34 problems

$p$-Parity conjecture. For every abelian variety $A$ over a number field $K$ and every prime number $p$,

Let $E$ be an elliptic curve over $\mathbf Q$ with conductor $N_E$, and let $d$ be a squarefree integer coprime to $2\cdot N_E$. Write $E_d$ for the quadratic twist of $E$ by $d$,…

Let $K$ be a number field, let $A/K$ be an abelian variety, and for each prime $p$ let $\selrk{p}(A/K)$ denote the $p$-Selmer rank. p-independence conjecture. The parity of…

Let $K/k$ be a quadratic extension of number fields, let $E/k$ be an elliptic curve, let $\mathfrak{n}_{E/k}$ be the conductor of $E/k$, let $\chi_{K/k}$ be the quadratic character…

$p$-parity conjecture. The $p^\infty$-Selmer rank $\operatorname{rk}_p E/K$ is even if and only if $w(E/K)=1$.

Let $E$ be an elliptic curve over a number field $K$. Let $w(E/K)\in\{\pm 1\}$ be the root number, defined as the conjectural sign in the functional equation for $L(E/K,s)$ under…

Let $F$ be a number field, let $L$ be a $p$-adic local field, and let $V$ and $V'$ be geometric symplectic self-dual $p$-adic representations of $G_F$ over $L$. For each representa…

Let $F$ be a number field, let $L$ be a $p$-adic local field, and let $V$ be a geometric $p$-adic representation of $G_F$ over $L$, meaning that it is unramified outside finitely m…

$p$-parity conjecture. The dimension $\operatorname{dim}_{\mathbb Q_p}V_p(E)$ is even if and only if

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$), and suppose that it has an even number of places of split multiplicative reduction. Parity conjecture. The parity conject…

Let $E$ be an elliptic curve defined over a number field $K$. The global root number $w(E/K)$ is the sign predicted by the functional equation of the $L$-function of $E$ over $K$,…

Let $X$ be a curve defined over a number field $K$, and let $G$ be a finite group of $K$-automorphisms of $X$. Assume that $\Omega^{1}(\operatorname{Jac}_X)$ is self-dual as a $G$-…

Let $A$ be an abelian variety over a number field $K$. Write $\operatorname{rk} A$ for its Mordell–Weil rank, and let $w(A/K_v)$ denote the local root number at each place $v$ of…

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $w(E)$ denote the sign of its functional equation. Parity conjecture. … This follows formally from the Birch–Swinnerton-Dyer…

Let $E^d$ denote the quadratic twist of the elliptic curve $E$ by $d$, let $\operatorname{rank}(E^d)$ be its Mordell–Weil rank, and let $w(E^d)$ be the root number of $E^d$. For…

$p$-parity conjecture. For every abelian variety $A$ over a number field $K$ and every prime $p$,

Parity conjecture. For every abelian variety $A$ over a number field $K$,

Let $C$ be a curve of genus $2$ or $3$ over a number field $K$, with an unramified double cover … and associated Prym variety $\operatorname{Prym}(D/C)$ over $K$. Let…

Let $A/K$ be an abelian variety over a number field $K$, and let $p$ be prime. Write $\operatorname{rk}_p(A/K)$ for the $p^\infty$-Selmer rank. For each place $v$ of $K$, let…

Let $A/K$ be an abelian variety over a number field $K$. For each place $v$ of $K$, let $w_{A/K_v}$ be the local root number of $A$. Parity conjecture. The parity of the Mordell–We…

The $p$-parity conjecture. One has

Let $A$ be an abelian variety over a global field $\mathcal{K}$. Write $\operatorname{rk} A/\mathcal{K}$ for the rank of its Mordell--Weil group, and let…

Let $\mathcal{F}$ be a family of elliptic curves over $\mathbb{Q}(t)$, with generic rank $r_{\mathcal{F}}$, and let $\mathcal{F}(t)$ be a nonsingular specialization with rank…

Let $E$ be an elliptic curve over $\mathbb{Q}$ with Mordell–Weil rank $r$, and let $\omega(E)$ be the sign of the functional equation of its Hasse–Weil $L$-function. Parity conject…

Parity conjecture. The parity of the Mordell–Weil rank is determined by the global root number: