Let $X_1$ be a projective, smooth and geometrically connected curve over $\mathbb{F}_q$ with $q=p^n$ elements where $p$ is a prime number, and let $X$ be its base change to an algebraic closure of $\mathbb{F}_q$. We give a formula for the number of irreducible $\ell$-adic local systems ($\ell \neq p$) with a fixed rank over $X$ fixed by the Frobenius endomorphism. We prove that this number behaves like a Lefschetz fixed point formula for a variety over $\mathbb{F}_q$, which generalises a result of Drinfeld in rank $2$ and proves a conjecture of Deligne. To do this, we pass to the automorphic side by Langlands correspondence, then use Arthur's non-invariant trace formula and link this number to the number of $\mathbb{F}_q$-points of the moduli space of stable Higgs bundles.

Soit $X_1$ une courbe projective lisse et géométriquement connexe sur un corps fini $\mathbb{F}_q$ avec $q=p^n$ éléments où $p$ est un nombre premier. Soit $X$ le changement de base de $X_1$ à une clôture algébrique de $\mathbb{F}_q$. Nous donnons une formule pour le nombre des systèmes locaux $\ell$-adiques ($\ell \neq p$) irréductibles de rang donné sur $X$ fixé par l'endomorphisme de Frobenius. Nous montrons que ce nombre est semblable à une formule de point fixe de Lefschetz pour une variété sur $\mathbb{F}_q$, ce qui généralise un résultat de Drinfeld en rang $2$ et prouve une conjecture de Deligne. Pour ce faire, nous passerons du côté automorphe, utiliserons la formule des traces d'Arthur non-invariante, et relierons le nombre cherché avec le nombre $\mathbb{F}_q$-points de l'espace des modules des fibrés de Higgs stables.

We prove that for each Hamiltonian function $H \in \mathcal{C}^\infty(\mathbb{R}^4, \mathbb{R})$ defined on the standard symplectic $(\mathbb{R}^4,\omega_0)$ for which $M:= H^{-1}(0)$ is a non-empty compact regular energy level, the Hamiltonian flow on $M$ is not minimal. That is, we prove there exists a closed invariant subset of the Hamiltonian flow in $M$ that is neither $\emptyset$ nor all of $M$. This answers the four-dimensional case of a more than twenty year old question of Michel Herman, part of which can be regarded as a special case of the Gottschalk Conjecture.

Our principal technique is the introduction and development of a new class of pseudoholomorphic curve in the "symplectization" $\mathbb{R}\times M$ of framed Hamiltonian manifolds $(M, \lambda,\omega)$. We call these feral curves because they are allowed to have infinite (so-called) Hofer energy, and hence may limit to invariant sets more general than the finite union of periodic orbits. Standard pseudoholomorphic curve analysis is inapplicable without energy bounds, and thus much of this paper is devoted to establishing properties of feral curves, such as area and curvature estimates, energy thresholds, compactness, asymptotic properties, etc.

Let $\lambda$ denote the Liouville function. We show that, as $X \rightarrow \infty$, $$ \int^{2X}_X \sup_{\begin{smallmatrix} P(Y)\in\mathbb{R}[Y] \\ \mathrm{deg}{P} \leq k\end{smallmatrix}} \left| \sum_{x\leq n \leq x+H} \lambda(n) e(-P(n))\right| \ dx=o (XH)$$for all fixed $k$ and $X^{\theta} \leq H \leq X$ with $0 \lt \theta \lt 1$ fixed but arbitrarily small. Previously this was only established for $k \leq 1$. We obtain this result as a special case of the corresponding statement for (non-pretentious) $1$-bounded multiplicative functions that we prove.

In fact, we are able to replace the polynomial phases $e(-P(n))$ by degree $k$ nilsequences $\overline{F}(g(n) \Gamma )$. By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result $$ \int_{X}^{2X} \| \lambda \|_{U^{k+1}([x,x+H])}\ dx = o ( X )$$in the same range of $H$.

We present applications of this result to patterns of various types in the Liouville sequence. Firstly, we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of $\lambda$ over short polynomial progressions $(n+P_1(m),\ldots , n+P_k(m))$, which in the case of linear polynomials yields a new averaged version of Chowla's conjecture.

We are in fact able to prove our results on polynomial phases in the wider range $H\geq \mathrm{exp}((\mathrm{log} X)^{5/8+\varepsilon})$, thus strengthening also previous work on the Fourier uniformity of the Liouville function.