Extreme values of quadratic Hecke $L$-functions

We study large values of quadratic Hecke $L$-functions in the conductor aspect. Let $K$ be a fixed number field, and assume GRH for its finite-order Hecke $L$-functions. In a fixed ray class component with conductor norm comparable to $X$, we prove that $$\underset{\chi }{\max }L(\frac{1}{2}+\frac{A}{{\log }_{2}X},\chi )\ge \exp \{(e^{-A}+o(1))\sqrt{\frac{\log X{\log }_{3}X}{{\log }_{2}X}}\}$$ for every fixed $A\ge 0$. Every fixed smaller constant is attained by at least $X^{1-o(1)}$ characters. The same count holds at a suitably slowly moving threshold approaching the displayed constant. The combinatorial input is a sparse squarefree Gál set of cardinality $N$, retaining the known leading constant $2$ and having square multiplicative energy $N^{2+o(1)}$. The energy bound reflects the low degree of the associated Boolean polynomial. Together with the resonance estimate, it yields the abundance bound. We also prove unconditional analogues for quadratic characters with prime conductor away from one fixed place over any global function field of odd characteristic. Finally, we give bounds in the fixed strip and at $s=1$, including the dependence on the residue of the Dedekind zeta function and the prescribed local factors.