  Hankel determinants and automatic sequences are two classical subjects widely
studied in Mathematics and Theoretical Computer Science. However, these two
topics were considered totally independently, until in 1998, when Allouche,
Peyri\`ere, Wen and Wen proved that all the Hankel determinants of the
Thue-Morse sequence are nonzero. This property allowed Bugeaud to prove that
the irrationality exponents of the Thue-Morse-Mahler numbers are exactly 2.
Since then, the Hankel determinants of several other automatic sequences, in
particular, the paperfolding sequence, the Stern sequence, the period-doubling
sequence, are studied by Coons, Vrbik, Guo, Wu, Wen, Bugeaud, Fu, Han, Fokkink,
Kraaikamp, and Shallit. On the other hand, it is known that the Hankel
determinants of a rational power series are ultimately zero, and the Hankel
determinants of a quadratic power series over finite fields are ultimately
periodic. It is therefore natural to ask if we can obtain similar results about
the Hankel determinants of algebraic series. In the present paper, we provide a
partial answer to this question by establishing the automaticity of the reduced
Hankel determinants modulo $2$ of $\pm 1$-automatic sequences with kernel of
cardinality at most $2$.
