Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79-95, 2011) proved that at every position of a Sturmian word starts an abelian power of exponent $k$ for every $k > 0$. We improve on this result by studying the maximum exponents of abelian powers and abelian repetitions (an abelian repetition is an analogue of a fractional power) in Sturmian words. We give a formula for computing the maximum exponent of an abelian power of abelian period $m$ starting at a given position in any Sturmian word of rotation angle $��$. vAs an analogue of the critical exponent, we introduce the abelian critical exponent $A(s_��)$ of a Sturmian word $s_��$ of angle $��$ as the quantity $A(s_��) = limsup\ k_{m}/m=limsup\ k'_{m}/m$, where $k_{m}$ (resp. $k'_{m}$) denotes the maximum exponent of an abelian power (resp.~of an abelian repetition) of abelian period $m$ (the superior limits coincide for Sturmian words). We show that $A(s_��)$ equals the Lagrange constant of the number $��$. This yields a formula for computing $A(s_��)$ in terms of the partial quotients of the continued fraction expansion of $��$. Using this formula, we prove that $A(s_��) \geq \sqrt{5}$ and that the equality holds for the Fibonacci word. We further prove that $A(s_��)$ is finite if and only if $��$ has bounded partial quotients, that is, if and only if $s_��$ is $��$-power-free for some real number $��$. Concerning the infinite Fibonacci word, we prove that: i) The longest prefix that is an abelian repetition of period $F_j$, $j>1$, has length $F_j( F_{j+1}+F_{j-1} +1)-2$ if $j$ is even or $F_j( F_{j+1}+F_{j-1} )-2$ if $j$ is odd, where $F_{j}$ is the $j$th Fibonacci number; ii) The minimum abelian period of any factor is a Fibonacci number. Further, we derive a formula for the minimum abelian periods of the finite Fibonacci words

To appear in Theoretical Computer Science