10. Let the set $M=\{1,2, \cdots, 2023\}$, for a subset $A=\left\{a_{1}, \cdots, a_{k}\right\}$ of $M$, let it correspond to a number $\alpha=(-1)^{a_{1}+\cdots+a_{k}}$ (the empty set $\varnothing$ corresponds to the number $\alpha=(-1)^{0}=1$). For all subsets $A$ of $M$, find the sum of the numbers they correspond to $\sum_{A \subseteq M} \alpha$.
For the set $M$, the number of odd subsets (the sum of all elements in the subset is odd) is equal to the number of even subsets (the sum of all elements in the subset is even, the sum of elements in the empty set is 0). 
Consider all subsets $A$ of $M$ that do not contain the element 1. Clearly, $A \bigcup\{1\}$ is also a subset of $M$, and different $A$ correspond to different $A \cup\{1\}$, and there are $2^{2022}$ such subsets $A$.
Since exactly one of $A$ and $A \bigcup\{1\}$ is an odd subset, and the other is an even subset, the number of odd subsets of $M$ is equal to the number of even subsets.

Noting that odd subsets correspond to $\alpha=-1$, and even subsets correspond to $\alpha=1$, so $\sum_{A \subseteq M} \alpha=0$.