17 problems

Large-prime existence conjecture. For all sufficiently large primes $p$, there exist an elliptic curve $E/\mathbb{Q}$ and an imaginary quadratic field $K_0$ such that the triple…

Fix a prime $p$. For imaginary quadratic fields $K$ in which $p$ is inert, let $\bb\b5_p(K)$ be the Iwasawa $\bb\b5$-invariant, and let $r$ be a nonnegative integer. Ellenberg–Jain…

Invariants Heuristics. Among such imaginary quadratic fields, one always has

Let $p$ be a prime number and $k$ a number field. Let $\widetilde{k}$ be the maximal multiple $\mathbb{Z}_p$-extension field of $k$, let $L_{\widetilde{k}}$ be its maximal unramifi…

Tower ground state conjecture. The fields with discriminants $d\in\{-225\,299,-343\,380,-423\,476,-486\,264\}$ of type $\mathrm{F}.7$,…

Let $F$) be an imaginary quadratic field, let $\mathfrak{m}$ be a non-Eisenstein maximal ideal of residue characteristic $p>2$ associated to a residual Galois representation…

Let $K$ be an imaginary quadratic field in which $p$ splits, let $\chi:\operatorname{Gal}(\overline{\mathbb{Q}}/K)\to\mathbb{Z}_p^\times$ be a finite continuous character, and let…

The authors' conjecture. For each such $p$, $q$, $n$, and $m$, there are infinitely many imaginary quadratic fields of this form whose class number is divisible by $n$. This extend…

Let $K=\mathbb{Q}(\mathbb{\sqrt{-D}})$ be an imaginary quadratic field, where $D>1$ is square-free, and write its class group as a direct product of cyclic groups. For integers…

Mennicke's modularity conjecture. Every such elliptic curve $E$ is modular.

For an odd prime $p$, an elementary abelian $p$-group is a group isomorphic to $(\mathbb{Z}/p\mathbb{Z})^r$. Nonexistence conjecture for elementary abelian class groups. No element…

Fix $n\in\mathbb{N}$ and a partition $\lambda=(n_1,\dots,n_r)$ of $n$, with $n_1\geq\cdots\geq n_r\geq1$, and define … For a finite abelian group $G$, let $\mathcal{F}(G)$ denote t…

The refined Soundararajan conjecture. As $h\to\infty$ through odd values,

Let $l$ be an odd prime, let $F$ be an imaginary quadratic field, and let $\mathcal{E}$ be an elliptic curve over $F$. Let $F^{\mathrm{anti}}$ be the anticyclotomic $\mathbb{Z}_l$-…

Infinitude conjecture. There are infinitely many imaginary quadratic fields $K$ for which the absolute abelian Galois group is isomorphic to $G$.

Existence conjecture. There exists a continuous Galois representation

Let $K$ be an imaginary quadratic field, let ${\mathcal O}_K$ be its ring of integers, and let $\Gamma_0({\mathfrak p})$ denote the corresponding congruence subgroup for a prime id…