603 problems

Let $n$ be a positive integer, and let $C_G(\mathbb{Z}_n)$ denote the number of isomorphism classes of generalized Weierstrass elliptic curves over $\mathbb{Z}_n$. Generalized Weie…

Let $\phi$ be a suitable test function, and let Theorem denote the density theorem established in the paper for the family of elliptic curves. Unrestricted-support conjecture. Theor…

Derickx–van Hoeij conjecture. There is a modular unit $f$ defined over $\mathbb{Q}$ on $X_1(p)$ such that

Let $p$ be an odd prime and $E$ an elliptic curve with good ordinary reduction at $p$. Let $… be the cyclotomic Iwasawa algebra, and let$L_p(E)\in\Lambda$be the$p$-adic$L…

Galbraith's conjecture. If $2\leq g^+_N\leq 5$, then $X^+_0(N)$ contains exceptional rational points if and only if

Let $K$ be an imaginary quadratic field. A Bianchi newform over $K$ is a weight-$2$ cuspidal eigenform of level $[?]$ with rational Hecke eigenvalues; here $[?]$ denotes the ideal…

Let $K$ be a number field, and let $E$ be a Legendre elliptic curve over $K$ with parameter $\lambda\in K$. For each even integer $n\geq 4$, consider the $n$ points constructed abo…

Let $p>2$ be a prime, let $E/{\mathbb Q}$ be an elliptic curve, and let $\Sigma_p$ denote the set of Dirichlet characters of order $p$. For $S\subset\Sigma_p$, say that $S$ has den…

Let $a,b,c,d,e,fbinom{}{} inmathbb{Z}$ and consider the elliptic surface … where the coefficients $a_{i,j}$ are given by (ais). For $t\in\mathbb{Q}$, let $\mathcal{E}_t$ denote the…

Let $C$ be an elliptic curve defined over $\mathbb Q$, embedded in projective space, and let $x_n$ be a fixed coordinate in a coprime integral representative $[x_0,\ldots,x_N]$ of…

Let $p$ be a prime with $p\geqslant 3$, let $F^{cyc}$ be the cyclotomic extension of $F$, and let $F^S$ be the maximal extension of $F$ unramified outside the specified set $S$. Fo…

Let $G_{\bold Q}=\operatorname{Gal}(\bar{\bold Q}/\bold Q)$, let $p$ be an odd prime, and let $\bar{\varepsilon}_p:G_{\bold Q}\to\bold F_p^\times$ be the character describing the a…

Let $\bold Q$ denote the field of rational numbers, and call an elliptic curve over $\bold Q$ semistable if it is semistable at every prime, meaning that at each prime it has eithe…

Let $\bold Q$ denote the field of rational numbers, and call an elliptic curve over $\bold Q$ modular if it admits a holomorphic map from some modular curve $X_0(N)$ onto it. Taniy…

Let $E_{/\mathbb{Q}}$ be an elliptic curve of conductor $N$, let $K$ be the quadratic imaginary field associated with a Heegner discriminant, and let $y_K\in E(K)$ be the correspon…

Let $E$ be an elliptic curve over a number field. Its essential dimension is denoted by $\operatorname{ed}E$. Infinite essential-dimension conjecture for elliptic curves. Then … Th…

Let $E$ be an elliptic curve over ${\mathbf Q}$, and let $d\geq 1$ be fixed. A prime $p$ is a local torsion prime for $E$ if there exists an extension $K/{\mathbf Q}_p$ of degree a…

BSD regulator-quotient square conjecture. Then

$p$-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…

Let $E_t^d$ be the CM elliptic curve and let $w_d$ denote the order of the relevant unit group. For a prime $p$, write $\left(\frac{a_p(E_t^d)}{p}\right)_m$ for the $m$th power res…

Let $f$ be a rational map with an invariant complex elliptic curve. A periodic point is repelling when the derivative of the corresponding return map is expanding. Repelling-point…

Let $f$ be a rational map of ${\mathbb P}^2({\mathbb C})$, let ${\mathcal C}\subset{\mathbb P}^2({\mathbb C})$ be an $f$-invariant elliptic curve, let ${\mathcal B}({\mathcal C})$…

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…