300 problems

Iwasawa main conjecture. The $p$-adic $L$-function is integral and generates the characteristic ideal:

Let $k$ be totally real, let $K/k$ be an abelian CM extension with group $G$, let $p$ be a rational prime, and let $\chi$ be a one-dimensional character of $G$. Define … Gross's hi…

Mazur–Tate–Teitelbaum conjecture. One has $L_p(f,1)=0$ and

Let $\mathcal{R}_\infty$ be the two-variable Iwasawa algebra, let $\mathcal{L}_p(f/K)$ be the associated two-variable $p$-adic $L$-function, and let…

Assume that $\mathcal{R}$ is regular. Let $K_\infty/K$ be the $p$-anticyclotomic $\mathbb{Z}_p$-extension, let $\mathcal{R}_\infty$ be the associated Iwasawa algebra, and define ……

Let $E/\mathbb{Q}$ be an elliptic curve with supersingular reduction at $p$, where $p$ is an odd prime and $a_p(E)=0$. Let $\mathbb{Q}_\mathrm{cyc}$ be the cyclotomic…

Let $K$ be an imaginary quadratic field in which the odd prime $p$ splits as $(p)=\mathfrak p\overline{\mathfrak p}$, let $\Lambda$ be the anticyclotomic Iwasawa algebra, let…

Boundedness conjecture. The function $\mathscr{L}_p(\pi)$ is a bounded analytic function on

Let $V\cong\mathcal K^{2d}$ be an even-dimensional $p$-adic representation of $G_{\mathbb Q}$ satisfying condition $\mathrm{(Geom)}$, with no Artin representation as a subquotient.…

For $\kappa\in\mathbb{Z}_p$, let … where $\pi_0(\bar{t})$ is the unit root associated with the closed point $\bar{t}$. Dwork's conjecture. The unit root $L$-function…

$p$-adic Birch–Swinnerton-Dyer conjecture.

Let $E_1$ and $E_2$ be elliptic curves satisfying the hypotheses stated in the paper, and write $g$ for their common rank in the setting under consideration. For each $i$, let…

Let $E$ be an elliptic curve over $\Q$, let $p>3$ be a prime, let $T_p(E)$ be its $p$-adic Tate module, and let $c_{\Q}^{\textup{BK}}\in H^1(\Q,T_p(E))$ be Kato's Beilinson-Kato cl…

Let $L_p(\textup{\bf f}_\kappa,\sigma)$ denote the Manin–Višik $p$-adic $L$-function for an arithmetic specialization $\kappa$, and consider its central critical points. Greenberg'…

Let $f\in S^{new}_{2r}(\Gamma_0(N))$ be a newform, and let $p\nmid N$ be an Eisenstein prime for $f$, meaning that the residual representation $\overline{\rho}_f$ is reducible. Let…

BDP characteristic-ideal conjecture. One has

In the abelian rank-one setting, let $\epsilon$ be the minus unit singled out by Stark's complex conjecture, let $\mathfrak q\mid p$ be the chosen prime, and let…

Let $\varepsilon_\infty^\chi$ be the inverse-limit Rubin–Stark element, let $\mathscr{C}_{\omega_p^+}^{\mathrm{str}}$ be the explicit reciprocity operator, and let…

Let $F$) be a totally real field, let $\mathbf{f}$ be a Hilbert modular form of parallel weight $2$ over $F$, and let $L_p(s,\mathbf{f})$ be its cyclotomic $p$-adic $L$-function. L…

Let $M/\mathbf Q$ be a pure motive of weight $-2$, let $V$ be its $p$-adic realization, and let $D\subset\mathbf{D}_{\mathrm{cris}}(V)$ be a regular $\varphi$-submodule. Define … a…

Schneider's vanishing conjecture. For all $j>1$,

Greenberg–Benois conjecture. If $\mathcal{L}_p(S,V)$ has $e_0+t$ trivial zeros, then $S^{e_0+t+r}$ divides it exactly and

Let $\mathcal{M}$ be the motive attached to the representation $T$, let $\mathfrak{I}_p$ index the subsets of basis vectors of size $g_-$, and let $\mathcal{H}_+$ be the relevant s…

Let $p$ be an ordinary prime for an elliptic curve, and let the cyclotomic $p$-adic $L$-function have $bc$-invariant $bc$. Under mild assumptions, the valuation of the $p$-adic per…

Let $E/\mathbb Q$ be the number field and $\bar G$ its relevant Galois group, let $E_0=\{x\in E:\operatorname{Tr}_{E/\mathbb Q}(x)=0\}$, and let $\mu_\infty$ and $\mu_p$ be the arc…