Let $E$ be an elliptic curve over $\mathbb Q$. Let $F$ be the rational function field of $E$.


The $K\_2$ group of $F$ [may be described](https://en.wikipedia.org/wiki/Algebraic_K-theory#Matsumoto%27s_theorem) by elements in $F^\times ⊗\_\mathbb{Z} F^\times$ quotiented by the relations $\langle f ⊗ (1 − f)\rangle (f ∈ F^\times, f\neq 0,1)$.


For $P ∈ C(\bar {\mathbb Q})$ and $\{g, h\}\in K\_2(F)$, the tame symbol is defined as


$$T\_P (\{g, h\}) = (−1)^{\text{ord}\_P(g)\text{ord}\_P(h)}\frac{g^{\text{ord}\_P (h)}}{h^{\text{ord}\_P (g)}}(P)$$


The tame kernel $K\_2^T(E)$ is the subgroup of $K\_2(F)$ whose elements $\{g,h\}$ satisfies $T\_p(\{g,h\})=1$ for every $P ∈ E(\bar {\mathbb Q})$.


**Question:** How to compute\* the torsion of


* $K\_2(E)$, or
* $K\_2^T(E)/K\_2(\mathbb Q)$?


\*It's not known whether the torsion of the groups above are finitely generated. So it could be possible that there is no algorithm to compute the torsion because the torsion is infinite. Having this in mind, the word "compute" should be interpreted as follows:


* Is there an algorithm to determine whether a symbol $\{g,h\}$ is trivial in one of the K-groups?
* What is the most efficient way to find as many as possible torsion elements in one of the K-groups?



Computation of the torsion of K-groups related to elliptic curves
The most efficient way I know to detect whether an element of $K\_2(E)$ is torsion is to use the elliptic dilogarithm. This relies however on a conjecture, and it is not an exact method, in the sense that it uses floating-point arithmetic.


Consider the map
\begin{align\*}
\beta : F^\times \otimes F^\times & \to \mathbb{Z}[E(\overline{\mathbb{Q}})] \\
g \otimes h & \mapsto \sum\_{P,Q \in E(\overline{\mathbb{Q}})} \mathrm{ord}\_P(f) \mathrm{ord}\_Q(g) [P-Q].
\end{align\*}


Bloch has defined a regulator map
\begin{equation\*}
\mathrm{reg}\_\infty \colon K\_2(E) \to \mathbb{R},
\end{equation\*}
which can be computed using the elliptic dilogarithm $D\_E : E(\mathbb{C}) \to \mathbb{R}$. Namely, given $x \in K\_2(E)$, whose image in $K\_2^T(E)$ is written as $\sum\_i \{g\_i,h\_i\}$, we have (up to a constant factor)
\begin{equation\*}
\mathrm{reg}\_\infty(x) = \sum\_i D\_E(\beta(g\_i \otimes h\_i))
\end{equation\*}
where $D\_E$ is extended by linearity by defining $D\_E(\sum n\_P [P]) = \sum n\_P D\_E(P)$. The elliptic dilogarithm can be computed very rapidly: writing $E(\mathbb{C}) = \mathbb{C}^\times/q^{\mathbb{Z}}$, one has $D\_E([x]) = \sum\_{n \in \mathbb{Z}} D(xq^n)$. Here $D$ is the Bloch-Wigner dilogarithm, implemented e.g. in PARI/GP as $\texttt{polylog(2,x,2)}$. Since $|q|<1$, the series for $D\_E$ converges exponentially fast.


One should also take into account the bad places of $E$. For each prime $p$, there is a residue map $K\_2(E) \to K'\_1(\mathcal{E}\_p)$, where $\mathcal{E}\_p$ is the fiber at $p$ of the minimal regular model of $E$ over $\mathbb{Z}$. It turns out that $V\_p := K'\_1(\mathcal{E}\_p) \otimes \mathbb{Q}$ is nonzero precisely when $E$ has split multiplicative reduction at $p$, in which case $\operatorname{dim}(V\_p)=1$. It is also possible to compute the residue map, see Bloch, Grayson, [$K\_2$ and $L$-functions of elliptic curves - Computer calculations](http://math.stanford.edu/%7Econrad/BSDseminar/refs/BlochK2.pdf) and Rolshausen, Schappacher, [On the second $K$-group of an elliptic curve](http://irma.math.unistra.fr/%7Eschappa/NSch/Publications_files/1998b_KRNSch.pdf).


Now, it is conjectured that the extended regulator map
\begin{equation\*}
\mathrm{reg} \colon K\_2(E) \to \mathbb{R} \oplus \bigoplus\_p (V\_p \otimes \mathbb{R})
\end{equation\*}
is an isomorphism after tensoring $K\_2(E)$ with $\mathbb{R}$. In particular, and conjecturally, an element $x$ of $K\_2(E)$ is torsion precisely when it is in the kernel of the extended regulator map.


If the image of $x$ appears numerically to be 0, then one may try to ascertain that $x$ is torsion in $K\_2^T(E)$ by finding Steinberg relations (this becomes a linear algebra problem in the group $F^\times \otimes F^\times$). On the other hand, if the image appears to be nonzero, this can in principle be proved by computing with enough accuracy.


Regarding the torsion in $K\_2(E)$ and $K\_2^T(E)/ K\_2(\mathbb{Q})$, it is known that $K\_2(\mathbb{Q})$ embeds in $K\_2(E)$ by means of the structural morphism $E \to \operatorname{Spec} \mathbb{Q}$, since this morphism has a section. The group $K\_2(\mathbb{Q})$ is infinitely generated (for a description, see Milnor's book *Introduction to algebraic $K$-theory*). So the torsion of $K\_2(E)$ is infinite. I don't know about the torsion of $K\_2^T(E)/ K\_2(\mathbb{Q})$ in general, but Goncharov and Levin have given a complex computing $K\_2^T(E)/K\_2(k)$ for an elliptic curve $E/k$, at least when $k$ is algebraically closed, see Theorem 1.5 in [Zagier's conjecture on $L(E,2)$](https://arxiv.org/abs/alg-geom/9508008). For $k=\overline{\mathbb{Q}}$, this shows (if my computation is correct) that $K\_2^T(E)/K\_2(\overline{\mathbb{Q}})$ contains a copy of the group $E(\overline{\mathbb{Q}})\_{\mathrm{tors}}$.


