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 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.

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.

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 $d \leq (\log N)^{\log 2 + \varepsilon}\kern-1.5pt$.

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.

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.

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.

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.

We show that for $\mathrm {C}^*$-algebras with the global Glimm property, the rank of every operator can be realized as the rank of a soft operator, that is, an element whose hereditary sub-$\mathrm {C}^*$-algebra has no nonzero, unital quotients. This implies that the radius of comparison of such a $\mathrm {C}^*$-algebra is determined by the soft part of its Cuntz semigroup.

Under a mild additional assumption, we show that every Cuntz class dominates a (unique) largest soft Cuntz class. This defines a retract from the Cuntz semigroup onto its soft part, and it follows that the covering dimensions of these semigroups differ by at most $1$.

Several structural results about permutation groups of finite rank definable in differentially closed fields of characteristic zero (and other similar theories) are obtained. In particular, it is shown that every finite rank definably primitive permutation group is definably isomorphic to an algebraic permutation group living in the constants. Applications include the verification, in differentially closed fields, of the finite Morley rank permutation group conjectures of Borovik-Deloro and Borovik-Cherlin. Applying the results to binding groups for internality to the constants, it is deduced that if complete types p and q are of rank m and n, respectively, and are nonorthogonal, then the $(m+3)$rd Morley power of p is not weakly orthogonal to the $(n+3)$rd Morley power of q. An application to transcendence of generic solutions of pairs of algebraic differential equations is given.

We study the quasi-ergodicity of compact strong Feller semigroups $U_t$, $t> 0$, on $L^2(M,\mu )$; we assume that M is a locally compact Polish space equipped with a locally finite Borel measue $\mu $. The operators $U_t$ are ultracontractive and positivity preserving, but not necessarily self-adjoint or normal. We are mainly interested in those cases where the measure $\mu $ is infinite and the semigroup is not intrinsically ultracontractive. We relate quasi-ergodicity on $L^p(M,\mu )$ and uniqueness of the quasi-stationary measure with the finiteness of the heat content of the semigroup (for large values of t) and with the progressive uniform ground state domination property. The latter property is equivalent to a variant of quasi-ergodicity which progressively propagates in space as $t \uparrow \infty $; the propagation rate is determined by the decay of . We discuss several applications and illustrate our results with examples. This includes a complete description of quasi-ergodicity for a large class of semigroups corresponding to non-local Schrödinger operators with confining potentials.

We show that, for any prime p, there exist absolutely simple abelian varieties over $\mathbb {Q}$ with arbitrarily large p-torsion in their Tate-Shafarevich groups. To prove this, we construct explicit $\mu _p$-covers of Jacobians of curves of the form $y^p = x(x-1)(x-a)$ which violate the Hasse principle. In the appendix, Tom Fisher explains how to interpret our proof in terms of a Cassels-Tate pairing.

Let $\eta $ be [-11pc] [-7pc]a closed real 1-form on a closed Riemannian n-manifold $(M,g)$. Let $d_z$, $\delta _z$ and $\Delta _z$ be the induced Witten’s type perturbations of the de Rham derivative and coderivative and the Laplacian, parametrized by $z=\mu +i\nu \in \mathbb C$ ($\mu ,\nu \in \mathbb {R}$, $i=\sqrt {-1}$). Let $\zeta (s,z)$ be the zeta function of $s\in \mathbb {C}$, defined as the meromorphic extension of the function $\zeta (s,z)=\operatorname {Str}({\eta \wedge }\,\delta _z\Delta _z^{-s})$ for $\Re s\gg 0$. We prove that $\zeta (s,z)$ is smooth at $s=1$ and establish a formula for $\zeta (1,z)$ in terms of the associated heat semigroup. For a class of Morse forms, $\zeta (1,z)$ converges to some $\mathbf {z}\in \mathbb {R}$ as $\mu \to +\infty $, uniformly on $\nu $. We describe $\mathbf {z}$ in terms of the instantons of an auxiliary Smale gradient-like vector field X and the Mathai–Quillen current on $TM$ defined by g. Any real 1-cohomology class has a representative $\eta $ satisfying the hypothesis. If n is even, we can prescribe any real value for $\mathbf {z}$ by perturbing g, $\eta $ and X and achieve the same limit as $\mu \to -\infty $. This is used to define and describe certain tempered distributions induced by g and $\eta $. These distributions appear in another publication as contributions from the preserved leaves in a trace formula for simple foliated flows, giving a solution to a problem stated by C. Deninger.