For a class of affine algebraic groups $\mathcal{C}$ over a field $\kappa$, we define the notion of $\mathcal{C}$-fundamental gerbe of a fibered category, generalizing what we did for finite group schemes in a 2015 paper.

We give necessary and sufficient conditions on $\mathcal{C}$ implying that a fibered category $X$ over $\kappa$ satisfying mild hypotheses admits a Nori $\mathcal{C}$-fundamental gerbe. We also give a tannakian interpretation of the gerbe that results by taking as $\mathcal{C}$ the class of virtually unipotent group schemes, under a properness condition on $X$.

Finally, we prove a general duality result, generalizing the duality between group schemes of multiplicative type and Galois modules, that yields a construction of the multiplicative gerbe of multiplicative type which is independent of the previous theory, and requires weaker hypotheses. This gives a conceptual interpretation of the universal torsor of Colliot-Thélène and Sansuc.

We provide the first steps towards a new relative trace formula proof of the celebrated formula of Waldspurger relating the square of a toric period integral on ${PGL}_2$ to the central value of an $L$-function.

We prove a transcendence theorem concerning values of holomorphic maps from a disk to a quasiprojective variety over $\overline{\mathbb{Q}}$ that are integral curves of some algebraic vector field (defined over $\overline{\mathbb{Q}}$). These maps are required to satisfy some integrality property, besides a growth condition and a strong form of Zariski-density that are natural for integral curves of algebraic vector fields.

This result generalizes a theorem of Nesterenko concerning algebraic independence of values of the Eisenstein series $E_2$, $E_4$, $E_6$. The main technical improvement in our approach is the replacement of a rather restrictive hypothesis of polynomial growth on Taylor coefficients by a geometric notion of moderate growth formulated in terms of value distribution theory.

We study generically split octonion algebras over schemes using techniques of ${\mathbb{A}}^1$-homotopy theory. By combining affine representability results with techniques of obstruction theory, we establish classification results over smooth affine schemes of small dimension. In particular, for smooth affine schemes over algebraically closed fields, we show that generically split octonion algebras may be classified by characteristic classes including the second Chern class and another “mod $3$” invariant. We review Zorn’s “vector matrix” construction of octonion algebras, generalized to rings by various authors, and show that generically split octonion algebras are always obtained from this construction over smooth affine schemes of low dimension. Finally, generalizing P. Gille’s analysis of octonion algebras with trivial norm form, we observe that generically split octonion algebras with trivial associated spinor bundle are automatically split in low dimensions.