1,258 problems

Let $p\equiv 4,7\, \mathrm{mod}\, 9$, let $i=1,2$, and let $E_{p^i}$ be the elliptic curve $y^2=x^3+\frac{p^{2i}}{4}$. Write $\Omega_{p^i}$ for its real period, let $\delta(p^i)=0$…

Let $E$ be an elliptic curve over $\mathbf{Q}$ and $E^d$ its quadratic twists. Let the analytic rank be the order of vanishing of $L(E_d,s)$ at the center of the critical strip, an…

Let $A_0$ be the optimal elliptic curve in the $\mathbb{Q}$-isogeny class of an elliptic curve $A$, and let $\pi_{A_0}:X_0(N)\to A_0$ be its modular parametrization. The Manin cons…

Let $u$ be a positive integer congruent to $4$ modulo $8$, such that $\operatorname{ord}_p(u(u^2-1))$ is zero or odd for every odd prime $p$. Assume conditions (1), (3), and (4) of…

Let $E$ be a fixed elliptic curve over $\mathbf{Q}$, and let $E^d$ range over its quadratic twists. The analytic rank is the order of vanishing of $L(E_d,s)$ at the center of the c…

Let $X$ be an elliptic curve, and for each finite place $v$ of good reduction let $a_X(v)$ be defined by $|X(k_v)|=p_v^{f_v}+1-a_X(v)$, where $p_v^{f_v}=|k_v|$. Sato–Tate conjectur…

Let $E$ be an elliptic curve and let $a_p$ denote its Frobenius trace at a good prime $p$. For a fixed integer $t$, Lang–Trotter conjecture. … The conjecture predicts that primes w…

Beilinson's conjecture. The vector space

Iwasawa main conjecture. The $p$-adic $L$-function is integral and generates the characteristic ideal:

Wisde-prime infinitude conjecture. The set $\mathcal{W}$ is infinite. The paper gives computational evidence for many such primes and notes that this is equivalent to the infinitud…

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…

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

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…

Let $S$ consist of the infinite place and the primes dividing $Np$, let $\mathbb{Z}_S=\mathbb{Z}[1/Np]$, let $\Lambda$ be the cyclotomic Iwasawa algebra, and let $\mathbf{T}$ be th…

Koblitz's conjecture. There exists a constant $C>0$ such that

Let $f$ be a weight-$2$ newform of level $\Gamma_0(p)$ with integer Fourier coefficients, let $E=E_f$ be the associated elliptic curve with multiplicative reduction at $p$, and let…

Let $p$ be a prime with $p\equiv 4,7,8\pmod 9$. A prime is a rational cube sum if there exist $a,b\in\mathbb Q$ such that $p=a^3+b^3$. Sylvester's rational cube-sum conjecture. It…

Let $E/\mathbb{Q}$ be an elliptic curve. An elliptic curve over $\mathbb{Q}$ is modular if there exists a primitive Hilbert modular newform over $\mathbb{Q}$ of parallel weight $2$…

Szpiro's conjecture. For every $\varepsilon>0$, there exists some constant $c>0$ depending only on $\varepsilon$, such that all elliptic curves $E$ over $\boldsymbol{\mathbb{Q}}$ s…

Frey–Mazur conjecture. For all sufficiently large primes $p$, every $p$-congruence of elliptic curves comes from an isogeny.

Let $K$ be the imaginary quadratic field in the setup, let $\mathcal{N}_{\rm Heeg}$ be the set of admissible squarefree products, and let…

Mazur–Tate conjecture. The following claims are valid:

Poonen–Rains conjecture. As $E$ varies over all elliptic curves over $F$ ordered by height,

Let $E$ be an elliptic curve over a number field $F$, and let $\Sh(E/F)$ denote its Tate–Shafarevich group. For a prime $p$, write $\Sh(E/F)[p^\infty]$ for its $p$-primary subgroup…

Let $x,y$ range over integers, and let $\epsilon>0$. Hall's conjecture. For any $\epsilon>0$, the inequality … has only finitely many solutions. This conjecture concerns the possib…