# Tagged Questions

**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 ...

**11**

votes

**3**answers

592 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, ...

**10**

votes

**0**answers

291 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 ...

**6**

votes

**1**answer

321 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 ...

**8**

votes

**1**answer

262 views

### What is the structure of the group of rational points of an abelian variety over a Laurent series field?

Let $K = \mathbb{F}_q((t))$, and let $A_{/K}$ be a nontrivial abelian variety. Then $A(K)$ is a compact $K$-adic Lie group. What can be said about its structure?
By way of comparison, if ...

**2**

votes

**0**answers

94 views

### Subgroups-ideals correspondence for abelian varieties over $\mathbf{F}_p$

The question I have arose while reading Waterhouse's Thesis (Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521–560.), and motivates another question I recently asked.
...

**2**

votes

**1**answer

171 views

### Purely additive reduction of Jacobian of Hyperelliptic curve

For general, let X be an abelian variety of dimension g.
We say that X has 'purely additive reduction' at prime p if the dimension of the unipotent radical of the special fiber of the Neron Model of ...

**2**

votes

**1**answer

190 views

### Duality for rank one modules over a number ring

Let $K$ be a number field, and $R$ an order of $K$. Consider the category $\mathcal{M}$ of all finitely generated $R$-submodules of $K$. If $X$ is an object of $\mathcal{M}$ such that ...

**5**

votes

**0**answers

262 views

### A frustrating cohomology class on the moduli of abelian surfaces

Here's a very frustrating question that I have been stuck on for some time. I believe that my question could fit in a general framework of what happens when you restrict $L^2$-cohomology classes on a ...

**4**

votes

**1**answer

252 views

### Conductor of abelian varieties

Let $A$ be a non-zero abelian variety defined over a number field $F$. Let $v$ be a finite place of $F$, and let $f_v(A)$ be the usual conductor exponent of $A$ at $v$ (defined e.g. on p.500 of the ...

**22**

votes

**3**answers

1k views

### Products of primitive roots of the unity

Let $m>2$ be an integer and $k=\varphi(m)$ be the number of $m$-th primitive roots of the unity. Let $\Phi = \{ \xi_1, \ldots, \xi_{k/2}\} $ be a set of $k/2$ pairwise distinct primitive $m$-th ...

**0**

votes

**0**answers

77 views

### field of definition of abelian varieties with extra endormorphism

Let $A$ be a complex abelian variety such that $\mathrm{End}(A)$ is strictly bigger than $\mathbb{Z}$.
Question: Is is true that $A$ is defined over $\overline{\mathbb{Q}}$?
This is of course what ...

**1**

vote

**0**answers

180 views

### a reference for Kummer theory, with proofs ?

What is a standard reference for Kummer theory of semi-Abelian varieties ?
I need a complete exposition with detailed proofs. Also in prime characteristic,
although I am not sure what the statement ...

**6**

votes

**1**answer

223 views

### Local Norm Mapping for Abelian Varieties

Let $A/K$ be an abelian variety defined over a nonarchimedean local field $K$ of characteristic $0$ and let $L$ be a finite extension of $K$. Consider the norm map $$A(L)\xrightarrow{N_{L/K}}A(K)$$ I ...

**2**

votes

**1**answer

208 views

### Is the canonical height of a totally p-adic point on an abelian variety bounded away from zero?

Inspired by the result of Schinzel and Smyth that a totally real number other than $0$ and $\pm 1$ has height at least $\frac{1}{2}\log \Big( \frac{1+\sqrt{5}}{2} \Big) = 0.240659\ldots$, Bombieri and ...

**7**

votes

**0**answers

351 views

### Points of minimum Arakelov height and harmonic arithmetical varieties

Added. (28/2) To put it less pompously (and more vaguely, less concretely), I wanted to relate the impression that it is the general rule that an Arakelov (i.e., geometric) height on an arithmetical ...

**12**

votes

**1**answer

389 views

### The torsion point count in higher dimension

It is an easy consequence of the Serre open image theorem that for the torsion point count on elliptic curves, the following possibilities arise.
If $E/\bar{\mathbb{Q}}$ is an elliptic curve without ...

**10**

votes

**0**answers

282 views

### Average ranks of abelian surfaces

Most people nowadays believe that over a fixed global field, $50$% of the elliptic curves have $0$ rank, $50$% have rank $1$, and $0$% have higher rank. A significant advance in this direction has ...

**9**

votes

**2**answers

492 views

### What are some consequences of the Mumford-Tate conjecture?

Let $A/K$ be an abelian variety over a number field. On the one hand we have the singular cohomology group
$$V := H^1(A(\mathbb{C}),\mathbb{Q})$$
with respect to some fixed embedding $K \subset ...

