We provide a complete classification of Teichmüller curves occurring in hyperelliptic components of the meromorphic strata of differentials. Using a non-existence criterion based on how Teichmüller curves intersect the boundary of the moduli space we derive a contradiction to the algebraicity of any candidate outside of Hurwitz covers of strata with projective dimension one, and Hurwitz covers of zero residue loci in strata with projective dimension two.

This paper studies twisted signature invariants and twisted linking forms, with a view toward obstructions to knot concordance. Given a knot K and a representation $\rho $ of the knot group, we define a twisted signature function $\sigma _{K,\rho } \colon S^1 \to \mathbb {Z}$. This invariant satisfies many of the same algebraic properties as the classical Levine-Tristram signature $\sigma _K$. When the representation is abelian, $\sigma _{K,\rho }$ recovers $\sigma _K$, while for appropriate metabelian representations, $\sigma _{K,\rho }$ is closely related to the Casson-Gordon invariants. Additionally, we prove satellite formulas for $\sigma _{K,\rho }$ and for twisted Blanchfield forms.

We show that Calabi–Yau fibrations over curves are uniformly K-stable in an adiabatic sense if and only if the base curves are K-stable in the log-twisted sense. Moreover, we prove that there are cscK metrics for such fibrations when the total spaces are smooth.

Given a general polarized $K3$ surface $S\subset \mathbb P^g$ of genus $g\le 14$, we study projections of minimal degree and their variational structure. In particular, we prove that the degree of irrationality of all such surfaces is at most $4$, and that for $g=7,8,9,11$ there are no rational maps of degree $3$ induced by the primitive linear system. Our methods combine vector bundle techniques à la Lazarsfeld with derived category tools and also make use of the rich theory of singular curves on $K3$ surfaces.

We study the behaviour of Kauffman bracket skein modules of 3-manifolds under gluing along surfaces. For this we extend this notion to $3$-manifolds with marking consisting of open intervals and circles in the boundary. The new module is called the stated skein module.

The first results concern non-injectivity of certain natural maps defined when forming connected sums along spheres or disks. These maps are injective for surfaces or for generic quantum parameter, but we show that in general they are not when the quantum parameter is a root of 1. We show that when the quantum parameter is a root of 1, the empty skein is zero in a connected sum where each constituent manifold has non-empty marking. We also prove various non-injectivity results for the Chebyshev-Frobenius map and the map induced by deleting marked balls.

We then interpret stated skein modules as a monoidal symmetric functor from a category of “decorated cobordisms” to a category of algebras and their bimodules. We apply this to deduce properties of stated skein modules as a Van-Kampen like theorem, a computation through Heegaard decompositions and a relation to Hochshild homology for trivial circle bundles over surfaces.

What proportion of integers $n \leq N$ may be expressed as $x^2 + dy^2$ for some $d \leq \Delta $, with $x,y$ integers? Writing $\Delta = (\log N)^{\log 2} 2^{\alpha \sqrt {\log \log N}}$ for some $\alpha \in (-\infty , \infty )$, we show that the answer is $\Phi (\alpha ) + o(1)$, where $\Phi $ is the Gaussian distribution function $\Phi (\alpha ) = \frac {1}{\sqrt {2\pi }} \int ^{\alpha }_{-\infty } e^{-x^2/2} dx$.

A consequence of this is a phase transition: Almost none of the integers $n \leq N$ can be represented by $x^2 + dy^2$ with $d \leq (\log N)^{\log 2 - \varepsilon }$, but almost all of them can be represented by $x^2 + dy^2$ with $d \leq (\log N)^{\log 2 + \varepsilon}\kern-1.5pt$.

Given a polarised abelian variety over a number field, we provide totally explicit upper bounds for the cardinality of the rational points whose Néron-Tate height is less than a small threshold. These imply new estimates for the number of torsion points as well as the minimal height of a non-torsion point. Our bounds involve the Faltings height and dimension of the abelian variety together with the degrees of the polarisation and the number field but we also get a stronger statement where we use certain successive minima associated to the period lattice at a fixed archimedean place, in the spirit of a result of David for elliptic curves.

We describe the structure of regular codimension $1$ foliations with numerically projectively flat tangent bundle on complex projective manifolds of dimension at least $4$. Along the way, we prove that either the normal bundle of a regular codimension $1$ foliation is pseudo-effective, or its conormal bundle is nef.

Let F be a non-archimedean locally compact field of residual characteristic p, let $G=\operatorname {GL}_{r}(F)$ and let $\widetilde {G}$ be an n-fold metaplectic cover of G with $\operatorname {gcd}(n,p)=1$. We study the category $\operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$ of complex smooth representations of $\widetilde {G}$ having inertial equivalence class $\mathfrak {s}=(\widetilde {M},\mathcal {O})$, which is a block of the category $\operatorname {Rep}(\widetilde {G})$, following the ‘type theoretical’ strategy of Bushnell-Kutzko.

