96 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…

The weak Birch–Swinnerton-Dyer conjecture. The integers $r_n$ and $R(n)$ are equal.

Let $E$ be an elliptic curve, let $\mathcal{L}(E)$ denote its algebraic $L$-value, let $\mathrm{BSD}(E)$ denote its Birch–Swinnerton-Dyer quotient, and let … be the $q$-adic valuat…

Let $E_1$ and $E_2$ be elliptic curves satisfying the hypotheses stated in the paper, and write $g$ for their common rank in the setting under consideration. For each $i$, let…

Let $A$ be the modular abelian variety in the paper, and let $L_A(s)$ be its $L$-function. Let $K_A$ be the coefficient of the leading term of the Taylor expansion of $L_A(s)$ at…

Let $E$ be an elliptic curve over $\mathbb{Q}$, let $H/\mathbb{Q}$ be a finite Galois extension, and let $\varrho\colon G_\mathbb{Q}\to\operatorname{GL}(V_\varrho)$ be an Artin rep…

Strong BSD conjecture. The group $\Sha(A/{\mathbf{Q}})$ is finite and

Let $A$ be an abelian variety over a number field $k$, and let $L(A,z)$ be its Hasse–Weil $L$-series. Let $\text{\russ\char88}(A_k)$ denote the Tate–Shafarevich group of $A$ over…

The rank 0 part of the Birch and Swinnerton-Dyer conjecture. For every $E\in\mathscr{E}$,

BSD conjecture for abelian varieties. The rank $r$ equals the analytic rank $\operatorname{ord}_{s=1}L(A,s)$, and

Refined Birch–Swinnerton-Dyer conjecture. The Birch–Swinnerton-Dyer conjecture holds, $\sha(J_X)$ is finite, and

BSD regulator-quotient square conjecture. Then

Let $E$ be the elliptic curve of conductor $N$, let $D$ be a negative fundamental discriminant satisfying the hypotheses above, and let $E_D$ be the quadratic twist of $E$ by $D$.…

Let $G$ be a finite abelian group, let $V\to W$ be a $G$-cover as above, and let $X$ satisfy the hypotheses that make ${\rm H}^1(X,{\mathcal O}_X)$ $G$-isomorphic to…

Let $G$ be a finite abelian group and let $V\to W$ be a $G$-cover of smooth projective geometrically connected curves of positive genus over ${\bf Q}$. Set $A={\rm Jac}(V)$, let…

Two-variable $p$-adic Birch–Swinnerton-Dyer conjecture. The element $\mathbf{L}$ lies in $\mathbf{I}^r$ and satisfies

Let $p$ be a prime with $p\equiv 4,7\pmod 9$, and define … Let $E_{2q}$ be the elliptic curve $Y^2=X^3-432(2q)^2$. The BSD parity conjecture. It predicts that … The source explains…

Let $n>2$ be a cube-free integer, and let $E_n$ be the elliptic curve … Write $L(s,E_n)$ for its complex $L$-function and $\operatorname{rank}_{\mathbb Z}E_n(\mathbb Q)$ for its Mo…

Let $K$ be the imaginary quadratic field in the setup, let $\boldsymbol{\kappa}^{\rm Heeg}$ be the Heegner-point Kolyvagin system, and let…

Let $\boldsymbol{\delta}=\{\delta_n\}_{n\in\mathcal{N}}$, and let $\mathscr{M}_\infty(\boldsymbol{\delta})$ be the limiting divisibility invariant defined from these quantities. Le…

Let $E$ be an elliptic curve, let $K[c]$ be the ring class extension of a real quadratic field $K$, and let $r_E(K[c])$ be the Mordell–Weil rank. Let…

Let $f\colon X\rightarrow C$ be a smooth projective surface fibration over a smooth projective curve over $k=\mathbb{F}_q$, with smooth generic fibre $X_\eta$ over $K=k(C)$, and le…

Let $E$ be an elliptic curve with good ordinary reduction at a prime $p$, let $r=\operatorname{rank}E(\mathbb{Q})$, and let $L_p(E,T)$ be the ordinary $p$-adic $L$-series. Write…

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good supersingular reduction at $p$, and let $r=\operatorname{rank}E(\mathbb{Q})$. Let $L_p(E,T)$ be the supersingular $p$-adic…