Arithmetic exponent pairs for algebraic trace functions and applications

We study short sums of algebraic trace functions via the $q$-analogue of van\nder Corput method, and develop methods of arithmetic exponent pairs that\ncoincide with the classical case while the moduli has sufficiently good\nfactorizations. As an application, we prove a quadratic analogue of\nBrun-Titchmarsh theorem on average, bounding the number of primes $p\\leqslant\nX$ with $p^2+1\\equiv0\\pmod q$. The other two applications include a larger\nlevel of distribution of divisor functions in arithmetic progressions and a\nsub-Weyl subconvex bound of Dirichlet $L$-functions studied previously by\nIrving.\n