an abelian variety $A$ over a field $k$ with characteristic different from $\ell$ and Galois group $G = Gal(\overline k/k)$, there is always an injective map of the form:
$$\mathbb Q_\ell\otimes ...
- 36.2k
4
votes
0
answers
95
views
Eta quotient and order of a cuspidal divisor
Let $X_0(N)$ be the modular curve associated to a congruence subgroup $\Gamma_0(N)$. If $N=p$ is a prime, then there are two cusps $0$ and $\infty$ on $X_0(N)$. Suppose that $p>7$ so that the genus ...
- 136
3
votes
0
answers
248
views
Rationality of Eisenstein series g2 and g3 for elliptic curves defined over numberfields
Let $K$ be a number field and let $E/K$ be an elliptic curve. (Fix an embedding of $K$ into the complex numbers $\mathbb{C}$). Let $\eta$ be the invariant differential of $E/K$. Let $\omega_1$ and $\...
- 139
3
votes
2
answers
777
views
Reference for Using Group Cohomology to calculate Etale Cohomology
I'm looking for a reference for the following statement:
Let $X$ be a variety (over an algebraically closed field $k$), and let $F$ be a locally constant etale sheaf. Let $x \in X(k)$. Then
$ \mathrm{...
- 139
0
votes
0
answers
114
views
Isomorphism between 2nd symmetric product and Jacobian
Let $X=X_0(N)$ be hyperelliptic with $g(X)\geq 2$ with $\infty$ as a cusp and $\iota$ as the hyperelliptic involution. Then the map
$$X^{(2)} \longrightarrow Jac(X)$$ $$D \longrightarrow [D-\infty -\...
- 39