40 problems

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…

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

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 $X\to\infty$,

Let $E:y^2=f(x)$ be an elliptic curve over $\mathbb{Q}$. For a squarefree integer $d$, let $E^{(d)}$ be the quadratic twist of $E$ by $d$, and let $n_{E,r}^{\ast}(X)$ count fundame…

Rank-one joint distribution conjecture. As $d$ ranges over $\mathcal{F}$, the joint distribution of $\log|L'(\frac12,E_d)|$ and $\log(|\Sha(E_d)|R(E_d)/\sqrt{|d|})$ is asymptotical…

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 $f$ be a newform of level $\Gamma_0(N)$ and even weight $k$ with $k\geq 6$, let $\pi$ be the cuspidal automorphic representation of $\operatorname{PGL}_2(\mathbb{A})$ generated…

Let $A/F$ be a modular elliptic curve, let $E/F$ be a quadratic CM extension of class number one, let $A^E/F$ be the quadratic twist of $A$ by $E/F$, and let $\psi$ be an optimal e…

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 $E^d$ 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…

Let $cal D$ be a family of fundamental discriminants in a fixed arithmetic progression, let $cal M_A(D;\ell)$ denote the corresponding smoothed average of…

Let $k,\beta\in\mathbb{N}$ and let $\underline{\theta}=(\theta_1,\dots,\theta_k)\in\mathbb{R}^k$. For the even-sign quadratic twists of an elliptic curve $E/\mathbb{Q}$, let…

Let $X$ be the set of quadratic fields, and let $\pi'$ be an irreducible cuspidal automorphic representation of $\operatorname{PGL}_2(\mathbb{A})$. For $E\in X$, let $\eta_E$ be th…

Let $K$ be a local field of characteristic zero and let $L_K$ be its Weil group when $K$ is archimedean and its Weil–Deligne group when $K$ is non-archimedean. Let…

For integers $A,B$ with $4A^3+27B^2\neq 0$, consider the quadratic twists … for positive square-free integers $d$, and let $\eta_d(A,B)$ be defined by … with $\eta_d(A,B)=+\infty$…

Automorphic Goldfeld conjecture. Among quadratic characters $\chi'$ satisfying

Let $E$ be any elliptic curve over ${\mathbb Q}$, let $r$ be a positive integer, and let $E^{(M)}$ denote the quadratic twist of $E$ by $M$. A square-free integer has exactly $r$ p…

Le Boudec's conjecture. For every $\alpha>0$, there exists a constant $c_{A,B}(\alpha)>0$ such that