50 problems

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

Perrin-Riou's conjecture. Equality should hold in this inclusion:

Let $E/\mathbb{Q}$ have good ordinary reduction at $p$, let $K_\infty/K$ be the anticyclotomic $\mathbb{Z}_p$-extension, let $\Lambda$ be its Iwasawa algebra, and let…

Let $D_i<0$ be fundamental discriminants, let $\operatorname{Cl}(D_i)$ be the class group of $\mathbb{Q}(\sqrt{D_i})$, let $\sigma_i\in\operatorname{Cl}(D_i)$, and let $q_i$ be the…

Let $E_{/\mathbb{Q}}$ be an elliptic curve of conductor $N$, let $K$ be the quadratic imaginary field associated with a Heegner discriminant, and let $y_K\in E(K)$ be the correspon…

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 $D$.…

$\Lambda$-adic Gross–Zagier conjecture.

Heegner $L$-function conjecture. Inside $\Gamma_{\mathrm{cycl}}\otimes\Lambda_{\mathrm{anti}}$, one has

Let $\Lambda^{\rm ac}$ be the anticyclotomic Iwasawa algebra, let $\mathbf{T}$ be the big Galois representation, and suppose ${\rm Sel}_{\rm Gr}(\mathbf{T})$ is torsion-free of…

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

Let $E/\bQ$ be an elliptic curve, let $p>2$ be a prime of good ordinary reduction, and let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis. Let $\Lambda$ be t…

Let $(E,p,K)$ be a triple in which $E$ is an elliptic curve over $\mathbf{Q}$ of analytic rank $2$, $p$ is an odd prime of anomalous good ordinary reduction for $E$, and $K$ is an…

Let $E/\mathbf{Q}$ be an elliptic curve of analytic rank $2$, let $p$ be an odd prime of good ordinary anomalous reduction, and let $K$ be an imaginary quadratic field satisfying t…

Gross–Zagier conjecture. If $P_L$ has infinite order in $E_L(L)$, then $P_L$ generates a subgroup of finite index in $E_L(L)$, and this index equals

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N_0$, and let $p\geq 5$ be a prime of good supersingular reduction. Let $K$ be an imaginary quadratic field with discriminant…

Signed Heegner point main conjecture. Both $\mathcal S^\pm$ and $\mathcal X^\pm$ have $\Lambda$-rank one, and

Let $z^{\rm Hg}$ be the $\Lambda$-adic Heegner element, let $\mathcal D$ be the derived descent map, let ${\rm Eul}_S$ be the product of Euler factors at the finite places in $S$,…

Let $G_c^+$ be the indicated character group, and let $\epsilon_{A,c,\psi}$ be the algebraic correction factor attached to the abelian variety $A$, the conductor parameter $c$, and…

Michel–Venkatesh's equidistribution of subgroups conjecture. For any fixed $\eta>0$, the sequence of orbits $G_D\cdot x_{0,D}$ becomes equidistributed with respect to $d\mu$ on…

Let $\mathcal{R}$ be a primitive branch of a Hida family. Its Big Heegner points determine proportionality coefficients in the fraction field of $\mathcal{R}$, and let $\mathbb{S}$…

Perrin–Riou–Howard's Heegner point main conjecture. The $\Lambda$-modules $\mathrm{Sel}_{\mathrm{Gr}}(K,\mathbf{T})$ and $\mathrm{Sel}_{\mathrm{Gr}}(K,\mathbf{A})$ have rank $1$ an…

Let $F$ be a totally real number field, $\mathbb{A}$ its ring of adeles, and $\pi$ a cuspidal cohomological automorphic representation of $\operatorname{GL}_{2}(\mathbb{A})$. For a…

Mazur–Rubin's question. As $K$ varies, what are the lines $L_K$, and how are they distributed in $E(\mathbb{Q})\otimes\mathbb{Q}_p$? This asks for a description of the shadow lines…

Let $E$ be ordinary at $p$, assume $\widetilde\varepsilon=-1$, and let $L_p(A)$ be the relevant $p$-adic $L$-function. Let $\mathscr Y^-$ be the anticyclotomic character space, let…

Consider the setup of the anticyclotomic family in the paper and let $\widetilde r=\dim A^{\dagger}(K)_{\mathbf Q}$. If $\widetilde\varepsilon=+1$, then $\widetilde r$ is even and…