**3**

votes

**0**answers

147 views

### Ordinary vs Non-ordinary for GL(2)-type Abelian Surfaces over Q

Let $A_f$ be an abelian surface over $\mathbf{Q}$ of $\mathbf{GL}_2$-type arising from a weight $2$ cuspidal eigenform $f\in S_2(\Gamma_0(N))$. What is known (or expected to be true) for the size of ...

**1**

vote

**0**answers

178 views

### Canonical forms for elliptic fibrations with Mordell-Weil group of rank 1 and zero torsion

Consider an elliptic fibration given by the following Weierstrass model:
$$
E: y^2 + a_1 x y + a_3 y =x^3 + a_2 x^2 + a_4 x + a_6,\quad a_6=a_2 a_4.
$$
( I work with characteristic zero).
With the ...

**4**

votes

**0**answers

245 views

### Dieudonné modules over rings of charateristic zero

Dear Colleagues,
would appreciate if you could recommend references, if such a theory exits, for the following question.
Let $A$ be an Abelian scheme over $\text{Spec}(R)$, where $R$ is a subring of ...

**7**

votes

**1**answer

376 views

### On simple factors of modular jacobians: endomorphism ring and simplicity of mod p reduction

Let $A_f$ be the abelian variety over $\mathbf{Q}$ arising as a $\mathbf{Q}$-simple factor of the Jacobian $J_0(N)$ of the modular curve associated to a normalized newform $f$ of weight $2$ on the ...

**17**

votes

**3**answers

1k views

### Over which fields does the Mordell-Weil theorem hold?

According to a well-known theorem of Mordell, the group of rational points $E(\mathbf{Q})$ of an elliptic curve $E/\mathbf{Q}$ is finitely generated. Weil generalized this theorem to abelian varieties ...

**1**

vote

**0**answers

132 views

### Construction of RM abelian variety from eigenform

Let $f$ be a normalized eigenform of weight $2$ level $N$. If the Fourier coefficients of $f$ generate a totally real field $F$, then we associate to $f$ a system of $\ell$-adic Galois representations ...

**1**

vote

**0**answers

264 views

### Component group of Neron model of a parametrized abelian variety

Let $A$ be an abelian variety of dimension $2$ over a $p$-adic field $K$ with (additive) valuation $v$. Assuming $A$ has multiplicative reduction, the theory of $p$-adic theta functions gives us an ...

**2**

votes

**1**answer

257 views

### Weil reciprocity on abelian varieties and biextensions?

I was once told, by someone who would likely be right about such things, that the version of Weil reciprocity for abelian varieties (as in Lang, Abelian Varieties) should come out of consideration of ...

**3**

votes

**1**answer

185 views

### upper bounds for the ranks of the minus parts of modular jacobians

Let $p$ be a prime, and $J^-(p)$ be the maximal quotient of the Jacobian of the modular curve $X_0(p)$ on which the involution acts by $-1$.
Is anything known or conjectured about upper bounds for ...

**3**

votes

**2**answers

360 views

### Quotients of Tate modules

Let $p$ be a prime number, let $K$ denote a finite extension $\mathbb{Q}_{p}$ and let
$\overline{K}$ be an algebraic closure of $K$. Let $A$ be an ellitpic curve over
$K$ and denote by $T_{p}A$ its ...

**4**

votes

**1**answer

305 views

### Albert classification of rational endomorphism rings of simple Abelian varieties over finite fields

Recall the Albert classification of rational endomorphism rings with involution of simple Abelian varieties over arbitrary fields:
Type I: totally real, trivial involution
Type II and III: ...

**4**

votes

**0**answers

222 views

### false elliptic curves and principal polarizations

Hi,
Let $\Delta$ be a quaternion algebra over $\mathbf Q$ and let $\mathcal O_\Delta$ be a maximal order in $\Delta$.
Recall that a false elliptic curve over a field $K$ is a pair $(A/K,i)$ ...

**3**

votes

**1**answer

307 views

### Construction of Kummer map for abelian variety

Let $A$ be an abelian variety over the rational numbers $\mathbf{Q}$. Let $V=T_p A \otimes \mathbf{Q}_p$ be the $\mathbf{Q}_p$-Tate module of $A$. Let $G$ be the absolute Galois group of $\mathbf{Q}$. ...

**2**

votes

**1**answer

429 views

### Generalization of singular moduli

$j$-invariants of CM curves $E$ over (say) the complex numbers are known as singular moduli. As the theory of complex multiplication explains, singular moduli are algebraic integers of great ...

**14**

votes

**0**answers

380 views

### Torsion points of abelian varieties in the perfect closure of a function field

The following is a problem, which was recently brought to my attention by H. Esnault and A. Langer.
Let $K$ be the function field of a smooth curve over the algebraic closure $k$ of the finite field ...

