# Tagged Questions

**9**

votes

**0**answers

161 views

### Lifting Abelian Varieties to p-adic fields

Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...

**9**

votes

**4**answers

278 views

### Is any quadric birational to a product of Brauer-Severi varieties?

Let $k$ be a field with algebraic closure $\bar k$. Assume that $k$ is perfect and not of characteristic $2$ for simplicity. Let
$$X: \quad Q(x)=0, \quad \subset \mathbb{P}^n_k,$$
be a non-singular ...

**2**

votes

**0**answers

71 views

### Is the group of rational points of an anisotropic absolutely quasi-simple algebraic group over a non-archimedean local field known to be perfect?

Suppose that $G$ is an algebraic group defined over a non-archimedean local field $k$ which is absolutely quasi-simple and anisotropic over $k$. Is it known whether the group $G(k)$ is necessarily ...

**4**

votes

**1**answer

204 views

### Fermat surface known to have very few rational integer solutions

The motivation for this question is the Selmer curve, given by
$$\displaystyle 3x^3 + 4y^3 + 5z^3 = 0.$$
One can show that this curve has no rational integer solutions, despite having a solution ...

**7**

votes

**1**answer

154 views

### Hasse principle and Brauer-Manin obstruction for forms of large degree

The Hasse principle is perhaps an at-first naive generalization of the Chinese remainder theorem; that if a linear equation can be solved modulo $p$ for any prime $p$, then it can be solved in the ...

**1**

vote

**0**answers

69 views

### degree of isogenies between Jacobians and Abelian Varieties

Let $K$ be a local field of characteristic zero and positive residual characteristic. Let $A$ be a simple abelian variety and assume we have an isogeny $f:Jac_C\rightarrow A$ with $C$ a smooth curve ...

**12**

votes

**2**answers

306 views

### Existence of local sections

I would like to know when the property of "having a section" for a morphism of varieties in characteristic $0$ can be detected by spreading out to characteristic $p$.
Take a number field $K$, and let ...

**0**

votes

**1**answer

114 views

### Arithmetic property of a surface of general type

In my previous post I asked about the hyperbolicity of the affine surface $S'=\{zw \neq o\}$ in the projective surface $z^2 = P(x) Q(y)$ in $\mathbb{P}^3$, where $P$ and $Q$ are two general ...

**0**

votes

**1**answer

157 views

### Kernel of a 3-isogeny between two elliptic curves

Suppose $E_1$ and $E_2$ are two elliptic curves defined over $\mathbb{Q}$ and there exists a 3-isogeny $\varphi$: $E_1 \longrightarrow E_2$. If $E_1$ has no $\mathbb{Q}$-rational point of order 3, ...

**3**

votes

**1**answer

199 views

### What is the relation between KC and height of rational number?

Roughly speaking,Kolmogorov Complexity of a bits string or a description is the minimal length of programs outputing a bits string,and height of rational number is logarithm of the largest numerator ...

**2**

votes

**1**answer

254 views

### Curves of high genus with many rational points

The seminal theorem of Faltings confirms Mordell's conjecture: that is, curves of genus at least 2 have at most finitely many rational points. The proof of Faltings' theorem is not effective, meaning ...

**0**

votes

**0**answers

45 views

### Bounding a sum of functions defined on effective divisors

The following is from a PhD thesis "Algebraic Circle Method" by Thibaut Pugin, which I am currently reading.
Let $k$ be a finite field of order $q$.
Let $X \subseteq \mathbb{A}^{n+1}_k$ be the ...

**1**

vote

**0**answers

114 views

### Ratio of periods for elliptic curves in an isogeny class

Let $E \to E^\prime $ be an isogeny of elliptic curves defined over $\mathbb{Q}$. Then what is the definition of ratio of periods for elliptic curves in the isogeny class and how to calculate the ...

**3**

votes

**0**answers

113 views

### Question related to the number of rational curves on a hypersurface

The following is from a PhD thesis "Algebraic Circle Method" by Thibaut Pugin, which I am currently reading.
Let $k$ be a finite field of order $q$.
Let $\mathfrak{X} \subseteq \mathbb{P}^n_k$ be ...

**4**

votes

**0**answers

118 views

### Continuity of the Hilbert pairing

I would like to know if the Kummer pairing (or the analogue of the Hilbert Symbol) for a one dimensional group defined over the ring of integers of a higher-dimensional local field is continuous (with ...

**2**

votes

**0**answers

124 views

### Number of rational curves on varieties over finite fields

Let $k$ be a finite field. Let $P_r$ be set of degree $r$ binary forms
over $k$.
We define
$$
\mathcal{M}_r = \{ (x_0, ..., x_n) \in P_r^{n+1} : x_0^d + ... + x_n^d = 0 \text{ and the }x_i \text{'s ...

