18 problems

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…

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

Parity conjecture. The Mordell–Weil rank $\operatorname{rk} E(K)$ is even if and only if the root number $w(E/K)$ is $+1$.

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…

Parity conjecture.

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…

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 $E^d$ 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…

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 $A$ 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 $E$ 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 $E/\mathbb Q$ be an elliptic curve with split multiplicative reduction at $2$. Cyclotomic positive-rank conjecture. The curve has infinitely many rational points over…

Let $K=\mathbb Q(\sqrt{-1},\sqrt{17})$. Biquadratic-field parity conjecture. Every elliptic curve $E/\mathbb Q$ has even Mordell–Weil rank over $K$: … The proposed statement follow…

Parity conjecture. Under these assumptions,

Let $A$ be an abelian variety over a number field $K$, let $F/K$ be a Galois extension, let $p$ be a prime, and let $\tau$ be a self-dual representation of…

Let $A$ be an abelian variety over a number field $K$, and let $\mathcal X_p(A/K)$ be the tensor product with $\mathbb Q_p$ of the Pontryagin dual of the $p^\infty$-Selmer group of…