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$\DeclareMathOperator\Sel{Sel}$Let $E$ be an elliptic curve defined over a number field $K$ with full $2$-torsion. The classical complete $2$-descent method tells that the $2$-Selmer group $\Sel_2(E/K)$ can be identified with the set of locally solvable everywhere homogeneous spaces. More precisely, we consider a quadratic twist $$E:y^2=x(x-na)(x+nb)$$ defined over $\mathbb Q$, where $n,a,b,\frac{a+b}{2}$ are positive odd integers. Then $\Sel_2(E/\mathbb Q)$ can be identified with $$\{\Lambda=(d_1,d_2)\in(\mathbb Q^\times/\mathbb Q^{\times2})^2: D_\Lambda(\mathbb{A}_K)\neq\emptyset\},$$ where $$D_\Lambda: d_1u_1^2-d_2u_2^2=na,\quad d_1u_1^2-d_1d_2u_3^2=-nb.$$

Assume that $n$ is a positive square-free integer prime to $2ab(a+b)$ and $\Lambda=(d_1,d_2)$ where $d_1,d_2$ are positive square-free odd divisors of $nab(a+b)$. Then we can show: if $D_\Lambda$ is locally solvable everywhere except $v=2$, then it is also locally solvable at $v=2$. The proof depends on complicated calculations on Hilbert symbols.

So my question is: when the following hold for any $\Lambda$? Does this relate the Selmer group or Shafarevich-Tate group of $E$?

Let $D_\Lambda$ be a homogeneous space in the form as above. If $D_\Lambda$ is locally solvable everywhere except a place $v$, then it is also locally solvable at $v$.

$\DeclareMathOperator\Sel{Sel}$Let $E$ be an elliptic curve defined over a number field $K$ with full $2$-torsion. The classical complete $2$-descent method tells that the $2$-Selmer group $\Sel_2(E/K)$ can be identified with the set of locally solvable everywhere homogeneous spaces. More precisely, we consider a quadratic twist $$E:y^2=x(x-na)(x+nb)$$ defined over $\mathbb Q$, where $n,a,b,\frac{a+b}{2}$ are positive odd integers. Then $\Sel_2(E/\mathbb Q)$ can be identified with $$\{\Lambda=(d_1,d_2)\in(\mathbb Q^\times/\mathbb Q^{\times2})^2: D_\Lambda(\mathbb{A}_K)\neq\emptyset\},$$ where $$D_\Lambda: d_1u_1^2-d_2u_2^2=na,\quad d_1u_1^2-d_1d_2u_3^2=-nb.$$

Assume that $n$ is a positive square-free integer prime to $2ab(a+b)$ and $\Lambda=(d_1,d_2)$ where $d_1,d_2$ are positive square-free odd divisors of $nab(a+b)$. Then we can show: if $D_\Lambda$ is locally solvable everywhere except $v=2$, then it is also locally solvable at $v=2$. This can be proved by the poduct formula on Hilbert symbols.

So my question is: when the following hold for any $\Lambda$? Does this relate the Selmer group or Shafarevich-Tate group of $E$?

Let $D_\Lambda$ be a homogeneous space in the form as above. If $D_\Lambda$ is locally solvable everywhere except a place $v$, then it is also locally solvable at $v$.

$\DeclareMathOperator\Sel{Sel}$Let $E$ be an elliptic curve defined over a number field $K$ with full $2$-torsion. The classical complete $2$-descent method tells that the $2$-Selmer group $\Sel_2(E/K)$ can be identified with the set of locally solvable everywhere homogeneous spaces. More precisely, we consider a quadratic twist $$E:y^2=x(x-na)(x+nb)$$ defined over $\mathbb Q$, where $n,a,b$ are positive odd integers. Then $\Sel_2(E/\mathbb Q)$ can be identified with $$\{\Lambda=(d_1,d_2)\in(\mathbb Q^\times/\mathbb Q^{\times2})^2: D_\Lambda(\mathbb{A}_K)\neq\emptyset\},$$ where $$D_\Lambda: d_1u_1^2-d_2u_2^2=na,\quad d_1u_1^2-d_1d_2u_3^2=-nb.$$

Assume that $n$ is a positive square-free integer prime to $2ab(a+b)$ and $\Lambda=(d_1,d_2)$ where $d_1,d_2$ are positive square-free odd divisors of $nab(a+b)$. Then we can show: if $D_\Lambda$ is locally solvable everywhere except $v=2$, then it is also locally solvable at $v=2$. The proof depends on complicated calculations on Hilbert symbols.

So my question is: does the following hold or not?

