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We present in this article a new method for dealing with automatic sequences. This method allows us to prove a Möbius randomness principle for automatic sequences from which we deduce the Sarnak conjecture for this class of sequences. Furthermore, we can show a prime number theorem for automatic sequences that are generated by strongly connected automata where the initial state is fixed by the transition corresponding to .
We establish a homology relation for the Deligne–Mumford moduli spaces of real curves which lifts to a Witten–Dijkgraaf–Verlinde–Verlinde (WDVV)-type relation for a class of real Gromov–Witten invariants of real symplectic manifolds; we also obtain a vanishing theorem for these invariants. For many real symplectic manifolds, these results reduce all genus real invariants with conjugate pairs of constraints to genus invariants with a single conjugate pair of constraints. In particular, we give a complete recursion for counts of real rational curves in odd-dimensional projective spaces with conjugate pairs of constraints and specify all cases when they are nonzero and thus provide nontrivial lower bounds in high-dimensional real algebraic geometry. We also show that the real invariants of the -dimensional projective space with conjugate point constraints are congruent to their complex analogues modulo .
We give a geometric proof that the Hasse principle holds for the following varieties defined over global function fields: smooth quadric hypersurfaces, smooth cubic hypersurfaces of dimension at least in characteristic at least , and smooth complete intersections of two quadrics, which are of dimension at least , in odd characteristics.