Annals of Mathematics

We develop a theory of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence, which also includes the possibility of dealing with several systems associated with sufficiently independent matrix transformations. Our main results go far beyond the existing literature, also surpassing those of two unpublished preprints the authors made available on the arXiv in 2018. The main new feature is that our results apply now without any restriction on the matrices defining the corresponding Mahler systems. As a consequence, we settle several problems concerning expansions of numbers in multiplicatively independent bases. For instance, we prove that no irrational real number can be automatic in two multiplicatively independent integer bases, and we give a new proof and a broad algebraic generalization of Cobham's theorem in automata theory. We also provide a new proof and a multivariate generalization of Nishioka's theorem, a landmark result in Mahler's method.

Let $G$ be a simple group over a global function field $K$, and let $\pi$ be a cuspidal automorphic representation of $G$. Suppose $K$ has two places $u$ and $v$ (satisfying a mild restriction on the residue field cardinality), at which the group $G$ is quasi-split, such that $\pi_u$ is tempered and $\pi_v$ is unramified and generic. We prove that $\pi_w$ is tempered at all unramified places $K_w$ at which $G$ is unramified quasi-split.

More generally, the set of unitary spherical representations is partitioned according to nilpotent conjugacy classes in the Lie algebra of $G$. We show that if $\pi_v$ is in the set corresponding to the nilpotent class $N$, and if $\pi_u$ satisfies an analogous hypothesis, then $\pi_w$ belongs to the same class $N$, where $w$ is as above. These results are consistent with conjectures of Shahidi and Arthur.

The proofs use the Galois parametrization of cuspidal representations due to V. Lafforgue to relate the local Satake parameters of $\pi$ to Deligne's theory of Frobenius weights. The main observation is that, in view of the classification of unitary spherical representations, due to Barbasch and the first-named author, the theory of weights excludes almost all complementary series as possible local components of $\pi$. This in turn determines the local Frobenius weights at all unramified places. In order to apply this observation in practice, we need a result of the second-named author with Gan and Sawin on the weights of discrete series representations.

We prove effective equidistribution theorems, with polynomial error rate, for orbits of the unipotent subgroups of $\mathrm{SL}_2 (\mathbb{R})$ in arithmetic quotients of $\mathrm{SL}_2 (\mathbb{C})$ and $\mathrm{SL}_2 (\mathbb{R}) \times \rm{SL}_2(\mathbb{R})$.

The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.

Unconditionally, we prove the Iwaniec--Sarnak conjecture for $L^4$-norms of the Hecke--Maass cusp forms. From this result, we can justify that for even Maass cusp form $\phi$ with the eigenvalue $\lambda_{\phi}=\frac{1}{4}+t_{\phi}^2$, for $a>0$, a sufficiently large $h>0$ and for any $0\lt\epsilon_1 \lt \epsilon/10^7 (\epsilon>0)$, for almost all $1\le k\lt t_{\phi}^{1-\epsilon}$, we are able to find $\beta_k= \{X_k+yi:a \lt y \lt a+h\}$ with $-\frac{1}{2}+\frac{k-1}{t_{\phi}^{1-\epsilon}}\le X_k\le-\frac{1}{2}+\frac{k}{t_{\phi}^{1-\epsilon}}$ such that the number of sign changes of $\phi$ along the segment $\beta_k$ is $\gg_{\epsilon} t_{\phi}^{1-\epsilon_1}$ as $t_{\phi}\to\infty$. Also, we obtain the similar result for horizontal lines. On the other hand, we conditionally prove that for a sufficiently large segment $\beta$ on $\mathrm{Re}(z)=0$ and $\mathrm{Im}(z)>0$, the number of sign changes of $\phi$ along $\beta$ is $\gg_{\epsilon} t_{\phi}^{1-\epsilon}$ and consequently, the number of inert nodal domains meeting any compact vertical segment on the imaginary axis is $\gg_{\epsilon} t_{\phi}^{1-\epsilon}$ as $t_{\phi}\to\infty$.

We prove that there exist closed, aspherical, smooth $4$-manifolds that are homeomorphic but not diffeomorphic. These provide counterexamples to a smooth analog of the Borel conjecture in dimension four.

We give the first examples of a pair of knots $K_1$, $K_2$ in the $3$-sphere for which their unknotting numbers satisfy $u(K_1\#K_2) \lt u(K_1)+u(K_2)$. This answers question 1.69(B) from Kirby's problem list, in the negative.