Let $G$ be a quasi-split reductive group over a finite field and let $\^G$ be the Langlands dual group. Assuming the derived subgroup of $G$ is almost simple, we use techniques from the geometric Langlands program to relate special values of Bessel functions for generic principal series representations of $G$ to the trace of Frobenius acting on Kloosterman sheaves of Heinloth-Ng\^o-Yun for $\^G$. We also give explicit examples in essentially all possible cases, including exceptional groups.
We develop an extension of Deligne--Lusztig theory to certain (possibly infinite type) disconnected reductive groups arising from the special fibers of point stabilizers in the Bruhat--Tits building, which we call \emph{paraductive}. We then compute explicit lower bounds for the Tate cohomology of representations of pa...
Let $\mathbf{f}$ be a Hida family of cusp forms over a standard (power series) Iwasawa algebra. We determine the pseudo-isomorphism class of the big Shafarevich-Tate group attached to $\mathbf{f}$ over a suitable imaginary quadratic field by describing all its higher Fitting ideals; this description is given in terms o...
Enrico Da Ronche, M. Longo, S. Vigni et al.· 0 citations
We prove the Tate conjecture for abelian fivefolds over finite fields. The proof constructs correspondences for a residual motive using a Moret--Bailly family, Gross--Schoen heights, and monodromy. We also prove standard conjecture~$D_\ell$ over $\overline{\mathbf F}_p$ and independence of $\ell$ of rational cycle clas...
Let $\mathcal G=H\rtimes\Gamma$ be a semidirect product of a finite group $H$ with $\Gamma=\mathbb Z_p^d$, the direct product of finitely many copies of the additive group of the $p$-adic integers. In the case $d=1$, Iwasawa algebras of such profinite groups were studied by Ritter--Weiss, Lau, Nickel, and the author. W...
Let $X$ be a smooth projective curve and let $G$ be a reductive group over an algebraically closed field of characteristic zero. We compute the image of arbitrary finite-rank local systems on $\mathrm{Bun}_G(X)$ under the geometric Langlands equivalence, confirming a conjecture of V. Lafforgue in the case of the consta...
We relate the theory of $q$-rational and $q$-real numbers introduced by Morier-Genoud and Ovsienko to the classical theory of Kleinian groups and their Teichm\"uller spaces. This provides a geometric point of view on several recent results about realisations of $q$-rationals for particular values of $ q \in \mathbb{C}...
Alexander Elzenaar· 0 citations
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