**5**

votes

**1**answer

488 views

### Main conjecture for elliptic curves

Suppose that $E$ is an elliptic curve defined over $\mathbb{Q}$, and that $p$ is a prime where $E$ has good ordinary reduction. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ ...

**0**

votes

**0**answers

90 views

### Consistency of the u-invariant under field extension

A algebraic field extension L/k induces of homomorphism between the Wittrings. We get
$\phi: W(k) -> W(L)$. If every anisotropic isometry class of $W(k)$ stays anisotropic, the kernel of $\phi$ ...

**7**

votes

**2**answers

320 views

### Forms of algebraic varieties

Let $X$ be an algebraic variety (say, projective, irreducible and smooth), defined over a field $K$, and let $L$ be a Galois extension. I am interested in algebraic varieties $Y$, defined over $K$, ...

**10**

votes

**1**answer

282 views

### Degeneration of the Hodge spectral sequence for scheme over truncated Witt ring?

Let $f:X\to S$ be a smooth proper morphism of schemes. If $S$ is of characteristic zero (i.e., $S$ is a $\mathbb{Q}$-scheme), then Deligne has shown:
The Hodge-De Rham spectral sequence ...

**10**

votes

**1**answer

415 views

### Examples of elliptic curves over $\mathbb{Q}$

I need examples of two non-isogenous elliptic curves $E_{1}, E_{2}$ over $\mathbb{Q}$ having the following 2 properties -
1) $E_{1}, E_{2}$ have no rational torsion points.
2) $E_1[9] \cong E_2[9]$ ...

**4**

votes

**1**answer

160 views

### What is the interpretation of this galois cohomology set?

Let $K$ be a field of characteristic zero. Let $G_K:=Gal(\bar{K}/K)$
The nontrivial elements of the set $H^1(G_K,PGL_2)$ correspond to $\bar{K}/K$-forms of $\mathbb{P}^1$; i.e. curves that are ...

**1**

vote

**0**answers

60 views

### Unbranched cover of a curve of CM type

Let $C$ be a curve of genus $g\geq 2$ over complex number. Assume that $C$ has complex multiplication (CM).
Does there exist such a curve $C$ such that $C'$ is also of CM type for any unbranched ...

**3**

votes

**1**answer

116 views

### On Universal Abelian surfaces over a Shimura curve.

Let ${\cal O}, {\cal O}'$ be two order in ${\mathrm M}_2({\Bbb R})$ that are sets of all $2 \times 2$ matrices over real number ${\Bbb R}$. Assume that we have the relation ${\cal O}' = a{\cal ...

**7**

votes

**0**answers

399 views

### Looking for a paper of Hartshorne

In a famous paper
Hartshorne - Varieties of small codimension,
Hartshorne formulates a conjecture, which roughly says that varieties of small codimension in projective space are complete ...

**4**

votes

**1**answer

135 views

### Conductor CM abelian variety

This is probably well known but I am not an expert in the subject.
Given an abelian variety $A$ of dimension $g$ with CM by $O_K$ where $K$
is a CM field of degree $2g$, let $N_A$ be the norm of the ...

**1**

vote

**0**answers

125 views

### Zariski density of Q-bar points

Let $X \subset \mathbb{C}^n$ be an affine variety which is defined over $\mathbb{Q}$ (i.e. the zero set of some finite collection of polynomials with coefficients in $\mathbb{Q}$). Define $X'$ to be ...

**2**

votes

**1**answer

138 views

### How to obtain the Period matrix from the Igusa Invariants of a genus two curve?

I am looking for an algorithm to obtain the period matrix tau= (tau1 & tau12\ tau12 & tau2) from the Igusa invariants of a genus two curve. More precisely:
I consider a family of genus two ...

**4**

votes

**1**answer

314 views

### Basics on anabelian geometry and Grothendieck's section conjecture

Even I can find similar questions and some answers on that questions, most of them are not quite unsatisfactory to me. Maybe this is a very stupid question, but there is no other place that I can ask ...

**7**

votes

**2**answers

310 views

### Is there a largest prime p such that J_0(p) completely splits into elliptic curves

The question in the title is related to a more general question. Namely does there exist an integer $N$ such that for all curves $C/\mathbb C$ of genus $> N$ one has that not all simple isogeny ...

**0**

votes

**1**answer

166 views

### how to see CM types as functions on the Galois group?

Let $K$ be a CM field, that is, an imaginary quadratic extension of a totally real number field. Its degree $[K: \mathbb{Q}]$ is a en even number $2n$.
(1) For me a CM type is a subset $\Phi \subset ...

**1**

vote

**0**answers

86 views

### another question on the Manin-Drinfeld theorem

A few days ago I asked a question about possible higher dimensional generalizations of the Manin-Drinfeld theorem. Let me come back to the classical statement.
It says that a divisor on the modular ...

**2**

votes

**1**answer

166 views

### A problem in intersection theorem

I'm reading the paper:
SGA 7 II, Intersections sur les surfaces regulieres.
In Papge 6 , I cannot understand why there is sign $-1$ in the formula (1.10.4):
Let $S$ be a trait, for any $\mathcal ...

