246 problems

Cohen–Lenstra conjecture. The proportion of discriminants $D\in S_X^{\pm}$ for which the Sylow $p$-subgroup of the class group of $\mathbb{Q}(\sqrt D)$ is isomorphic to $B$ satisfi…

Let $p$ be an odd prime, let $K_p^+=\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ be the maximal real subfield of the $p$th cyclotomic field, and let $h_p^+$ be its class number. The plus part…

Let $K$ be a number field, let $K_\infty^{\mathrm{cycl}}$ be its cyclotomic $\mathbb{Z}_p$-extension, and let $T_p^{\mathrm{cycl}}(K)$ be the associated Kuz'min–Tate module. Write…

Brumer's conjecture. If $G$ is abelian, then

Let $k \geq 1$ be an integer and let $d$ be an integer with $d \not\equiv 2 \bmod 4$. For a prime $p$, consider the quadratic field $\mathbb{Q}(\sqrt{dp})$. Cohn–Lagarias conjectur…

Let $m$ and $d$ be positive integers, and let $K$ range over all number fields of degree $d$. The $m$-rank of the class group of $K$ is the largest integer $r$ such that…

Let the Stickelberger–Brumer elements be the special values of the $L$-function evaluators associated with an abelian extension of global fields, and let the corresponding ideal cl…

Let $L/K$ be an abelian extension of number fields with Galois group $G$. Let $S$ be a finite set of places of $K$ containing all archimedean places and all places ramified in…

Let $K$ be a quadratic number field, and write $\operatorname{Cl}(K)[2^{\infty}]$ for the $2$-primary part of its ideal class group. Gerth's conjecture. The distribution of…

Let $G$ be an even group and let $H\leq G$ have index $2$. For a quadratic field $K$, let $f(K)$ count the unramified extensions $L/K$ whose Galois group over $\mathbb{Q}$ is isomo…

Let $F_n$ be the cyclotomic field in question, let ${\mathbb A}(F_n)$ be its ring of adeles, and let $K_{p^{n+1}-1}$ be the subgroup … of $GL(1,{\mathbb A}(F_n))$, where the produc…

Generalized Gross conjecture. The coinvariant group $X'(k^\mathrm{c}_\infty)_{\operatorname{Gal}(k^\mathrm{c}_\infty/k)}$ is finite.

Let $S$) be a finite set of primes not containing any divisor of $2|\Gamma|$, let $\Gamma_\infty\subset\Gamma$ have order $1$ or $2$, and let $\mathcal{F}(\Gamma_\infty,X)$ be the…

Let $p$) be a fixed prime, let $d$ be a positive integer, and let … be a $\mathbb{Z}_p^d$-extension tower in which $K_n/K$ corresponds to the subgroup $(p^n\mathbb{Z})^d$. Write…

Let $k$ be a number field, let $G$ be a transitive permutation group, and let $p$ be a prime such that $p\nmid \lvert G\rvert$. For a real number $X>1$, let $E_k(G,X)$ be the set o…

Let $H/F$ be a finite abelian CM extension of a totally real number field $F$, let $G={\rm Gal}(H/F)$, let $p>2$ be prime, and let $S,T$ be as in the source. Write…

Let $k$ be a number field and let $G$ be a finite group. Write … The Cohen–Lenstra–Martinet heuristics give a distribution on $\mathbb{Z}[G]'$-modules. Cohen–Lenstra–Martinet conje…

Let $K=\mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic field, and let $C_1$ be the cone whose elements have angle between $\pi/36$ and $2\pi/36$. Let $P_M(-d)$ count primes less t…

For each integer $q\teq1$, let $E(q)$ denote the number of isomorphism classes of elliptic curves over $Q$ with conductor $q$. Brumer–Silverman’s conductor conjecture. For every…

Let $K$ be a nowhere-evenly-ramified number field of even degree $n$ with real signature $(r_1,r_2)$. Enumerate such fields by discriminant. Cohen–Lenstra–Martinet–Malle conjecture…

Cohen and Martinet's conjecture. For every “reasonable” non-negative function $f$ on the isomorphism classes of finite $eZ_S[Gamma]$-modules,

Moment-determination conjecture. The phenomenon that the moments

Let $L/K$ be a Galois CM-extension with Galois group $G$, and let $S$ be a finite set of places of $K$ containing all archimedean places and all places ramified in $L/K$. Let…

Let $L/K$ be an abelian CM-extension with Galois group $G$, and let $S$ be a finite set of places of $K$ containing all archimedean places and all places ramified in $L/K$. Let…

Burns's conjecture. One has $\Theta(\phi)\in\mathbb{Z}[G]$, moreover $\Theta(\phi)\in\operatorname{Ann}_{\mathbb{Z}[G]}(Cl_S(K))$, and if $S'$ satisfies…