**3**

votes

**0**answers

502 views

### level structures and moduli of abelian varieties

Hello,
In the definition of level structure of level $n$ for an elliptic curve $A$, there are two versions:
an isomorphism of group schemes $(\mathbf Z/n\mathbf Z)^2 \to A[n]$.
an isomorhpism of ...

**5**

votes

**1**answer

288 views

### O-linear Weil-pairing on abelian varieties with real multiplication

Let $A/k$ be an abelian variety with real multiplication by some ring of integers $\mathcal O \subset F$. Let $n$ be an integer prime to the characteristic of $k$.
We have the standart $e_n$ pairing ...

**5**

votes

**3**answers

480 views

### Reference for a theorem of Tate on the endomorphism rings of AVs over finite fields

Let $k$ be a finite field, and $A$, $B$ abelian varieties over $k$. Let $T_p(A)$, $T_p(B)$ be
the (contravariant) Dieudonn\'e modules associated to the p-divisible groups attached to $A$ and $B$, ...

**8**

votes

**0**answers

198 views

### Mod m versions of the toric part of Tate modules

Let $A$ be a polarized abelian variety over a local field $K$ with residue characteristic $p$. In the course of proving that a polarized abelian variety $A/K$ has semi-stable reduction iff for all ...

**2**

votes

**0**answers

192 views

### structure of $T_\ell A$ for $A/\mathbf{F}_q$ an abelian variety

Can someone give me references for the structure of the $G_{\mathbf{F}_q}$-module $T_\ell A$, $A/\mathbf{F}_q$ an abelian variety?

**-5**

votes

**1**answer

528 views

### If an abelian variety has an m-torsion point, is the set of all Galois conjugates of the m-torsion all the m-torsion?

I believe this is the case, but I couldn't come up with a proof off the top of my head, so I want to make sure.
If $A$ is an abelian variety over some field $K$ (I'm in fact interested only in ...

**4**

votes

**1**answer

386 views

### Tamagawa numbers of abelian varieties and torsion.

Let $A$ be an abelian variety defined over a number field $K$. Fix a prime $v \subset \mathcal{O}_K$, with underlying rational prime $p$. What relationship, known or conjectural (if any), should there ...

**4**

votes

**1**answer

352 views

### Selmer of an abelian variety versus that of its dual.

What is the precise relationship between the Selmer group of an abelian variety and that of its dual? For instance, does the vanishing of one not imply the same for the other?
To fix ideas, let $A$ ...

**9**

votes

**1**answer

408 views

### Is an abelian variety with a Galois invariant, rank one submodule of its Tate module, CM?

Let $A$ be an absolutely simple abelian variety over a number field $K$. Assume that, for some prime $p$, the Tate module $T_p A$ has a submodule of rank one, invariant under the absolute Galois group ...

**5**

votes

**1**answer

593 views

### moduli space of abelian varieties of CM-type

Fix a CM-field $K$ of degree $2g$, and a natural number $n$ which is a multiple of $g$. Write
$\tau_1, \tau_2, \ldots, \tau_g, \rho \tau_1, \rho \tau_2, \ldots, \rho \tau_g$
for the different ...

**8**

votes

**3**answers

583 views

### possible CM-types of abelian varieties

Fix a CM-field $K$ of degree $2g$. Let $A$ be a polarized abelian variety of dimension $n$ over $\mathbb{C}$, with an isomorphism $\theta : K \to End_{\mathbb{C}}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$. ...

**11**

votes

**1**answer

537 views

### Galois action on one-dimensional quotients of l-adic cohomology

Let $A$ be an abelian variety of dimension $g$ over a number field $K$, and $\ell$ be a rational prime. Suppose that the Galois action on the $\ell$-adic cohomology $H^k(A, \mathbb{Z}_\ell) ...

**25**

votes

**3**answers

2k views

### In which ways can the isogeny theorem fail for local fields?

Fix a field $K$ with absolute Galois group $G$. By an isogeny theorem over $K$, I mean the statement that the map $\operatorname{Hom}(A,B)\otimes\mathbb{Z}_l \to \operatorname{Hom}_G(T_l A, T_l B)$ is ...

**6**

votes

**1**answer

499 views

### Tate models for semistable algebraic varieties with mixed reduction over a local field

It's known that if $A$ is an abelian variety of totally multiplicative reduction over a p-adic field K, then, after taking a finite field extension, it becomes isomorphic, as a rigid analytic group, ...

**7**

votes

**1**answer

701 views

### Abelian varieties and Selberg class

Hello everyone,
I would like to know whether, assuming Selberg's orthonormality conjecture, it would be possible to establish a "natural" correspondence between abelian varieties and functions ...

**7**

votes

**5**answers

2k views

### Generalizations of Belyi's theorem

Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:
1) $X$ is defined over $\overline{\mathbb{Q}};$
2) There exists a meromorphic ...