**4**

votes

**1**answer

220 views

### Torsors and the fpqc topology

Fix a scheme $S$, a group scheme $G/S$ (let us say smooth, maybe even affine with some finiteness conditions if you like), and suppose I have some other $S$-scheme $P$ with a right $G$-action. We want ...

**11**

votes

**3**answers

594 views

### Why study CM abelian varieties?

I know that abelian varieties of CM type have central importance in algebraic geomtry and number theory. There are many conjectures and concepts related to them like Andre-Oort, Coleman conjecture, ...

**2**

votes

**1**answer

247 views

### Pencil with desired Jet in Algebraic geometry(new!)

Let $k$ be an algebraic closed field.
Let $n$ be a positive integer.
Let $X$ be an irreducible, proper and smooth scheme over $k$ with an immersion $i:X\hookrightarrow E:=\mathbb P^N_k$ with $N$ ...

**1**

vote

**1**answer

172 views

### Weierstrass points on modular curves

What is knowns about Weierstrass points on modular curves? Are there any explicit formulas of them, or any information about Weierstrass gaps? I am interested in (compactifications of) the quotients ...

**2**

votes

**1**answer

171 views

### A problem on Jets in algebraic geometry

Let $k$ be a perfect field, let $n$ and $m$ be two positive integers.
Consider $X=\mathbb P_k^n\times \mathbb P_k^m$. Let $x_0=(1,0,\cdots,0;1,0,\cdots,0)\in X$ be fixed.
For any pair of integers ...

**3**

votes

**0**answers

99 views

### Curves on hypersurfaces generated by diagonal sums

This is related to an earlier question of mine ((Non-)Existence of curves of low degree on affine and projective varieties). It seems that the question is too difficult for specific surfaces, although ...

**2**

votes

**2**answers

538 views

### L-functions and algebraic geometry

Robert Langlands commented in a letter to Deligne that perhaps some of the deepest problems of algebraic geometry lie in L-functions. I want to understand the general philosophy and the connection ...

**11**

votes

**3**answers

289 views

### (Non-)Existence of curves of low degree on affine and projective varieties

I am interested in papers that investigates the existence or non-existence of curves of low degree (relative to the degree of the ambient variety). The starting example is that of surfaces and ...

**10**

votes

**0**answers

294 views

### Katz--Mazur for abelian varieties

Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.
Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac ...

**3**

votes

**1**answer

286 views

### Automorphisms of complete discrete valuation ring

Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb ...

**1**

vote

**2**answers

319 views

### A neat monodromy group of a family of Kaehler manifolds

Let $X\rightarrow B$ be a family of Kaehler manifolds with possibly singular fibers. Let $G$ be the monodromy group on $H^n(X_b,\mathbb{Z})$, where $n=\dim X_b$ with the smooth fiber $X_b$ over some ...

**6**

votes

**1**answer

322 views

### What is the motivation for defining the conductor of an abelian variety?

Let $K$ be a $p$-adic field, and let $A$ be an abelian variety over $K$. The conductor of the abelian variety is often defined as $2u+t+\delta$, where $u$, $t$ and $\delta$ are invariants related to ...

**2**

votes

**1**answer

240 views

### Mordell-Weil and finiteness of rational points

Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order, that is, $\mathrm{End}_K(E)\cong \mathcal{O}_K$. Suppose the class number of $K$ is equal to $1$. Let ...

**2**

votes

**1**answer

204 views

### Is Mazur's deformation ring R integral?

Consider the absolutely irreducible Galois representation
$\overline{\rho} \colon G_{\Bbb Q} \to {\mathrm{GL}}_2({\Bbb F}_p)$. We apply the Mazur's deformation theory on the lift ${\rho} \colon ...

**0**

votes

**0**answers

113 views

### Action on C[[X,Y]]/f(X,Y) giving complete intersection quotients

Let $R \colon\!= {\Bbb C}[[X,Y]]/(f(X,Y))$ be a complete local ring of Krull-dimension $1$. Assume that we have an action of $\Bbb Z$ on $R$ such that fixed elements by $\Bbb Z$ in $R$ are only ...

**0**

votes

**1**answer

218 views

### Iwasawa theory for Mazur's deformation ring R

The ideal class group $\mathrm{Cl}({\cal O}_K)$ and Mazur's deformation ring $R(\overline{\rho})$ for a number field $K$ are said to be similar to each other.
Let ${\Bbb Q}_{\infty}$ be the unique ...

**9**

votes

**1**answer

442 views

### The Sato-Tate conjecture for hypersurfaces?

The Sato-Tate conjecture for elliptic curves $E$ predicts the distribution of the eigenvalues of Frobenius at $p$ on the Tate module of $E$ as $p$ varies in terms of the distribution of the ...