11 problems

Let $M$ be an $L$-admissible premotivic structure over $K$, let $\tau:K\to\mathbb{C}$ be an embedding, and let $\lambda$ be any finite prime of $K$. Write…

Let $M$ be an object of the category of premotivic structures over $K$ that is $L$-admissible at every prime. Let $L(M,s)$ be its $K\otimes\mathbb{C}$-valued $L$-function, let…

Fix number fields $K,L$ and a pure motive $M$ of weight $w$ over $K$ with coefficients in $L$. Let $L(s,M)$ be its motivic $L$-function, let $L_{\infty}(s,M)$ be its archimedean fa…

Let $M$ be a critical motive, let $\mathbb{Q}(M)$ be its field of coefficients, and let $\sigma:\mathbb{Q}(M)\to\mathbb{C}$ be an embedding. Write $L(s,\sigma,M)$ for the motivic…

With $S=\eta_{!*}\eta^*M$, $D(S)$ its Verdier dual, and the determinant lines and $\mathbb Q$-structures described in the paper, the Beilinson determinant-form conjecture predicts…

Let $X$ be regular, projective, and equidimensional over $\mathbb Z$, let $M=\operatorname{M}(X)(m)$, and let the compactly supported motivic cohomology and duality data be as in t…

Let $M$ be a geometric motive over $\mathbb Z$, with $DM$ its dual, and let $\chi(DM)$ denote the relevant Euler characteristic. The special $L$-values conjecture predicts … Assumi…

Let $M$ be a motive over $\mathbb Q$. Its $\ell$-independence conjecture predicts that the $L$-series of $M$ is independent of the choice of the prime $\ell$ and the choice of…

Let $X$ be a smooth projective scheme over $\mathbb{Q}$ of dimension $d$, let $H^{2i-1}(X)$ denote its relevant motive, and define … Here $CH^i(X)^0$ is the space of codimension-…

Let $X$ be a smooth projective scheme over $\mathbb{Q}$, and let $H^w(X)$ be its weight-$w$ motive with completed $L$-function $\hat{L}(H^w(X),s)$. Artin's conjecture. The function…

Let $X$ be a smooth projective scheme over $\mathbb{Q}$ of equidimension $\delta$, and let $L(H^w(X),s)$ and its completed function $\hat{L}(H^w(X),s)$ be the corresponding motivic…