Let $G$ be a reductive algebraic group over a number field $k$. Weil's conjecture on Tamagawa numbers (now a theorem) tells us that the Tamagawa number $\tau(G)$ of $G$ is 1 if $G$ is semisimple and simply-connected. Are there known bounds for $\tau(G)$ for the general case?


Presumably one might be able to use the formula $\tau(G) = |\text{Pic}(G)|/|\text{Sha}(G)|$ and try to estimate the numerator, but I am not familiar with the literature in this respect.



Bounds on Tamagawa numbers of reductive groups
Yes, the formula is correct, see Sansuc, J.-J. Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres. J. Reine Angew. Math. 327 (1981), 12–80, (10.1.2).


In the extreme cases (when G is semisimple or a torus) there are formulas for ${\rm Pic}(G)$ in Lemma 6.9 of Sansuc's paper.
Namely, when $G$ is semisimple with fundamental group $B=\pi\_1(G\_\bar k)$, we have
$${\rm Pic}(G)=X^\*(B)^\Gamma,$$
where $X^\*$ denotes the character group and $\Gamma={\rm Gal}(\bar k/k)$.
It follows that the order of the Picard group is bounded by the order of the fundamental group of $G\_{\bar k}$.
When $G$ is a torus, we have
$${\rm Pic}(G)=H^1(k, X^\*(G)).$$


For a general reductive group $G$, set
$G^{\rm ss}=[G,G]$ and $G^{\rm tor}=G/G^{\rm ss}$,
then for the semisimple part of $G$ we obtain $$\#{\rm Pic}(G^{\rm ss})\le\#\pi\_1(G^{\rm ss}\_{\bar k})$$
and for the toric part we obtain
$$\#{\rm Pic}(G^{\rm tor})=\# H^1(k,X^\*(G)).$$
From the short exact sequence
$$ 1\to G^{\rm ss}\to G\to G^{\rm tor}\to 1$$
we obtain an exact sequence
$${\rm Pic}\ G^{\rm tor}\to{\rm Pic}\ G\to {\rm Pic}\ G^{\rm ss},$$
see Sansuc's paper, Corollary 6.11. We obtain the following bound:
$$\#\tau(G)\ \le\ \#{\rm Pic}(G)\ \le \# H^1(k,X^\*(G))\cdot \#\pi\_1(G^{\rm ss}\_{\bar k}).$$


