100 problems

Let $K$ be a complex cubic field, let $K(\mathfrak{f})$ be the relevant narrow ray class field, let $e_{\mathfrak{f}}$ be the number of roots of unity in $K(\mathfrak{f})$, and fix…

Fix an odd prime number $p$ and let $K$ be an imaginary quadratic field. Let $G_K$ be the Galois group of the maximal unramified pro-$p$ extension of $K$, called the $p$-tower grou…

Stark–Heegner point conjecture. The points $P_\tau^+$ and $P_\tau^-$ are defined over $H_{\mathcal O_\tau}$ and $H_{\mathcal O_\tau}^+$, respectively, and $P_\tau^-$ belongs to the…

Let $K$ be a real quadratic number field, and let $X(\mathfrak f)$ denote Shintani's invariant associated with an ideal or modulus $\mathfrak f$, defined in terms of the double sin…

Let $t=(d,r,Q)$ be an admissible tuple with associated real quadratic field $K$, let $d=d_{j,m}$, and let $f$ be the conductor of $Q$. Let $s=(d,r,Q,G,g)$ be a fiducial datum. Deno…

Let $F$ be totally real, let $C_{\mathfrak f}$ be the narrow ray class group modulo $\mathfrak f$, and let $s_\iota$ denote complex conjugation classes at the real embeddings. Let…

Let $K$ be an imaginary quadratic field of discriminant $d_K$ other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, let $\mathfrak{n}$ be a proper nontrivial ideal of its…

Nonvanishing determinant conjecture. $\det(N_\ell)\neq 0$ for every odd prime $\ell$.

Stark's conjecture. The coefficient $R_K(\chi)$ has the form of a regulator: it is the determinant of a matrix whose entries are linear combinations of logarithms of units of $K_0$…

Let $K$ be a real quadratic field, let $\mathfrak{C}$ be a narrow ideal class of $K$, and define … where $\mathfrak{C}_1$ and $\mathfrak{C}_2$ are the ideal classes used in this co…

Let $X$ be a regular connected scheme, flat and proper over $\operatorname{Spec}({\mathbb Z})$, with generic fibre $X\otimes_{\mathbb Z}{\mathbb Q}$ projective over ${\mathbb Q}$.…

Let $p$ be an odd prime splitting completely in $k$, let $K/k$ and the other notation be as in the paper, and let $G=\operatorname{Gal}(K/k)$. Let $\eta^+_{K/k}$ be the image of th…

Let $k$ be a totally real number field, let $p$ be an odd prime splitting completely in $k$, and let $K$ be as in the paper. Write $G=\operatorname{Gal}(K/k)$ and let $\delta(K/k)$…

Let $K$ be a real quadratic field, and let Stark numbers and Galois action refer to the algebraic quantities and Galois-theoretic data sought in Hilbert's twelfth problem. Manin's…

Let $K$ be a local or global field, and consider the zeta-regularized numbers arising from a family of zeta functions associated with $K$. Stark's conjecture. For more general fiel…

Let $k$ be a field and let $q,r,q',r'$ be nonnegative integers. The condition $(N_q^r,k)$ asserts that, for every $k'\in\operatorname{\mathcal E}(k)$ and every discrete torsion abe…

Hilbert's conjecture. The class number of $H_K$ is always odd.

Darmon-Greenberg-Liu reciprocity conjecture. The extension $E_\tau(j)/E_\tau$ is conjectured to be an abelian extension unramified outside $2$. This is the concise field-theoretic…

For squarefree $D>1$, let $\mathcal{O}$ be the ring of integers in $\mathbb{Q}(\sqrt{D})$, and let $\mathcal{S}_D$ consist of the elements $u=A+B\sqrt{D}$ with $A,B\in\mathbb{Z}$ s…

Let $H/F$ be an abelian extension of number fields, let $W$ be the number of roots of unity in $H$, and let $S$ be a finite set of places of $F$ containing the ramified and infinit…

Stark's conjecture. Except in the trivial case $\e_\A=1$, one has

Fix $d>3$, let $\Delta=(d+1)(d-3)=f^2\Delta_0$ with $\Delta_0$ a fundamental discriminant, and for each positive divisor $f'$ of $f$ let $\mathcal O'=\mathcal O_{(f')^2\Delta_0}$.…

Let $d\geq 4$, set $\Delta_d=(d+1)(d-3)$, and let $K_d=\Q(\sqrt{\Delta_d})$. For a Weyl–Heisenberg line-SIC $S$ in dimension $d$, let $\fieldvec{S}$ denote its SIC field, and let…

Let $K$ be an imaginary quadratic field with discriminant $d_K$, suppose $(5)=\wp_5^2$, let $\Sigma=\textsf{K}_1$, and let $\rho\in\Sigma$. Let…

Let $d$ be an integer with quadratic discriminant $d_k=\prod_{p\mid d}p^*$, where $p^*=(-1)^{(p-1)/2}p$ for odd $p$, and let $2^*\in\{-4,8,-8\}$. Let $\textsf{K}_{\wp_5'}$ be the i…