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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 $E/K$ 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 $E/\Bbb{Q}$ be an elliptic curve. Let $D$ be a square free negative integer. It is conjectured that 50％ of twist of elliptic curve $E_D$ has rank $0$ and $50％$ has rank $1$. But is some particular …

Let $C$ be a singular curve defined over a local field $K$. Let $\tilde{C}$ be its smooth compactification(maybe this is not normalization). Why $\tilde{C}(K)\neq \emptyset$ implies ${C}(K)\neq \empty …

Let $E:y^2=x^3+17x$ be an elliptic curve. In this MO page(Infinitely many elliptic curve with twist rank more than $1$ in specific case), Nulhomologous's and other's comment reads from parity conjectu …

Let $K$ be a quadratic number field. Let $\sigma$ be a generator of Galois group of $K/\Bbb{Q}$. Let $E$ be an elliptic curve defined over $\mathbb{Q}$. Let $\mathrm{Sha}(E/K)$ denote the Tate-Shafare …

This is a question related to the definition of Brauer-Manin obstruction. Let $K$ be a number field. $X/K$ be an algebraic variety over $K$. Let $O_{X,P}$ be a local ring of $X$ at $P$. Let $Br(X)=\bi …

Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$. Tate-Shafarevich group $\mathit{Sha}(E/\Bbb{Q})$ is defined as follows. $$\mathit{Sha}(E/\Bbb{Q})\stackrel{\text{def}}{=} \operatorname{Ker …

The result of LMFDB claims (https://www.lmfdb.org/EllipticCurve/Q/1640/c/1 ) that (2-part of) Tate-Shafarevich group $\mathrm{Sha}(E/\Bbb{Q})$ of elliptic curve $y^2=x^3-8747x-314874$ has order $16$. …

$\DeclareMathOperator\Sha{Sha}\DeclareMathOperator\Gal{Gal}$Let $L/K$ be a quadratic extension of number field $K$. Let $\sigma$ be a generator of $\Gal(L/K)$. Let $E/K$ be an elliptic curve defined o …

How can I calculate the numerator of congruent zeta function of given hyperelliptic curve ? For example, let $C:y^2=(x^2+1)(x^4-8x^3+2x^2+8x+1)$. numerator of congruent zeta function mod$23$ of this i …