607 problems

Let $F(b,c)$ denote the lattice sums used in the paper, and let $m(\alpha)$ and $g(\alpha)$ be the Mahler measures defined by … and … where $y=e^{2\pi i\theta_1}$ and…

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…

Goldfeld's conjecture. For every elliptic curve $E/\mathbb Q$, those $\chi\rein \Sigma_2$ for which

Mennicke's modularity conjecture. Every such elliptic curve $E$ is modular.

Let $C/\mathbb{Q}$ be the specific plane quartic defined in Conjecture 1.6 of Furio and Lombardo. The conjecture asserts that $C(\mathbb{Q})$ consists of exactly four rational poin…

Let $E$ be an elliptic curve with good reduction at $p$, let $K_\infty/K$ be the anticyclotomic $\mathbb{Z}_p$-extension, and let $K_n$ be a finite layer. Under the hypotheses…

The supplied sources identify a Lang–Trotter conjecture concerning the frequency of primes at which reductions of points on an elliptic curve satisfy a primitive-point condition. T…

For every elliptic curve $E/\mathbb{Q}$ with complex multiplication and $\operatorname{rank}E(\mathbb{Q})=2$, and for every good ordinary prime $p$ of $E$, is…

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 $p$ be a prime satisfying … For an elliptic curve $E/\mathbb Q$, write $E(\mathbb Q(\sqrt{-p}))_{\mathrm{tors}}$ for its torsion subgroup over $\mathbb Q(\sqrt{-p})$. Existence…

Let $E$ be an elliptic curve over $\mathbb{Q}(t)$, let $E(\mathbb{A}^1)$ denote the elliptic curve over the function field $\mathbb{Q}(t)$, and let $E(t)$ denote its specialization…

Shimura–Taniyama conjecture. Any elliptic curve over $K$ is modular.

Let $x,y$ be positive integers with $x^3\ne y^2$. Weak Hall's conjecture. There exists an absolute constant $\kappa>0$ such that … This weaker form replaces Hall's exponent $1/6$ b…

Modularity conjecture. Every elliptic curve over a totally real field $F$ is modular.

Birch and Swinnerton-Dyer conjecture. One has

Let $K$ be a number field, let $\mathfrak N$ be an ideal of $\mathcal O_K$, and let $\mathfrak f$ be a non-trivial, new, weight two complex eigenform over $K$ of level…

Lang–Trotter conjecture for pairs. There exists a non-negative constant $c_{E_1,E_2}$ such that

Serre's uniformity question. There exists a bound $C$ such that for every non-CM elliptic curve $E/\mathbb{Q}$ and every prime $\ell>C$, the representation $\rho_{E,\ell}$ is surje…

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…

Mazur's conjecture. The $\Lambda$-module ${\rm Sel}_{E}(K_\infty)[p^\infty]^{\lor}$ is a finitely generated torsion $\Lambda$-module.

Beilinson's conjecture. One has

Sato–Tate conjecture. The angles $\varphi_p$ are equidistributed in $[0,\pi]$ with respect to the Sato–Tate density measure

Let $E/\mathbb{Q}$ be an elliptic curve with $j$-invariant $j_E$ and minimal discriminant $\Delta_E$. Let $\hat h$ denote the canonical height and $h$ the relevant height on ration…

Coates–Sujatha's pseudo-nullity conjecture. The module