Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p$ an odd prime. Is it possible that $G$ contains a maximal torus of order $2^m$ for some positive integer $m$?
Would you please some one help me to find a reference containing size of maximal tori in finite simple groups of Lie type?
Let me know if you have a more specific question, and I can try to answer it. Here are some attempts:
Existence: For every positive integer $n\geq 2$, there is a finite simple group of Lie type with a maximal torus of order $2^n$: If $n$ is odd, then $\operatorname{PSL}(n,3)$ has a maximal torus of type $\Phi_1^{n-2} \Phi_2$ and order $(3-1)^{n-2} (3+1)=2^n$. If $n$ is even, then $\operatorname{PSL}(n+2,3)$ has a maximal torus of type $\Phi_1^{n+1}/(n,q-1)$ and order $(3-1)^{n+1}/\gcd(n+2,3-1)=2^n$.
A-type case: Maximal Tori for GL, SL, PGL, and PSL can be described fairly explicitly. Things are easiest for GL where a maximal torus is the group of units of a $n$-dimensional semisimple commutative $k$-algebra, that is, a the group of units of a direct product of field extensions of the underlying field. If the underlying field is finite, then field extensions are uniquely parameterized by their dimension. Hence we find a partition $d_1 \leq d_2 \leq \ldots \leq d_m$ of $n$, so that $n=d_1 + \ldots + d_m$. Then the maximal torus are the block diagonal matrices where the $i$th block is chosen from $\newcommand{\GL}{\operatorname{GL}}\newcommand{\GF}{\operatorname{GF}}\GL(1,q^{d_i}) \leq \GL(d_i,q)$ where nonzero elements of the field with $q^{d_i}$ elements are written as $k$-linear transformations of that field, where $k$ is the subfield with $q$ elements. Such a torus has order $(q^{d_1}-1)(q^{d_2}-1)\cdots(q^{d_n}-1)$. In SL and PGL one takes the subgroup or the quotient group, and in each case one removes a factor of $q-1$. In PSL one additionally removes a factor of $\gcd(n,q-1)$. In particular, choosing the partition $d_1=d_2=\ldots=d_{n-2}=1, d_{n-1}=2$, that is, $1+1+\ldots+1+2=n$, we get a maximal torus in GL of order $(q-1)^{n-2} (q^2-1)$ and a maximal torus in SL or PGL of order $(q-1)^{n-2}(q+1)$ and a maximal torus in PSL of order $(q-1)^{n-2}(q+1)/\gcd(n,q-1)$. Specializing this to $q=3$ and $n$ odd gives $2^{n-2} 4 / 1 = 2^n$. Similarly, choosing the partition $d_1=d_2=\ldots=d_{n+2}=1$, that is, $1+1+\ldots+1=n+2$ we get a maximal torus in GL of order $(q-1)^{n+2}$, and of SL and PGL of order $(q-1)^{n+1}$, and of PSL of order $(q-1)^{n+1}/\gcd(n,q-1)$. Choosing $q=3$ and $n$ even we get $2^{n+1}/2 = 2^n$. Since PSL(2,3) is not simple we run into trouble trying to get very tiny tori in a simple group.
Table: Here is a table for the orders of maximal tori in simply connected finite groups of untwisted Lie type of rank up to 4. $\Phi_1=q-1$, $\Phi_2=q+1$, $\Phi_3 = q^2+q+1$, $\Phi_4 =q^2+1$, $\Phi_5=q^4+q^3+q^2+q+1$, $\Phi_6=q^2-q+1$, $\Phi_8 = q^4+1$, $\Phi_{12}=q^4-q^2+1$.
