9 problems

Rank–conductor conjecture. One has

Let $E^d$ denote the quadratic twist of the elliptic curve $E$ by $d$. Consider quadratic twists with $|d|$ prime. Rank 0-or-1 conjecture. $100\%$ of quadratic twists $E^d$ where…

Fifth-moment rank bias conjecture. The average value of the main term of the fifth moment is $-5rp^3$.

For positive integers $m$ and $n$, let $\mathcal{S}_{\ell;m,n}$ be the finite set of elliptic surfaces over $\mathbb{F}_\ell$ of the form $y^2=x^3+A(T)x+B(T)$ with $\deg(A)=m$ and…

Let $G$ be a torsion group, and let $\mathcal{E}_G$ denote the family of elliptic curves over $\mathbb{Q}$ with torsion subgroup prescribed by $G$. Folklore conjecture on ranks wit…

Let elliptic curves over $\mathbb{Q}$ be ordered by height, and let algebraic rank denote Mordell–Weil rank. Park–Poonen–Voight–Wood conjecture. (i) The proportion of elliptic curv…

Let $E$ range over elliptic curves over $\mathbb{Q}$, ordered by naive height, and let rank mean the algebraic or analytic rank as appropriate. Goldfeld's minimalist conjecture. Th…

Uniform boundedness conjecture for ranks of quadratic twists. There is a positive integer $R=R(E,k)$ such that every elliptic curve $E'$ over $k$ that is a quadratic twist of $E$ s…

Honda's conjecture. There is $c=c(A,k)>0$ such that for every finite extension $L/k$,