Precisely, first we construct a ‘maximal simple type’ $(\widetilde {J_{M}},\widetilde {\lambda }_{M})$ of $\widetilde {M}$ as an $\mathfrak {s}_{M}$-type, where $\mathfrak {s}_{M}=(\widetilde {M},\mathcal {O})$ is the related cuspidal inertial equivalence class of $\widetilde {M}$. Along the way, we prove the folklore conjecture that every cuspidal representation of $\widetilde {M}$ could be constructed explicitly by a compact induction. Secondly, we construct ‘simple types’ $(\widetilde {J},\widetilde {\lambda })$ of $\widetilde {G}$ and prove that each of them is an $\mathfrak {s}$-type of a certain block $\operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$. When $\widetilde {G}$ is either a Kazhdan-Patterson cover or Savin’s cover, the corresponding blocks turn out to be those containing discrete series representations of $\widetilde {G}$. Finally, for a simple type $(\widetilde {J},\widetilde {\lambda })$ of $\widetilde {G}$, we describe the related Hecke algebra $\mathcal {H}(\widetilde {G},\widetilde {\lambda })$, which turns out to be not far from an affine Hecke algebra of type A, and is exactly so if $\widetilde {G}$ is one of the two special covers mentioned above.

We leave the construction of a ‘semi-simple type’ related to a general block $\operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$ to a future phase of the work.

Mukai’s program in [16] seeks to recover a K3 surface X from any curve C on it by exhibiting it as a Fourier–Mukai partner to a Brill–Noether locus of vector bundles on the curve. In the case X has Picard number one and the curve $C\in |H|$ is primitive, this was confirmed by Feyzbakhsh in [11, 13] for $g\geq 11$ and $g\neq 12$. More recently, Feyzbakhsh has shown in [12] that certain moduli spaces of stable bundles on X are isomorphic to the Brill–Noether locus of curves in $|H|$ if g is sufficiently large. In this paper, we work with irreducible curves in a nonprimitive ample linear system $|mH|$ and prove that Mukai’s program is valid for any irreducible curve when $g\neq 2$, $mg\geq 11$ and $mg\neq 12$. Furthermore, we introduce the destabilising regions to improve Feyzbakhsh’s analysis in [12]. We show that there are hyper-Kähler varieties as Brill–Noether loci of curves in every dimension.

We provide two constructions of Gaussian random holomorphic sections of a Hermitian holomorphic line bundle $(L,h_{L})$ on a Hermitian complex manifold $(X,\Theta )$, that are particularly interesting in the case where the space of $\mathcal {L}^2$-holomorphic sections $H^{0}_{(2)}(X,L)$ is infinite dimensional. We first provide a general construction of Gaussian random holomorphic sections of L, which, if $H^{0}_{(2)}(X,L)$ is infinite dimensional, are almost never $\mathcal {L}^2$-integrable on X. The second construction combines the abstract Wiener space theory with the Berezin–Toeplitz quantization and yields a Gaussian ensemble of random $\mathcal {L}^2$-holomorphic sections. Furthermore, we study their random zeros in the context of semiclassical limits, including their distributions, large deviation estimates, local fluctuations and hole probabilities.

The global analogue of a Henselian local ring is a Henselian pair – a ring R and an ideal I which satisfy a condition resembling Hensel’s lemma regarding lifting coprime factorizations of monic polynomials over $R/I$ to factorizations over R. The geometric counterpart is the notion of a Henselian scheme, which can serve as a substitute for formal schemes in applications such as deformation theory.

In this paper, we prove a GAGA-style cohomology comparison result for Henselian schemes in positive characteristic, making use of a ‘Henselian étale’ topology defined in previous work in order to leverage exactness of finite pushforward for abelian sheaves in the étale topology of schemes. We will also discuss algebraizability of coherent sheaves on the Henselization of a proper scheme, proving (without a positive characteristic restriction) algebraizability for coherent subsheaves. We can then deduce a Henselian version of Chow’s theorem on algebraization and the algebraizability of maps between Henselizations of proper schemes.

A central question in Arithmetic geometry is to determine for which polynomials $f \in \mathbb {Z}[t]$ and which number fields K the Hasse principle holds for the affine equation $f(t) = \mathbf {N}_{K/\mathbb {Q}}(\mathbf {x}) \neq 0$. Whilst extensively studied in the literature, current results are largely limited to polynomials and number fields of low degree. In this paper, we establish the Hasse principle for a wide family of polynomials and number fields, including polynomials that are products of arbitrarily many linear, quadratic or cubic factors. The proof generalises an argument of Irving [27], which makes use of the beta sieve of Rosser and Iwaniec. As a further application of our sieve results, we prove new cases of a conjecture of Harpaz and Wittenberg on locally split values of polynomials over number fields, and discuss consequences for rational points in fibrations.