$$\begin{array}{rcc|c|c|} r & G & k & T_1 & T_2 & T_3 & T_4 & T_5 \\ \hline A_1 & SL_2 & 2 & \Phi_1 & \Phi_2 \\ \hline %{}^2A_2 & SU_3 & %{}^2B_2 & Sz & %{}^2G_2 & Ree & A_2 & SL_3 & 3 & \Phi_1^2 & \Phi_1 \Phi_2 & \Phi_3 \\ \hline B_2 & Sp_4 & 5 & \Phi_1^2 & \Phi_2^2 & \Phi_1 \Phi_2 & \Phi_1 \Phi_2 & \Phi_4 \\ \hline G_2 & & 6 & \Phi_1^2 & \Phi_2^2 & \Phi_1 \Phi_2 & \Phi_1 \Phi_2 & \Phi_3 \\ &&& \Phi_6 \\ \hline %{}^2A_3 & SU_4 & %{}^2A_4 & SU_5 & %{}^3D_4 & & %{}^2F_4 & Ree & A_3 & SL_4 & 5 & \Phi_1^3 & \Phi_1 \Phi_2^2 & \Phi_1^2 \Phi_2 & \Phi_1 \Phi_3 & \Phi_2 \Phi_4 \\ \hline % {}^2 A_3 & SU_3 & S_4 B_3 & O_7 & 10 & \Phi_1^3 & \Phi_2^3 & \Phi_1\Phi_2^2 & \Phi_1^2 \Phi_2 & \Phi_1\Phi_2^2 \\ && & \Phi_1^2\Phi_2 & \Phi_1\Phi_3 & \Phi_2\Phi_4 & \Phi_1\Phi_4 & \Phi_2\Phi_6 \\ \hline C_3 & Sp_6 & 10 & \Phi_1^3 & \Phi_2^3 & \Phi_1\Phi_2^2 & \Phi_1^2 \Phi_2 & \Phi_1\Phi_2^2 \\ && & \Phi_1^2\Phi_2 & \Phi_1\Phi_3 & \Phi_2\Phi_4 & \Phi_1\Phi_4 & \Phi_2\Phi_6 \\ \hline %{}^2A_5 & SU_6 & %{}^2A_6 & SU_7 & %{}^2D_4 & O^-_8 & A_4 & SL_5 & 7 & \Phi_1^4 & \Phi_1^3\Phi_2 & \Phi_1^2\Phi_2^2 & \Phi_1^2 \Phi_3 & \Phi_1\Phi_2\Phi_4 \\ &&& \Phi_5 & \Phi_1\Phi_2\Phi_3 \\ \hline B_4 & O_9 & 20 & \Phi_1^4 & \Phi_2^4 & \Phi_1\Phi_2^3 & \Phi_1^3\Phi_2 & \Phi_1^2\Phi_2^2 \\ &&& \Phi_1\Phi_2^3 & \Phi_1^2\Phi_2^2 & \Phi_1^3\Phi_2 & \Phi_1^2\Phi_2^2 & \Phi_1^2\Phi_3 \\ C_4&Sp_8&(same)& \Phi_4^2 & \Phi_2^2\Phi_4 & \Phi_1^2\Phi_4 & \Phi_1\Phi_2\Phi_4 & \Phi_1\Phi_2 \Phi_4 \\ &&& \Phi_1\Phi_2\Phi_4 & \Phi_2^2\Phi_6 & \Phi_1\Phi_2\Phi_6 & \Phi_1\Phi_2\Phi_3 & \Phi_8 \\ \hline D_4 & O^+_8 & 13 & \Phi_1^4 & \Phi_2^4 & \Phi_1^2\Phi_2^2 & \Phi_1^2\Phi_2^2 & \Phi_1^2\Phi_2^2 \\ &&& \Phi_1\Phi_2^3 & \Phi_1^3\Phi_2 & \Phi_1^2\Phi_3 & \Phi_4^2 & \Phi_1\Phi_2\Phi_4 \\ &&& \Phi_1\Phi_2\Phi_4 & \Phi_1\Phi_2\Phi_4 & \Phi_2^2\Phi_6 & \\ \hline F_4 & & 25 & \Phi_1^4 & \Phi_2^4 & \Phi_1^3\Phi_2 & \Phi_1^3\Phi_2 & \Phi_1\Phi_2^3 \\ &&& \Phi_1\Phi_2^3 & \Phi_1^2\Phi_2^2 & \Phi_1^2\Phi_2^2 & \Phi_3^2 & \Phi_1^2\Phi_3 \\ &&& \Phi_1^2\Phi_3 & \Phi_4^2 & \Phi_1^2\Phi_4 & \Phi_2^2\Phi_4 & \Phi_1\Phi_2\Phi_4 \\ &&& \Phi_1\Phi_2\Phi_4 & \Phi_6^2 & \Phi_2^2\Phi_6 & \Phi_2^2\Phi_6 & \Phi_1\Phi_2\Phi_6 \\ &&& \Phi_1\Phi_2\Phi_3 & \Phi_1\Phi_2\Phi_3 & \Phi_1\Phi_2\Phi_6 & \Phi_8 & \Phi_{12} \\ \hline %{}^2A_7 & SU_8 %{}^2A_8 & SU_9 %{}^2D_5 & O^-_{10} %{}^2E_6 & % Rank 5+ now \end{array}$$