34 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…

Let $g\in S_2(\Gamma_0(N))$ be a newform, let $p>2$ be a prime of good ordinary reduction for $g$, and let $K$ be an imaginary quadratic field satisfying the residual irreducibilit…

Smooth-complexity conjecture. One should have

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…

Let $p$ range over primes, and let $h_2(-p)$ denote the order of the 2-Sylow subgroup of the ideal class group of $\mathbb{Q}(\sqrt{-p})$. For a positive integer $k$, consider the…

Let $K$ be an imaginary quadratic field, let $p$ split as $(p)=\mathfrak q\overline{\mathfrak q}$, and let $\chi$ be the nontrivial character of $\operatorname{Gal}(K/\mathbb Q)$.…

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

Let $\lambda$ be a partition, let $G_{\lambda}(p)$ denote the corresponding finite abelian $p$-group, and let $c(\lambda)$ be the associated partition parameter. Holmin et al.'s co…

Let $\lambda$ be a partition, let $G_{\lambda}(p)$ denote the corresponding finite abelian $p$-group, and let $c(\lambda)$ and $n$ be the associated partition parameters. Let…

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 $k \ge 2$ be an integer, let $E$ be an imaginary quadratic field of discriminant $D$, and let $\ell=1$ or a prime with $\ell\nmid D$. Let $\lambda_q=\lambda_q(k)$ be the quanti…

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…

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.

Let $K$ be a global field, and let $\mathcal G_K^{ab}$ denote the abelianization of its absolute Galois group. For an imaginary quadratic field $K$, let $\widehat{\mathbb Z}$ denot…

Holmin–Jones–Kurlberg–McLeman–Petersen conjecture. As $h\to\infty$ through odd values,

Let $\mathcal{F}(h)$ denote the number of imaginary quadratic fields with class number $h$. Soundararajan's conjecture. For large $h$, one has … The variation in size depends on th…

Let $K$ be an imaginary quadratic field, let $\mathfrak{f}$ be a nonzero proper integral ideal of $K$, and let $K_{\mathfrak{f}}$ be the ray class field of $K$ modulo…

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 $\mathbb{Z}_l$-…

Monotonicity conjecture. The farthest distance reachable with a given step bound $k$ increases as $d$ decreases, subject to the constraint that $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$…