15 problems

Let $X$ be a smooth, projective, geometrically connected curve of genus $g$ defined over $\mathbb{Q}$. Let $K_4^{(3)}(X)$ denote the third Adams eigenspace of…

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

Let $E=\overline{E}\subset\mathbb{C}$ be a conjugation-stable field, let $\mathcal B(E)$ be its Bloch group, and let $\rho:\mathcal B(E)\to\mathbb{C}/\mathbb{Q}(2)$ be the Bloch re…

Let $F$ be a subfield of $\mathbb{C}$ with embedding $\sigma:F\hookrightarrow\mathbb{C}$. Let $\mathcal B(F)$ and $\mathcal B(\mathbb{C})$ denote the corresponding Bloch groups, an…

Let $k$ be an algebraically closed field of characteristic zero, and for $n\geq 1$ let … be the regulator map defined above. Regulator isomorphism conjecture. The regulator maps…

Beilinson's conjecture. Equality always holds:

Let $X$ be a smooth, projective scheme over ${\cal O}_K[1/S]$, and let $\bar{X}:=X\times_{{\cal O}_K[1/S]}\bar{K}$ be its base change to an algebraic closure. For integers $i$ and…

Let $X$ be a smooth, projective scheme over ${\cal O}_K[1/S]$, and let $\bar{X}:=X\times_{{\cal O}_K[1/S]}\bar{K}$ be its base change to an algebraic closure. For integers $i$ and…

Let $K$ be a number field, let $v$ be a finite place, let $\Lambda$ be the completion of the ring of integers at $v$, and consider the local case with $q=1$ and $a=0$. Let the regu…

Let $N\geq 3$, let $(a,b)\in I_N$, where $I_N\subset(\mathbb{Z}/N\mathbb{Z})^{\oplus 2}$ consists of pairs with $a,b,a+b\neq 0$, and set $k:=\varphi(N)/2$. Let $X_N^{[a,b]}$ be the…

Let $S$ be a smooth projective surface. Write ${{\rm CH}}^2(S,1)_{{\rm ind}}$ for the indecomposable part of the higher Chow group and $J^2(S)_{{\rm tr}}$ for the transcendental in…

Exact-locus finiteness conjecture. The exact locus is finite in $S$, and hence the $A$-factor locus is finite in $S$.

Let $\mathbb L(F)$ be the Lie coalgebra of multiple polylogarithms over a field $F$, with cobracket $\delta$, and let $\mathbb L^{\operatorname{symb}}(\mathbb C)$ be its symbolic v…

Let $p$ be a fixed prime, let $S$ be a finite set of places of $K$ containing all Archimedean places and the places dividing $p$, and let…

Let $K/k$ be a Galois extension of number fields, let $r$ be a negative integer, and let $f_{r,K}$ be a chosen ${\mathbb Q}[\operatorname{Gal}(K/k)]$-isomorphism from…