# Tagged Questions

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### Divisors on an abelian surface

Let $A$ be an abelian surface given by the quotient of a product of two generic elliptic curves $E_1 \times E_2$ by the product $T_1 \times T_2$ of two translations by $2$-torsion points. Then $A$ ...
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### Elliptic curve E and Galois representation

Assume that an elliptic curve $E$ over $\Bbb Q$ has a reducible mod $p$ representation. Then Q: Why is the semi-simplification of $E[p]$ the direct sum of ${\Bbb Z}/p{\Bbb Z}$ and $\mu_p$? Next ...
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### Torsion group of the following elliptic curve

Let $p_1=2, p_2 = 3,\ldots,$ be the prime numbers, and define $n_i = \prod_{j=1}^i p_j$. Moreover, let $E_i$ be the elliptic curve defined by $y^2 = x^3 + n_i$. Can one compute the torsion group ...
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### 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 ...
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### 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 $2 \cdot \mathrm{dim} (A)$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $A\to ... 1answer 439 views ### Generalization of singular moduli$j$-invariants of CM curves$E$over (say) the complex numbers are known as singular moduli. As the theory of complex multiplication explains, singular moduli are algebraic integers of great ... 3answers 643 views ### Elliptic curves on abelian surface Let$Y$be an abelian surface. Is it true that for every general point$P \in Y$, there exists an elliptic curve passing through$P$? 1answer 497 views ### Serre's open image theorem for products of elliptic curves over function fields via specialization In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6 ′′, p. 325): Let$K$be a number field and let ... 3answers 904 views ### Why can projective varieties just have abelian group operations? I just started to read Shimura - Automorphic forms and number theory (Lecture notes in mathematics, 54). On page 20 or so, he mentions that every projective variety which is an algebraic group, is ... 5answers 2k views ### Generalizations of Belyi's theorem Belyi's theorem states that the following properties of a nonsingular projective algebraic curve$X$are equivalent: 1)$X$is defined over$\overline{\mathbb{Q}};$2) There exists a meromorphic ... 1answer 789 views ### isogeny of elliptic curves Let$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} ...
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 $0$ ...
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 $A_p$, ...