53 problems

Let $\mathcal E$ be an elliptic curve over $\mathbb{Q}(T)$ and let $A_{\mathcal E}(p)$ denote the average of the Frobenius traces over its fibers, as defined above. Nagao's conject…

Let $K$ be a number field, and let elliptic curves over $K$ be ordered by height. For an elliptic curve $E/K$, write $r$ for its Mordell–Weil rank. Goldfeld–Katz–Sarnak conjecture.…

Let $E$ be an elliptic curve over $Q$. By Mazur's theorem, its torsion subgroup $E(Q)_{\rm tors}$ is one of the fifteen possible torsion groups … or … Here the rank of…

Let $E$ range over elliptic curves over $\mathbb{Q}$, and let $\operatorname{rank} E(\mathbb{Q})$ denote the Mordell–Weil rank of $E$ over $\mathbb{Q}$. Park–Poonen–Voight–Wood's c…

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 $D$ be a positive square-free integer and let $E_D:y^2=x^3-D^2x$ be the corresponding congruent number elliptic curve. A positive integer is a congruent number when it is the a…

Rank conjecture. $E_t(\mathbb Q)$ has rank at least $4$. This conjecture concerns the search for elliptic curves with torsion subgroup…

Let $E$ range over elliptic curves defined over $\mathbb{Q}$, ordered by any counting convention implicit in the phrase “almost all.” The folklore conjecture. Almost all elliptic c…

Let $K$ be a number field, and let $E$ range over elliptic curves over $K$, ordered by naive height $\operatorname{Ht}(E)$. Average-rank conjecture. The average rank of elliptic cu…

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…

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$), and suppose that it has an even number of places of split multiplicative reduction. Parity conjecture. The parity conject…

Let $E$ be an elliptic curve over a number field $L$, and let $w(E / L) \in \\{\pm1\\}$ be its global root number. Parity conjecture. The root number determines the parity of the M…

Let $E/\mathbb Q$ be the elliptic curve in the preceding construction, and let $W(E)\in\{\pm1\}$ be its global root number. Parity conjecture. The rank of $E$ is even if and only i…

Let $E/\mathbb{Q}$ be an elliptic curve and let $p$ range over primes of good supersingular reduction. Let $\lambda^\pm$ denote either the analytic or algebraic signed Iwasawa…

Consider rational elliptic curves, ordered by height, and let the Mordell–Weil rank mean the rank of $E(\mathbb{Q})$. Rank distribution conjecture. A density of $100\%$ of rational…

Smith's rank-greater-than-one error-term conjecture. For every $\epsilon>0$, this error term is conjectured to satisfy

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

Parity conjecture.

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

Generalised Sylvester's conjecture. If $n\not\equiv\pm1\pmod 9$, then $r_n\leq2k_1+k_2-1$ and $r_n\equiv k_2-1\pmod2$. If $n\equiv\pm1\pmod9$, then $r_n\leq2k_1+k_2$ and…

Let ${\mathcal E}$ be a nontrivial elliptic surface over ${\mathbb Q}$ defined by … where each $a_i(T)\in{\mathbb Q}[T]$ has degree at most $2$, and let ${\mathcal E}_t$ denote its…

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…

Order elliptic curves over ${\mathbb Q}$ by their naive height. Rank distribution conjecture. The proportion of elliptic curves of rank $0$ is $50\%$, and the proportion of ellipti…

Let $p\geq 7$ be a prime. Let $E$ be an elliptic curve over ${\mathbb Q}$, and let a ${\mathbb Z}/p{bb}{\mathbb Z}$-extension $L/{\mathbb Q}$ be a Galois extension with Galois grou…