16 problems

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…

Parity conjecture.

Let $E$ be an elliptic curve over a number field $F$. Write \begin{displaystyle}\operatorname{rk}(E/F)\end{displaystyle} for its algebraic rank and $w(E/F)$ for its global root n…

Let $E/\a0\mathbb{Q}$ be an elliptic curve of Mordella0a0Weil rank $r$, and let $\{P_1,\ldots,P_N\}$ be a sequence of rational points whose $x$-coordinates form an arithmetic progr…

Let $E$ be an elliptic curve over $\mathbb{Q}$, let $F / \mathbb{Q}$ be a finite Galois extension with $G = \operatorname{Gal}(F / \mathbb{Q})$, and let $\rho$ be an irreducible re…

Minimalist conjecture for twists. For $100\%$ of elliptic curves $E/\mathbb Q$,

Generalised Sylvester's conjecture. If $n\not\equiv\pm1\pmod9$, then

Let $\mathcal{E}$ be an elliptic curve over the function field $\Q(T)$ with nonconstant $j$-invariant. For $t\in\Q$, let $\mathcal{E}_t$ denote its specialization when it is an ell…

Let $E$ be an elliptic curve over an imaginary quadratic field $F$ satisfying the conditions stated in the source, and let $F^{(n)}$ and $F_{\operatorname{cyc}}$ denote the corresp…

Poonen–Rains rank conjecture. For each $r\in\{0,1\}$, $50\%$ of elliptic curves over $F$ have Mordell–Weil rank $r$.

Generalized Nagao conjecture.

Let $\pi:\mathcal{E}\to\mathbf{P}^1_{\mathbf{Q}}$ be an elliptic K3 surface, and for $t\in\mathbf{P}^1_{\mathbf{Q}}$ let $\mathcal{E}_t=\pi^{-1}(t)$ be the fibre. Mazur's conjectur…

Let $a\geq 1$ be an integer and $0\neq d\in\mathbb Q$ such that $f(X)+d^2$ has distinct roots, where $f$ and its roots $\rho,\rho_2,\rho_3$, the splitting field $L_f$, and the elli…

Let $K\subset L$ be an extension of number fields. Let $\alpha\in\mathcal O_K$ and nonzero $d\in\mathcal O_K$ satisfy the hypotheses of Theorem--in particular, the hypotheses refer…

Let $K\subset L$ be number fields. Let $d\in\mathcal O_K$ be nonzero and satisfy the hypotheses of Theorem--in particular, the hypotheses referred to as -- in the source. For the c…

The rank-preserving elliptic curves conjecture. There exists such a $d$ for which $(0,d)$ has infinite order and generates $E(L)\otimes\mathbb Q$. This would provide a rank-preserv…