13 problems

Beilinson's conjecture. The vector space

Beilinson–Parshin conjecture. One should have

Let $C$ be a smooth projective geometrically irreducible algebraic curve of genus $g$ over a number field $k$. Let $H^2_{\mathscr{M}}(C,\mathbb{Q}(2))_{\mathbb{Z}}$ be the integral…

Beilinson's linear independence conjecture. The values $L^{(N)'}(-1,x)$, for $x\in U$, are linearly independent over ${\mathbb Q}$. This is presented as a consequence of the Beilin…

Let $X$ be a singular $K3$ surface with Picard rank $20$ over $\mathbb{Q}$, let $H^3_\mathcal{M}(X,\mathbb{Q}(4))_{\operatorname{tr}}$ be the transcendental motivic cohomology, and…

Let $E$ be an elliptic curve over a number field $F$, and let $K_2^T(E)_{\textup{int}}$ denote the subgroup of $K_2$ of the function field of $E$ consisting of elements with trivia…

Let $C$ be a smooth projective curve over $\mathbb Q$ of genus $1$, let $E$ be its Jacobian, and let $\alpha$ be a nontrivial element of $H^2_\mathcal M(C,\mathbb Q(3))$. Let…

Let $f\in S_k(N_f,\chi_f)$ and let $g,h$ be weight-one modular forms with $\chi_f\chi_g\chi_h=1$. Let $W_{k-2}$ be the relevant Kuga–Sato variety, let $H$ be the coefficient field,…

Beilinson's conjecture. There is an integral element $\xi\in H^2_{\mathscr M}(E,\mathbb Z(2))$ in the sense of Scholl such that

Let $E$ be an irreducible algebraic representation of $G$, let $\pi=\pi_\infty\otimes\pi_f$ be an irreducible cuspidal automorphic representation stable at infinity, and let…

Weak Beilinson conjecture. There exists a $g$-dimensional subspace of the motivic cohomology having the desired properties.

Soit ${\mathbb F}$ un corps fini et soit $X$ une variété projective et lisse sur ${\mathbb F}$. Pour $i\geq 0$, notons $A^{i}(X)$ le groupe des cycles de codimension $i$ sur $X$ mo…

Beilinson's special-value conjecture. Assuming a suitable functional equation for $L(H^{i-1}(X),s)$, the two rational structures differ by the leading term in the Taylor expansion…