91 problems

Let $E$ be an elliptic curve over $\mathbf{Q}$ and $E^d$ its quadratic twists. Let the analytic rank be the order of vanishing of $L(E_d,s)$ at the center of the critical strip, an…

Let $E$ be a fixed elliptic curve over $\mathbf{Q}$, and let $E^d$ range over its quadratic twists. The analytic rank is the order of vanishing of $L(E_d,s)$ at the center of the c…

Watkins's conjecture. The set of quadratic twists $C^{(d)}$ with rank at least $1$ has logarithmic density

Weak Goldfeld conjecture. For each $r\in\{0,1\}$,

Let $C$ be a hyperelliptic curve over $\mathbb{Q}$ of genus $g \geq 2$, defined by a model … for $f\in\mathbb{Z}[x]$ without repeated roots. For each squarefree integer $d$, let…

For a newform $f$ and the cuspidal automorphic representation $\pi$ of $\operatorname{PGL}_2(\mathbb{A})$ generated by $f$, let $\Delta_E$ be the discriminant of a quadratic field…

Let $E$ be a fixed elliptic curve over $\mathbf{Q}$, and let $E^d$ range over its quadratic twists. Minimalist conjecture. The proportion of twists $E^d$ with Mordell–Weil rank at…

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 $E$ be an elliptic curve as above, let $N$ be a prime with $N\equiv 3\pmod{4}$, and suppose $\beta_{-N}\ne 0$. Let $E^{-N}$ denote the quadratic twist of $E$ associated with th…

Let $E$ be an elliptic curve of conductor $Q$, let $q\nmid Q$ be prime, and let $S^{\pm}(X)$ be the relevant families of fundamental discriminants with sign $\pm$. For…

Let $E$ be an elliptic curve of conductor $Q$, let $d$ range over fundamental discriminants, and let $L_E(1,\chi_d)$ be the central value of the quadratic twist of the $L$-function…

Let $C(N,p)$ denote the elliptic curves considered above, and let $p$ range over primes with $p\equiv -1\pmod 4$. Positive-rank conjecture. The curve $C(N,p)$ has positive rank for…

Elliptic-twist vanishing conjecture.

Katz–Sarnak conjecture. For every integer $j\geq 1$ and every compactly supported complex-valued function $h$ on ${\mathbf R}$,

Katz–Sarnak's conjecture. Random matrix models predict that half of the quadratic twists $E_D$ have analytic rank $0$ and the other half have analytic rank $1$; consequently, the a…

Let $E$ be an elliptic curve over ${\mathbb Q}$, let $E^{(d)}$ be a quadratic twist, and write … where $r_d$ is the analytic rank of $E^{(d)}$. Nonvanishing conjecture. For every d…

Let $C$ be a hyperelliptic curve defined by $y^2=f(x)$. For $k\in\mathbb{Q}/(\mathbb{Q}^\times)^2$, let $C^{(k)}$ be the hyperelliptic twist defined by … The analytic rank of…

Let $C/\mathbb Q$ be a genus-one hyperelliptic curve, let $\mathcal F_b$ be a twist family, and suppose that for every $d\in\mathcal F_b$ the class $[C]$ lies in…

Let $C/\mathbb Q$ be a genus-one hyperelliptic curve, let $E$ be its Jacobian, let $\Sigma$ be the specified finite set of places, and let $\mathcal F_b$ be the corresponding twist…

Let $E/\mathbb Q$ be an elliptic curve of rank $1$, let $\mathcal F_b$ be the twist family specified by the local data $b$, and write…

Let $C/\mathbb Q$ be a genus-one hyperelliptic curve, let $E$ be its Jacobian, and let $\mathcal F_b$ be the twist family specified by the local data $b$. Let…

Let ${\mathcal {E}}$ be a family of type (B) or (C). For a $\Sigma$-equivalence class ${\mathfrak{X}}$, let $r({\mathfrak{X}})$ denote its parity parameter and let…

Elliptic-curve multiple-Dirichlet-series pole conjecture. The function

CFKRS conjecture. If $|\operatorname{Re}(s)-\frac12|\ll1/\log X$ for all $s\in S$, then as $X\to\infty$,

CFKRS conjecture. If $|\operatorname{Re}(s)-\frac12|\ll1/\log X$ for all $s\in S$, then as $X\to\infty$,