1
vote
1answer
209 views

Example of non-modular elliptic surface?

In "On elliptic modular surfaces", Shioda proves some interesting theorems on smooth elliptic surfaces (admitting a section); he then focuses on "modular elliptic surfaces" and proves some more ...
1
vote
1answer
238 views

complex multiplication

For an abelian variety $A$, it is said to be have $complex \ multiplication$ if $\mathrm{End}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$ contains a number filed $F$ of degree $2 \cdot \mathrm{dim} (A)$. ...
1
vote
0answers
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 ...
3
votes
3answers
424 views

Another question related to the isogeny theorem for elliptic curves

I was reading the following question: About isogeny theorem for elliptic curves and was interested in the following statement at the end of Torsten Ekedahl's answer: "Note also that the situation is ...
17
votes
3answers
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 ...
18
votes
1answer
1k 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 ...
2
votes
1answer
215 views

Are abelian varieties degree two covers of some projective space

Let $A$ be an abelian variety over a field $k$ of dimension $g\geq 2$. There exists a finite morphism $A\to \mathbf{P}^g_k$. Here's the question. Does there exist a finite morphism $A\to ...
2
votes
1answer
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 ...
2
votes
3answers
620 views

Elliptic curves on abelian surface

Let $Y$ be an abelian surface. Is it true that for every general point $P \in Y$, there exists an elliptic curve passing through $P$?
2
votes
1answer
482 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 ...
9
votes
3answers
863 views

Why can projective varieties just have abelian group operations?

I just started to read Shimura - Automorphic forms and number theory (Lecture notes in mathematics, 54). On page 20 or so, he mentions that every projective variety which is an algebraic group, is ...
7
votes
5answers
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 ...
3
votes
1answer
782 views

isogeny of elliptic curves

Let $E$ and $F$ be two abelian varieties of dimension 1 over $\mathbb{C}$. Let $f : E \to F$ be a surjective homomorphism of abelian varieties ($f(0) = 0$). If $\ker (f) \cong \mathbb{Z}/2\mathbb{Z} ...
2
votes
4answers
957 views

reduction of CM elliptic curves

Can someone indicate how to prove the following equivalences for a CM elliptic curve $E$: (i) $p$ is inert in End($E$) (ii) $E_p$ is supersingular (iii) The trace of the Frobenius at $p$ is $0$ ...
9
votes
4answers
1k views

Torsion of an abelian variety under reduction.

Let $p$ be a prime. Suppose you have an Abelian scheme $A$ over $Spec\ \mathbb{Z}_p$. How do you prove that if $q$ is another prime, then the $q$-torsion of $A$ injects into the torsion of $A_p$, ...