30 problems

Let $K$ be the imaginary quadratic field, let $(\tilde{f},\chi_\varepsilon)$ be a Heegner pair, and let $\Lambda$ be the relevant Iwasawa algebra. Let $\bT_\varepsilon$ and…

Michel–Venkatesh adelic mixing conjecture. If $|D_i|\to\infty$ and $q_i\to\infty$ as $i\to\infty$, then

Let $d\to\fty$ along ${\mathbb D}^\flat$, let ${\sf R}_d^\Delta([\mathfrak s])$ be the shifted diagonal Heegner packet, let $\mu$ be the quotient measure on ${\sf S}^2$, and let…

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 $D$.…

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 $K$ be an imaginary quadratic field, let $K_\infty$ be the relevant anticyclotomic extension, and let $\Lambda$ be its Iwasawa algebra. Let $\mathfrak{X}$ denote the relevant P…

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 $z^{\rm Hg}$ 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 $\mathcal{R}$ 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…

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…

Let $K$ be an imaginary quadratic field, let $p\geq 5$ be a prime, let $E$ be an elliptic curve over $\mathbb{Q}$ with modular parametrization $\pi$, and let $K_{\infty}/K$ be the…

Let $E$ be an elliptic curve over $\mathbf{Q}$, let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis, let $P_K\in E(K)$ be the Heegner point, let $C$ be the pr…

Let $\mathbf f$ be the Hida family over $K$, let $L_p^{\rm alg}(\mathbf f/K)\in\mathbb I[[\Gamma_\infty]]$ be its algebraic anticyclotomic $p$-adic $L$-function, and let $\kappa$ d…

Let $E$ be an elliptic curve over ${\mathbb{Q}}$, let $K$ be an imaginary quadratic field, let $p$ be a prime, and let $K_{\infty}/K$ be the anticyclotomic ${\mathbb{Z}_p}$-extensi…

Horizontal nonvanishing conjecture. The cohomology class $\mathfrak{Z}_0$ is not $R$-torsion.

The two-variable Heegner point main conjecture. Assuming that $R$ is regular,

Yuan–Zhang–Zhang-type conjecture. There exists $b\in\widehat{B}^{\times}$ with $k(b,\omega_{\varphi})\neq0$ such that

Mean-size conjecture.