Let $D_\Lambda$ be a homogeneous space in the form as above. If $D_\Lambda$ is locally solvable everywhere except a place $v$, then it is also locally solvable at $v$.

$\DeclareMathOperator\Sel{Sel}$Let $E$ be an elliptic curve defined over a number field $K$ with full $2$-torsion. The classical complete $2$-descent method tells that the $2$-Selmer group $\Sel_2(E/K)$ can be identified with the set of locally solvable everywhere homogeneous spaces. More precisely, we consider a quadratic twist $$E:y^2=x(x-na)(x+nb)$$ defined over $\mathbb Q$, where $n,a,b,\frac{a+b}{2}$ are positive odd integers. Then $\Sel_2(E/\mathbb Q)$ can be identified with $$\{\Lambda=(d_1,d_2)\in(\mathbb Q^\times/\mathbb Q^{\times2})^2: D_\Lambda(\mathbb{A}_K)\neq\emptyset\},$$ where $$D_\Lambda: d_1u_1^2-d_2u_2^2=na,\quad d_1u_1^2-d_1d_2u_3^2=-nb.$$

Assume that $n$ is a positive square-free integer prime to $2ab(a+b)$ and $\Lambda=(d_1,d_2)$ where $d_1,d_2$ are positive square-free odd divisors of $nab(a+b)$. Then we can show: if $D_\Lambda$ is locally solvable everywhere except $v=2$, then it is also locally solvable at $v=2$. The proof depends on complicated calculations on Hilbert symbols.

So my question is: when the following hold for any $\Lambda$? Does this relate the Selmer group or Shafarevich-Tate group of $E$?

Let $D_\Lambda$ be a homogeneous space in the form as above. If $D_\Lambda$ is locally solvable everywhere except a place $v$, then it is also locally solvable at $v$.

Let $E$ be an elliptic curve defined over a number field $K$ with full $2$-torsion. The classical complete $2$-descent method tells that the $2$-Selmer group $Sel_2(E/K)$ can be identified with the set of locally solvable everywhere homogeneous spaces. More precisely, we consider a quadratic twist $$E:y^2=x(x-na)(x+nb)$$ defined over $\mathbb Q$, where $n,a,b$ are positive odd integers. Then $Sel_2(E/\mathbb Q)$ can be identified with $$\{\Lambda=(d_1,d_2)\in(\mathbb Q^\times/\mathbb Q^{\times2})^2: D_\Lambda(\mathbb{A}_K)\neq\emptyset\},$$ where $$D_\Lambda: d_1u_1^2-d_2u_2^2=na,\quad d_1u_1^2-d_1d_2u_3^2=-nb.$$

Assume that $n$ is a positive square-free integer prime to $2ab(a+b)$ and $\Lambda=(d_1,d_2)$ where $d_1,d_2$ are positive square-free odd divisors of $nab(a+b)$. Then we can show: if $D_\Lambda$ is locally solvable everywhere except $v=2$, then it is also locally solvable at $v=2$. The proof depends on complicated calculations on Hilbert symbols.

So my question is: does the following hold or not?

Let $D_\Lambda$ be a homogeneous space in the form as above. If $D_\Lambda$ is locally solvable everywhere except a place $v$, then it is also locally solvable at $v$.

$\DeclareMathOperator\Sel{Sel}$Let $E$ be an elliptic curve defined over a number field $K$ with full $2$-torsion. The classical complete $2$-descent method tells that the $2$-Selmer group $\Sel_2(E/K)$ can be identified with the set of locally solvable everywhere homogeneous spaces. More precisely, we consider a quadratic twist $$E:y^2=x(x-na)(x+nb)$$ defined over $\mathbb Q$, where $n,a,b$ are positive odd integers. Then $\Sel_2(E/\mathbb Q)$ can be identified with $$\{\Lambda=(d_1,d_2)\in(\mathbb Q^\times/\mathbb Q^{\times2})^2: D_\Lambda(\mathbb{A}_K)\neq\emptyset\},$$ where $$D_\Lambda: d_1u_1^2-d_2u_2^2=na,\quad d_1u_1^2-d_1d_2u_3^2=-nb.$$

Assume that $n$ is a positive square-free integer prime to $2ab(a+b)$ and $\Lambda=(d_1,d_2)$ where $d_1,d_2$ are positive square-free odd divisors of $nab(a+b)$. Then we can show: if $D_\Lambda$ is locally solvable everywhere except $v=2$, then it is also locally solvable at $v=2$. The proof depends on complicated calculations on Hilbert symbols.

So my question is: does the following hold or not?

Let $D_\Lambda$ be a homogeneous space in the form as above. If $D_\Lambda$ is locally solvable everywhere except a place $v$, then it is also locally solvable at $v$.