We initiate the study of deformation theory in the context of derived and higher log geometry. After reconceptualizing the ‘exactification’-procedures in ordinary log geometry in terms of Quillen’s approach to the cotangent complex, we construct an “tangent bundle’ over the category of log ring spectra. The fibers recover the categories of modules over the underlying ring spectra, and the resulting cotangent complex functor specializes to log topological André–Quillen homology on each fiber. As applications, we characterize log square-zero extensions and derive a log variant of étale rigidity, applicable to some tamely ramified extensions of ring spectra.

We investigate the sums $(1/\sqrt {H}) \sum _{X < n \leq X+H} \chi (n)$, where $\chi $ is a fixed non-principal Dirichlet character modulo a prime q, and $0 \leq X \leq q-1$ is uniformly random. Davenport and Erdős, and more recently Lamzouri, proved central limit theorems for these sums provided $H \rightarrow \infty $ and $(\log H)/\log q \rightarrow 0$ as $q \rightarrow \infty $, and Lamzouri conjectured these should hold subject to the much weaker upper bound $H=o(q/\log q)$. We prove this is false for some $\chi $, even when $H = q/\log ^{A}q$ for any fixed $A> 0$. However, we show it is true for ‘almost all’ characters on the range $q^{1-o(1)} \leq H = o(q)$.

Using Pólya’s Fourier expansion, these results may be reformulated as statements about the distribution of certain Fourier series with number theoretic coefficients. Tools used in the proofs include the existence of characters with large partial sums on short initial segments, and moment estimates for trigonometric polynomials with random multiplicative coefficients.

We determine the cohomology of the closed Drinfeld stratum of p-adic Deligne–Lusztig schemes of Coxeter type attached to arbitrary inner forms of unramified groups over a local non-archimedean field. We prove that the corresponding torus weight spaces are supported in exactly one cohomological degree and are pairwise non-isomorphic irreducible representations of the pro-unipotent radical of the corresponding parahoric subgroup. We also prove that all Moy–Prasad quotients of this stratum are maximal varieties, and we investigate the relation between the resulting representations and Kirillov’s orbit method.

Let X be a smooth, projective and geometrically connected curve defined over a finite field ${\mathbb {F}}_q$ of characteristic p different from $2$ and $S\subseteq X$ a subset of closed points. Let $\overline {X}$ and $\overline {S}$ be their base changes to an algebraic closure of ${\mathbb {F}}_q$. We study the number of $\ell $-adic local systems $(\ell \neq p)$ in rank $2$ over $\overline {X}-\overline {S}$ with all possible prescribed tame local monodromies fixed by k-fold iterated action of Frobenius endomorphism for every $k\geq 1$. In all cases, we confirm conjectures of Deligne predicting that these numbers behave as if they were obtained from a Lefschetz fixed point formula. In fact, our counting results are expressed in terms of the numbers of some Higgs bundles.

Motivated by the study of algebraic classes in mixed characteristic, we define a countable subalgebra of ${\overline {\mathbb {Q}}}_p$ which we call the algebra of André’s p-adic periods. The classical Tannakian formalism cannot be used to study these new periods. Instead, inspired by ideas of Drinfel’d on the Plücker embedding and further developed by Haines, we produce an adapted Tannakian setting which allows us to bound the transcendence degree of André’s p-adic periods and to formulate the p-adic analog of the Grothendieck period conjecture. We exhibit several examples where special values of classical p-adic functions appear as André’s p-adic periods, and we relate these new conjectures to some classical problems on algebraic classes.

In this paper, we study the twisted Ruelle zeta function associated with the geodesic flow of a compact, hyperbolic, odd-dimensional manifold X. The twisted Ruelle zeta function is associated with an acyclic representation $\chi \colon \pi _{1}(X) \rightarrow \operatorname {\mathrm {GL}}_{n}(\mathbb {C})$, which is close enough to an acyclic, unitary representation. In this case, the twisted Ruelle zeta function is regular at zero and equals the square of the refined analytic torsion, as it is introduced by Braverman and Kappeler in [6], multiplied by an exponential, which involves the eta invariant of the even part of the odd-signature operator, associated with $\chi $.

A famous theorem of Shokurov states that a general anticanonical divisor of a smooth Fano threefold is a smooth K3 surface. This is quite surprising since there are several examples where the base locus of the anticanonical system has codimension two. In this paper, we show that for four-dimensional Fano manifolds the behaviour is completely opposite: if the base locus is a normal surface, and hence has codimension two, all the anticanonical divisors are singular.