9
votes
0answers
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
4answers
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
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
2answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
1answer
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
3answers
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
1answer
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
1answer
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
1answer
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
0answers
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
2answers
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
3answers
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
0answers
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
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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 ...