Toroidal families and averages of $L$-functions, II: cubic moments

Generalizing our previous work on ``toroidal averages’', we study the average of special values of $L$-functions of the form $L(1\mathrm{/}2,{\chi }^a)L(1\mathrm{/}2,{\chi }^b)L(1\mathrm{/}2,{\chi }^c)$ for integers $a$, $b$ and $c$, where $\chi$ varies over Dirichlet characters of a given prime modulus. We highlight connections with estimates for bilinear forms of trace functions and with bounds for the number of solutions of monoidal equations in three variables in small boxes over finite fields.