Bloch groups, algebraic K-theory, units, and Nahm's Conjecture
Frank Calegari, Stavros Garoufalidis, Don Zagier
备注：
Several minor inaccuracies have been corrected and the exposition has been improved at several points. The motivational section on quantum topology has been shortened and moved to the introduction. The new version also contains a more careful treatment of the prime 3 (where versions of the Bloch group in the literature differ), permitting us to improve the statements of several of our results
给定数域~ $F$和自然数~ $n$的Bloch群的一个元素，我们在域$F_n=F(e^{2 πi/n})$中构造一个明确的单位，它定义到~ $F_n$的非零元素的$\nn$次幂.利用循环量子双对数，在~ $F$的Bloch群与$K$ -群$K_3(F)$的同一性下，给出了~ $K_3(F)$的某个抽象Chern类的\changed{(up to an unidentified invertible scalar)}和\changed{formula}.我们定义的单位被证明与量子模块化猜想中出现的结的Kashaev不变量（这是我们研究的原始动机）的数字相一致，并且也出现在单位根附近的Nahm和的径向渐近中。这后一种连接是用来证明纳姆的猜想有关的模块某些$q$ -超几何级数的消失有关的元素在布洛赫群的~ $\overline{\mathbb Q}$。
英文摘要：
Given an element of the Bloch group of a number field~$F$ and a natural number~$n$, we construct an explicit unit in the field $F_n=F(e^{2 πi/n})$, well-defined up to $\nn$-th powers of nonzero elements of~$F_n$. The construction uses the cyclic quantum dilogarithm, and under the identification of the Bloch group of~$F$ with the $K$-group $K_3(F)$ gives \changed{(up to an unidentified invertible scalar)} a \changed{formula} for a certain abstract Chern class from~$K_3(F)$. The units we define are conjectured to coincide with numbers appearing in the quantum modularity conjecture for the Kashaev invariant of knots (which was the original motivation for our investigation), and also appear in the radial asymptotics of Nahm sums near roots of unity. This latter connection is used to prove Nahm's conjecture relating the modularity of certain $q$-hypergeometric series to the vanishing of the associated elements in the Bloch group of~$\overline{\mathbb Q}$.
