6 problems

Let $\chi = \chi_0 \chi_p$, with $\chi_0 \ne 1$, be an even irreducible rational character and let $K=K_\chi$. For every irreducible $p$-adic character $\varphi$ dividing $\chi$, s…

Let $F/\mathbb{Q}$ be a totally real abelian extension of order $m$, and suppose … For primes $p\gg 0$ satisfying $p\nmid m\,\operatorname{disc}_F$, write $F$-regular for the condi…

Strong $\varepsilon$-conjecture. For each $0<\varepsilon<1$, for almost all $K$ in ${\mathcal K}_G^s$, the sequence

Successive-maxima conjecture. (i) As $f$ increases from its minimum, the successive maxima of $C_{\varepsilon}({\mathcal F}_{G,f}^s)$, $C'_{\varepsilon}({\mathcal F}'{}_{G,f}^s)$,…

Annihilation conjecture. The element ${\mathcal A}'_K(c)$ annihilates ${\mathcal T}_K$.

Solomon's conjecture. The element $\operatorname{sol}_F$ annihilates