1
vote
1answer
75 views

What is Rosati Form

I was reading a paper and they mentioned the Rosati form. Particularly, what they said was: Let $A$ be an abelian surface defined over $k$ such that $ST_A^0$ (the connected component of the Sato-Tate ...
0
votes
2answers
126 views

Poincaré bundle and Weil pairing for Abelian schemes

In which situations is there a Poincaré bundle for Abelian schemes? In [Mumford, Abelian varieties] only the case of Abelian varieties is treated. The same question for the Weil pairing ...
1
vote
0answers
102 views

algebraicity of Néron-Tate canonical height for Abelian varieties over global function fields

(transcendence of canonical heights) Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always ...
1
vote
0answers
66 views

Bound for field of definition (vs field of moduli) of an abelian variety

Let $A$ be a principally polarised abelian variety over $\mathbb{C}$ of dimension $g$. Let $K$ be the field of moduli of $A$. Proposition. $A$ has a model defined over an extension $L$ of $K$ such ...
1
vote
2answers
291 views

Moduli Spaces of Higher Dimensional Complex Tori

I know that the space of all complex 1-tori (elliptic curves) is modeled by $SL(2, \mathbb{R})$ acting on the upper half plane. There are many explicit formulas for this action. Similarly, I have ...
13
votes
1answer
737 views

Status of Grothendieck's conjecture on homomorphisms of abelian schemes

In [1] Grothendieck posits the following: Conjecture. Let $S$ be a reduced connected scheme, locally of finite type over Spec($\mathbf{Z}$) or a field $k$, $A$ and $B$ two abelian schemes over $S$, ...
0
votes
1answer
144 views

Reference for Complex Abelian Varieties

I am looking for a reference which explains how theta functions, algebraically independent meromorphic functions, and line bundles all fit together in the context of complex tori. More explicitly, ...
9
votes
2answers
518 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 ...
18
votes
1answer
2k views

Modern proof of Serre's open image theorem?

Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic ...
1
vote
0answers
135 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 ...
2
votes
1answer
798 views

books (or notes) on complex multiplication

This would be a vague question, but I still want to ask here. Do you have any recommended book on complex multiplicaton. I know only 2 books: Shimura's book ...
1
vote
0answers
193 views

Where can I find a copy of Serre's Cours au college de France 1985-1986?

Hi, I was wondering: where might I be able to find a copy of this work online? And are there any other resources for the proof of the open image theorem for abelian varieties with endomorphism ring ...
2
votes
1answer
439 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 ...
2
votes
0answers
224 views

modern reference for Néron's “Quasi-fonctions et Hauteurs sur les Varietes Abeliennes”

Is there a modern reference for Néron's "Quasi-fonctions et Hauteurs sur les Varietes Abeliennes" http://www.jstor.org/pss/1970644 i.e. using Grothendieck's language of schemes and in English?
6
votes
1answer
523 views

Mumford-Tate group and Galois representations

Could someone please point me towards a proof of why the image of a Galois representation on the Tate-module of an abelian variety is limited by its Mumford-Tate group?
2
votes
1answer
499 views

Serre's open image theorem for products of elliptic curves over function fields via specialization

In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6 ′′, p. 325): Let $K$ be a number field and let ...
5
votes
3answers
492 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 $B$, ...
2
votes
0answers
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
0answers
260 views

Why is Pic^0(C) of a curve C a variety?

Let $C$ be an abstract non-singular curve. I'm having a hard time finding a reference for why $\text{Pic}^0(C)$ is a variety. Any pointers towards a reference would be appreciated.
5
votes
1answer
773 views

Abelian subvarieties of abelian varieties — reference request

This question may be too naive, in which case I apologise in advance. Anyway, it is a well-known fact (see e.g. Milne's notes) that any abelian variety A has only finitely many direct factors up to ...
1
vote
3answers
413 views

projective subvarieties of the moduli space of abelian varieties

I know that the fibre of $A_{g,n}$ over $\mathbf{F}_p$ is quasi-projective (of what dimension?). Can one exhibit some smooth projective subvarieties of high dimension in it? What are references for ...
6
votes
1answer
688 views

Chow Ring of Moduli Space of Abelian Varieties

Is there a good reference for the structure of the Chow ring of $\mathcal{A}_g$, the moduli space of complex principally polarized abelian varieties? More generally, references for the intersection ...
4
votes
3answers
937 views

Modular forms reference

If f is a weight 2 newform on $\Gamma_1(N)$ then there exists an abelian variety Af whose endomorphism algebra is isomorphic to the field generated by the coefficients of f. I've seen this proven in ...