Let $\mathscr{B}$ be the Bruhat--Tits building of $\mathrm{PGL}_n(F)$, where $F$ is a non-Archimedean local field. We introduce geometric $k$-geodesics in $\mathscr{B}$ by means of CAT(0) convexity and combinatorial $k$-geodesics by a local successor relation on pointed $k$-facets. We prove that the two notions coincide. This allows us to use the local combinatorial definition on quotients $\Gamma\backslash\mathscr{B}$, without referring to the universal covering. When $\Gamma$ is discrete, torsion-free, cocompact, and type-preserving, the primitive closed $k$-geodesics define zeta functions $Z_k$ and their $\epsilon$-twisted variants $Z_k^\epsilon$. Our main result identifies an alternating product of these zeta functions with the unramified $L$-function of $L^2(\Gamma\backslash \mathrm{PGL}_n(F))$: $(1-u^n)^{\chi(\Gamma\backslash\mathscr{B})}L(\Gamma,q^{(n-1)/2}u)=\prod_{k=1}^{n-1} Z_k^\epsilon(\Gamma\backslash\mathscr{B},u)^{(-1)^{k+1}}$. This gives a uniform Ihara-type identity for all $\mathrm{PGL}_n$. We also extend the construction and the identity to $\mathrm{PGL}_n(D)$, where $D$ is a central division algebra over $F$; in that setting the residue parameter is $Q=|\mathcal{O}_D/\mathfrak{p}_D|$.
Let $K$ be an imaginary quadratic field and let $p \ge 5$ be a prime that is unramified in $K$. Let $\mathcal{A}_f/\mathbb{Q}$ be an abelian variety of $\mathrm{GL}_2$-type associated with a weight-two modular form $f$, with good non-ordinary reduction at $p$, and suppose that $(f,K)$ satisfies the generalized Heegner...
Let $\mathfrak L$ be a Frobenius maximal parabolic subalgebra of $\mathfrak{sl}_n$. For any $F\in\mathfrak L^*$ for which the Kirillov form $B_F(x,y)=F([x,y])$ is non-degenerate, let $\widehat F$ denote the associated principal element. We prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat...
A. Giaquinto, J. Irving, A. Lauve et al.· 2 citations
Let $p$ and $\ell$ be distinct odd primes. For a finite extension $F/\mathbb{Q}_p$, the local Langlands correspondence states that there is a canonical bijection between irreducible, smooth representations of $\text{GL}_n(F)$ and $n$-dimensional, $\Phi$-semisimple Weil--Deligne representations of the Weil group $W_F$....
Let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $A,B,C \in \mathcal{O}_K \setminus\{0\}$. Denote by $S_K'$ the set of prime ideals of $\mathcal{O}_K$ dividing $2ABC$. Assuming two standard conjectures concerning the modularity of mod-$p$ Galois representations and the Eichler-Shimura correspond...
Let $K=\mathbb{F}_q(\theta)$, and let $\mathcal{Z}$ be the $K$-algebra generated by Thakur's multiple zeta values $\zeta_A(\mathfrak{s})$. We prove that $\mathcal{Z}$ is a polynomial algebra and construct an explicit polynomial basis of $\mathcal{Z}$ over $K$. We also determine the transcendence degree of $K[\zeta_A(\m...
Let $\pi$ and $\pi'$ be unitary cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ over a number field $F$. Let $\mathfrak{C}_{\pi}$ be the analytic conductor of $\pi$. We develop a new approach to zero-free regions for $L$-functions via lower bounds for power sums, proving for all $\varepsi...
Jesse Thorner· 0 citations
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