Faisceau Automorphe Unipotent pour $\mathrm G_2$, Nombres de Franel, et Stratification de Thom-Boardman

We generalise to the equivariant case a result of J. Denef and F. Loeser about trigonometric sums on tori; on the other hand, we study the Thom-Boardman stratification associated to the multiplication of global sections of line bundles on a curve. We prove a subtle inequaliity about the dimensions of these strata. Our motivation comes from the geometric Langlands program. Based on works of W. T. Gan, N. Gurevich, D. Jiang and S. Lysenko, we propose, for the reductive group $G$ of type $\mathrm G_2$, a conjectural construction of the automorphic sheaf whose Arthur parameter is unipotent and sub-regular. Using our two results above, we determine the generic ranks of all isotypic components of an $S_3$-equivaraint sheaf which appears in our conjecture, this $S_3$ being the centraliser of the sub-regular $\mathrm{SL}_2$ inside the Langlands dual group of $G$.