249 problems

Beilinson's conjecture. The vector space

Multiple zeta values are real numbers obtained by shuffle-regularised iterated sums or integrals, and motivic multiple zeta values are their algebraic counterparts in the ring…

Let $F$ be a field, and let $K^{(q)}_{2q}(F)$ and $\operatorname{CH}^q(\operatorname{Spec}F,2q)$ have the meanings used in the Beilinson–Soulé vanishing conjecture. Beilinson–Soulé…

Let $X$ be a smooth projective variety over a finite field, let $i,n$ be integers, and let $\ell$ be a prime invertible in the finite field. Parshin's conjecture. The motivic cohom…

Let $X$ be a smooth variety over a field of characteristic zero, and let $H^3(\Gamma(X,4))$ denote the third cohomology of Goncharov's weight-four polylogarithmic complex on $X$. L…

Beilinson–Lichtenbaum conjecture. There are complexes of Zariski or étale sheaves

Let $X$ be an arithmetic scheme, meaning a separated scheme of finite type over $\operatorname{Spec}\mathbb{Z}$, and let $n<0$. Write $X_{\text{ét}}$ for its étale site and…

Pour $n\geq 1$, soit $\iota^\bullet(n):\mathcal{B}^\bullet(F,n)\rightarrow\mathrm{CE}^\bullet(\mathcal{C}(F))_n$ le morphisme de complexes induced by motivic polylogarithms. Conjec…

Beilinson–Parshin conjecture. One should have

Let ${\rm Cor}_{\cal M}$ be a motivic correlator defined over a number field $F$, and suppose that in the situation described in the source the correspondence principle holds and t…

Lower-weight conjecture. The only nonzero lower weight is

Jannsen's generalized Tate and Beilinson conjectures. For all $i,j\in{\mathbb Z}$, the regulator map induces an isomorphism

Let $F$ be a field, and let $\mathcal{H}^{\mathcal{M}}(F)$ and $\mathcal{H}^{\mathrm{f}}(F)$ denote the motivic and formal Hopf algebras of multiple polylogarithms. Quotient each b…

Let $A^j_{\eta}$ be the $j$-fold power of the universal elliptic-curve family over the generic point $\eta$, and let…

Simpson's motivicity conjecture. Rigid integrable connections are motivic.

Simpson's standard conjecture. By spreading out, the $\overline{\mathbb{Q}}$-points in the intersection

Let $t\in\mathbb{Q}\setminus\{1\}$, let $X_t$ be the Hesse cubic curve over $\mathbb{Q}$, and let $L(X_t,s)$ be the $L$-function of its first cohomology. The regulator map is … Her…

Let $G$ be an algebraic group, and let $\mathrm{Char}_G$ be the character variety parametrizing $G$-local systems. A local system is motivic when it is of geometric origin. Esnault…

Let $X$ be a proper regular arithmetic scheme of dimension $d$, let $n<0$, and let $H^i(X_{\text{\it \'{e}t}},\mathbb{Z}^c(n))$ denote negative-weight motivic cohomology. Geisser’s…

Let ${\mathcal X}$ be a regular scheme of finite type over $\mathbb Z$, and suppose that its zeta function has a meromorphic continuation to $s=\dim({\mathcal X})-1$. Soulé's conje…

Let ${\mathcal X}$ be a regular scheme of pure dimension $d$ proper over $\operatorname{Spec}(\mathbb Z)$, and let $n\in\mathbb Z$. Assume that ${\mathcal X}$ satisfies Assumptions…

Perfectness conjecture. The pairing is conjectured to be perfect.

Let $K$ be an imaginary quadratic field, let $f$ be a modular form without CM by $K$, let $M_f$ be its motive, and let $\beta$ be an $L$-valued algebraic Grössencharacter of $K$. W…

Termination conjecture for the motivic filtration. The filtration terminates, meaning

Let $\alpha^*$ be the inverse-image functor from etale to Zariski motivic complexes, let $j^*{\Bbb Z}_l(n)_S$ be the indicated arithmetic etale Tate-twist complex, and let $n\ge0$.…