20 problems

Let $K$ be a number field, and order elliptic curves over $K$ by height. Rank distribution conjecture. Half of all elliptic curves over $K$ have Mordell–Weil rank zero, while the r…

Let $E_m$ denote the cubic twist considered in the paper, and let $m$ be prime. For primes $m\equiv 4,7\pmod 9$, Elkies proved that the rank of $E_m$ is exactly $1$. Rank conjectur…

Density conjecture. Except for a set of $t\in\mathbf P^1(\mathbf Q)$ of density $0$, the rank of ${\mathscr E}_t(\mathbf Q)$ is either

Let $N$ denote the conductor of an elliptic curve, and let $r$ denote its rank. Ulmer's conjecture. In the number field case, the maximal rank should grow at the rate … possibly wi…

Let $K$ be a number field, and order elliptic curves over $K$ by their height. The rank of an elliptic curve $E$ is the rank of the finitely generated abelian group $E(K)$. Minimal…

DFK conjecture. The following predictions hold: if $p=3$, then, as $X\to\infty$,

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 $-5rp^3$.

Let $E$ range over elliptic curves defined over $\mathbb{Q}$, ordered in a suitable asymptotic sense, and let $r$ denote the rank of the finitely generated abelian group…

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 $E/\mathbb{Q}$ be an elliptic curve with Hasse–Weil $L$-function $L(E,s)$, and let $\epsilon\in\{\pm1\}$ be the sign of its functional equation. The parity conjecture. … The co…

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…

Sarnak's conjecture. The counting function satisfies

Let ${\mathcal F}_1$ be the family of elliptic curves with a marked point, and order its curves by the height used for this family. Minimalist conjecture analogue. The average Mord…

Unbounded-rank conjecture. The set of values $\{r_{\text{al}}(E): E/\mathbb Q\text{ an elliptic curve}\}$ is unbounded.

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let its quadratic twists range over quadratic characters of $\mathbb{Q}$. Rank-zero density conjecture. Half of all quadratic tw…

Let $E$ be an elliptic curve over $\mathbb{Q}$, and write $\operatorname{rank}(E)$ for the rank of its finitely generated Mordell–Weil group $E(\mathbb{Q})$. Unbounded-rank conject…

Let $\mathcal{F}_{r,t}^+(X)$ denote the subfamily with root number $+1$ and define … Let $R_{q,-1/2}$ denote the corresponding ratio predicted by the moment model at exponent…

Let $\mathcal{F}$ be the family of elliptic curves ordered by the weighted Weierstrass models and notation described above, and let the average rank mean the limiting average of th…