48 problems

BSD regulator-quotient square conjecture. Then

Let $p$ be an odd prime, let $K=\mathbb{Q}(\mu_p)$ and let $F$ be the false Tate extension layer used in the paper. Let $E/\mathbb{Q}$ be an elliptic curve with $E(K)$ finite, let…

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 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

Parity conjecture.

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 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 $f\colon X\rightarrow C$ 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…

Beilinson's conjecture. If $L(1,A)\neq0$, then

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…

Let $E/\mathbb{Q}$ have split multiplicative reduction at a prime $p$, put $G_p=\left(\mathbb{Z}/p\mathbb{Z}\right)^*/\langle-1\rangle$, and let $\lambda(a,p)$ be the even modular…

Let $E/\mathbb{Q}$ have split multiplicative reduction at a prime $p$, put $G_p=\left(\mathbb{Z}/p\mathbb{Z}\right)^*/\langle-1\rangle$, and let $\lambda(a,p)$ be the even modular…

Let $E/\mathbb{Q}$ be an elliptic curve, let $G_M=\left(\mathbb{Z}/M\mathbb{Z}\right)^*/\langle-1\rangle$, and let $\theta_M=\sum_{a\in G_M}\lambda(a,M)[a]$ be the Mazur–Tate eleme…

Let $A/\mathbb{Q}$ be an abelian variety with $L(A,1)\neq 0$ that is an optimal quotient of $J_0(N)/\mathbb{Q}$ attached to a newform. Let $c_{\infty}(A)$ be the number of connecte…

Let $A/\mathbb{Q}$ be an abelian variety with $L$-function $L(A,s)$, and let $\Omega(A)$ denote its real period. For each prime $p$, let $c_p(A)$ be the Tamagawa number of…

Let $E/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$, let $L_p(E)$ be its $p$-adic $L$-function, and choose a topological generator $\gamma$ of the Galois gr…

Let $E$ be an elliptic curve over $\mathbb{Q}$ of conductor $N$, let $p\geq5$ be a prime, and let $\widetilde{\boldsymbol{\delta}}$ be the collection of Kurihara numbers. Assume th…

Gross's conjectural formula. The order of the Tate–Shafarevich group of the Gross curve over $H$ is

Agashe's conjecture. Up to a power of $2$, the square of the order of $E^{-D}(\mathbb{Q})$ divides

Let $A$ be an abelian variety of dimension $g$ over a number field $F$, assume that $A$ has good reduction everywhere, and let $P_{A,\infty}(\eta)$ and $\mathfrak a_\eta$ be the ar…

Let $A$ be an abelian variety of dimension $g$ over a number field $F$, let $A^t$ be its dual abelian variety, and let $L^*(A,1)$ denote the leading coefficient of $L(A,s)$ at…

Let $X$ be a smooth projective geometrically connected variety over a finitely generated field $K$, with a smooth projective model $f:{\mathcal X}\to{\mathcal Y}$ as above, and let…