32 problems

Let $K/\mathbb{Q}$ be Galois of degree $n$ with group $G$, let $\eta\in K^\times$ generate a multiplicative $\mathbb{Z}[G]$-module of $\mathbb{Z}$-rank $n$, and let…

Perrin-Riou's integrality conjecture. One has

Let $K\subset\mathbb C_p$ be a discrete valuation subfield, meaning that its valuation is induced from the valuation on $\mathbb C_p$, and let $R$ be the valuation ring of $K$. Ass…

Regulator image conjecture. If $L(E/K,\chi,1)=0$, then $\mathcal{Q}^\chi$ belongs to the image of this regulator. This predicts a direct relation between the mock plectic invariant…

Let $K$ be a quadratic imaginary field, let $E/\mathbb Q$ be an elliptic curve, let $T_p(E)$ be its $p$-adic Tate module, and let $\delta_\infty$ and $\log_E^{a_p}$ be the maps def…

Let $\mathcal O$ be the coefficient ring, let $I$ be the relevant augmentation ideal, let $c\in\bigcap_{\Lambda_I}^r H^1_\Sigma(\mathcal O_{K,S},\mathbb T)_I$ be the special elemen…

Let $\beta^{(k)}$ be the $k$-th derived Bockstein map in the source's Selmer-complex setting, and let $r^+$ and $r^-$ be the relevant plus and minus ranks. Mazur–Bertolini–Darmon a…

Let $K/k$ be a CM Galois extension with group $G$ and complex conjugation $c$. Let $U^{-1}$ be the group of minus $p$-units of $K$, let $X^{-1}$ be the minus part of the degree-zer…

The $p$-adic Borel regulator injectivity conjecture. For every number field $k$, every rational prime $p$, and every odd dimension $2m-1\geq3$, this map is injective.

Let $E/\mathbb{Q}$ be an elliptic curve, let $r={\rm rank}(E(\mathbb{Q}))$, let $I_\infty$ be the augmentation ideal associated with the cyclotomic $\mathbb{Z}_p$-extension…

Let ${\mathcal K}$ be the set of totally real number fields. For $K\in{\mathcal K}$, let $D_K$ be its discriminant, let ${\mathcal T}_K$ be the relevant $p$-torsion group, and let…

Equidistribution conjecture. The following assertions hold:

Let $E$ be an elliptic curve over $\mathbb Q$, and let $p>2$ be a prime at which $E$ has good ordinary reduction. Let $E_{\mathbb Q_p}:=E\times_{\mathbb Q}\mathbb Q_p$. Let…

Let $\Delta$ act on $U_K\otimes_{\mathbf Z}\mathcal F$ and on $\mathcal U_p\otimes_{\mathbf Z_p}\mathcal F$, and let $\chi$ be a character of $\Delta$. Write … for the induced char…

Let $K/\mathbb{Q}$ be a real Galois extension with Galois group $G$, and let $\eta\in E_K$ be a Minkowski unit, meaning a unit generating a sub-$G$-module of finite index in $E_K$.…

Let $q\in\{2,3\}$, let $p>q$ be prime, and let $t$ be an integer. For a totally real number field $K$ with $\mathrm{Gal}(K)\cong(\mathbb{Z}/q\mathbb{Z})^t$, let $R'_{F,p}$ denote t…

Let $p>3$ be a prime, let $K$ range over cyclic cubic fields, and let $R_{K,p}$ denote the normalized $p$-adic regulator. Hofman-Zhang's conjecture. … The source says that this pre…

Let $R=\{\mathfrak p_1,\ldots,\mathfrak p_r\}$, let $E_R^*$ be the group of totally positive $R$-units, let $\vartheta\in H_{n+r-1}(E_R^*,\mathbf Z)\cong\mathbf Z$ be a generator,…

The Eisenstein-cocycle regulator conjecture. For every subset $J\subset R$,

Let $K/\mathbb{Q}$ be Galois of degree $n$ with group $G$, and let $\eta\in K^\times$ generate a multiplicative $\mathbb{Z}[G]$-module $F$ of $\mathbb{Z}$-rank $n$. An element of…

Let $K/\mathbb{Q}$ be Galois of degree $n$ with group $G$. For an irreducible $p$-adic character $\theta$ and an absolutely irreducible constituent $\varphi\mid\theta$, assume…

Let $K/\mathbb{Q}$ be Galois with group $G$, let $\eta\in K^\times$ generate a multiplicative $\mathbb{Z}[G]$-module of $\mathbb{Z}$-rank $n$, and let…

Let $K/\mathbb{Q}$ be Galois, let $G=\operatorname{Gal}(K/\mathbb{Q})$, and let $F$ be the multiplicative $\mathbb{Z}[G]$-module generated by $\eta$. Suppose that $F$ is monogenic.…

Let $p>3$ be a prime, and let $M_p^{\mathrm{inert}}$ be the measured space of matrices associated with the inert case. Define the random variable … where $f=X^2+3Y^2$. Inert-case v…

Let $p>3$ be a prime. Let $\mathbb Q_p^{(3)}$ be the unique unramified cubic extension of $\mathbb Q_p$, with ring of integers $\mathbb Z_p^{(3)}$ and cyclic Galois group generated…