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This is from Silverman's 'the arithmetic of elliptic curves', exercise 5.5. Let $K$ be an imaginary quadratic field, and let $R_1...R_n$ be orders in $K$. I would like to prove that there are more tha …

I have a question about proof that Tamagawa number, $[E (K): E_0 (K)]$ is finite. Could you please tell (correct) me any strange parts about my understanding of the outline of the proof ? My understan …

In Silverman's The arithmetic of elliptic curves, p. 201, theorem $7.1$ (Criterion of Neron-Ogg-Shafarevich), he applies the theorem "When $K$ is complete with respect to it's discrete value, then, $[ …

Let $E$ be an elliptic curve on $\Bbb Q_p$. $E_0(\Bbb Q_p)$ is points of $E(\Bbb Q_p)$ reduced to nonsingular points. How to prove #$E(\Bbb Q_p)$＝$∞$ directly ? According to Silverman's book 'the arit …

Let $K$ be a imaginary quadratic field, and $E/K$ be elliptic curve which has CM over $K$. Let $ψ_E$ be Hecke(Grossencharacter) character of $E/K$. Let fix prime ideal $I$ of $K$. Then, why $ψ_E(I)$ …

Let $K$ be a imaginary quadratic field, $R_K$ be ring of integers of $K$, and $E/K$ be elliptic curve which has CM over $K$. Let $\psi_E$ be Hecke (Grössencharakter) character of $E/K$. Let fix prime …

Let $E/ \Bbb{C}$ be an elliptic curve which has complex multiplication over a number field $K$. Then it is widely known that $j(E) \in \overline { \Bbb{Z}}$. What is the known generalization of this s …

For abelian variety $A/K$, divisibility problem (i.e. $\forall n≧1$, $Ш(A/K)⊂p^nH^1(G_K,A)$ holds for fixed prime $p$?) was asked by Cassels in 1962 and even now discussed. On the other hand, once gro …

Let $\phi ：C_1\to C_2$ be morphism of projective singular curve. Let $\tilde{C}_1$ and $\tilde{C}_2$ be their smooth compactification. Then $\phi$ extends to $\tilde{\phi} : \tilde{C}_1\to \tilde{C}_ …

Let $E:y^2=x^3-17$ be an elliptic curve. It is known that rank$(E/\Bbb{Q})=0$. (For example, prop $6.5$, $362$p in Silverman's book 'The arithmetic of elliptic curves') Over $K=\Bbb{Q}(i)$, what is t …

Let $L/K$ be a quadratic number field extension. Let $\operatorname{Sha}(E/L)$ be Tate-Shafarevich group of elliptic curve $E/L$. How $\sigma \in \operatorname{Gal}(L/K)$ acts on $\operatorname{Sha}(E …

Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$. Let $D\in \Bbb{Z}$ be a square free integer and $E_D/\Bbb{Q}$ be its quadratic twist. It is widely known that for all $E/\Bbb{Q}$: elliptic …

Let $E/K$ be an elliptic curve over number field $K$. Let $M_K$ be a set of all places of $K$. My question is, Does there exist a finite set $S\subset M_K$ such that $\forall C$: $E/K$-torsor, $\fo …

Let $E/\mathbb{Q}$ be an elliptic curve defined by $E: y^2 = x^3 + b$ (where $b \in \mathbb{Q}$). Let $E_D$ be an elliptic curve defined by $E_D: y^2 = x^3 + bD^2$. $E$ and $E_D$ are isomorphic over $ …

Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$. Let $E_D$ be a quadratic twist of $E$ also defined over $\Bbb{Q}$. Is there a known elliptic curve, $E/\Bbb{Q}$, such that the rank of $E_D …