We resolve parts (A) and (B) of Problem 1.100 from Kirby’s list [Problems in low-dimensional topology, in Geometric topology, AMS/IP Studies in Advanced Mathematics, vol. 2 (American Mathematical Society, Providence, RI, 1997), 35–473] by showing that many nontrivial links arise as cross-sections of unknotted holomorphic disks in the four-ball. The techniques can be used to produce unknotted ribbon surfaces with prescribed cross-sections, including unknotted Lagrangian disks with nontrivial cross-sections.

Let $k$ be a perfect field of characteristic $p>0$ and let $\operatorname{W}$ be the ring of Witt vectors of $k$. In this article, we give a new proof of the Frobenius descent for convergent isocrystals on a variety over $k$ relative to $\operatorname{W}$. This proof allows us to deduce an analogue of the de Rham complexes comparison theorem of Berthelot [$\mathscr{D}$-modules arithmétiques. II. Descente par Frobenius, Mém. Soc. Math. Fr. (N.S.) 81 (2000)] without assuming a lifting of the Frobenius morphism. As an application, we prove a version of Berthelot’s conjecture on the preservation of convergent isocrystals under the higher direct image by a smooth proper morphism of $k$-varieties.

We show that the completed Hecke algebra of $p$-adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$-adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$. This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge–Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proof, independent of the classical Jacquet–Langlands correspondence, that Galois representations can be attached to classical and $p$-adic quaternionic eigenforms.

For an abelian variety $A$ over a number field $k$ we discuss the maximal divisible subgroup of ${\mathrm{H} }^{1} (k, A)$ and its intersection with the subgroup Ш$(A/ k)$. The results are most complete for elliptic curves over $ \mathbb{Q} $.

The eta invariant of the Dirac operator over a non-compact cofinite quotient of PSL(2,ℝ) is defined through a regularized trace following Melrose. It reduces to the standard definition in terms of eigenvalues in the case of a totally non-trivial spin structure. When the S1-fibers are rescaled, the metric becomes of non-exact fibered-cusp type near the ends. We completely describe the continuous spectrum of the Dirac operator with respect to the rescaled metric and its dependence on the spin structure, and show that the adiabatic limit of the eta invariant is essentially the volume of the base hyperbolic Riemann surface with cusps, extending some of the results of Seade and Steer.

We demonstrate that a pair of additive quintic equations in at least 34 variables has a nontrivial integral solution, subject only to an 11-adic solubility hypothesis. This is achieved by an application of the Hardy–Littlewood method, for which we require a sharp estimate for a 33.998th moment of quintic exponential sums. We are able to employ p-adic iteration in a form that allows the estimation of such a mean value over a complete unit square, thereby providing an approach that is technically simpler than those of previous workers and flexible enough to be applied to related problems.

Let $ {\mathbb {C}}^{n+1}_o$ denote the germ of $ {\mathbb {C}}^{n+1}$ at the origin. Let $V$ be a hypersurface germ in $ {\mathbb {C}}^{n+1}_o$ and $W$ a deformation of $V$ over $ {\mathbb {C}}_{o}^{m}$. Under the hypothesis that $W$ is a Newton non-degenerate deformation, in this article we prove that $W$ is a $\mu$-constant deformation if and only if $W$ admits a simultaneous embedded resolution. This result gives a lot of information about $W$, for example, the topological triviality of the family $W$ and the fact that the natural morphism $(\operatorname {W( {\mathbb {C}}_{o})}_{m})_{{\rm red}}\rightarrow {\mathbb {C}}_{o}$ is flat, where $\operatorname {W( {\mathbb {C}}_{o})}_{m}$ is the relative space of $m$-jets. On the way to the proof of our main result, we give a complete answer to a question of Arnold on the monotonicity of Newton numbers in the case of convenient Newton polyhedra.

Given $E$, an elliptic curve defined over $K$, a field of positive characteristic, provided that $j$, the Weierstrass $j$-invariant, is not an element of $K^p$, we construct explicitly, that is, we give by a closed form formula, a non-trivial homomorphism, $\mu:E(K) \rightarrow K^+$, from the group of $K$-rational points of $E$ to $K^+$, the additive group of $K$. In the course of our analysis we discover a canonical differential, $\omega_q \in \Omega; K|{\bb F}_p$, associated to $E$ and we relate it to the differential $dq/q$ associated to the Tate curve. If the transcendence degree of $K$ over ${\bb F}_p$ is equal to one, as for example is the case for function fields in one variable, then $\mu$ is a $p$-descent map, that is, its kernel is equal to $pE(K)$ and the explicit formula for $\mu$ can be used to provide effective proofs of analogues of classical theorems on elliptic curves. For example, in the author's thesis at The University of Texas at Austin the analogue of Siegel's Theorem on the finiteness of integral points of $E(K)$ is proved effectively.

In earlier work, the first named author generalized the construction of Darmon-style $\mathcal {L}$-invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial coefficient system and Steinberg at a fixed prime. In this paper, assuming that the Archimedean component of the group has discrete series we show that these automorphic $\mathcal {L}$-invariants can be computed in terms of derivatives of Hecke eigenvalues in $p$-adic families. Our proof is novel even in the case of modular forms, which was established by Bertolini, Darmon and Iovita. The main new technical ingredient is the Koszul resolution of locally analytic principal series representations by Kohlhaase and Schraen. As an application of our results we settle a conjecture of Spieß: we show that automorphic $\mathcal {L}$-invariants of Hilbert modular forms of parallel weight $2$ are independent of the sign character used to define them. Moreover, we show that they are invariant under Jacquet–Langlands transfer and, in fact, equal to the Fontaine–Mazur $\mathcal {L}$-invariant of the associated Galois representation. Under mild assumptions, we also prove the equality of automorphic and Fontaine–Mazur $\mathcal {L}$-invariants for representations of definite unitary groups of arbitrary rank. Finally, we study the case of Bianchi modular forms to show how our methods, given precise results on eigenvarieties, can also work in the absence of discrete series representations.

We study the interplay between the minimal representations of the orthogonal Lie algebra $\mathfrak{g}=\mathfrak{so}(n+2,\mathbb{C})$ and the algebra of symmetries$\mathscr{S}(\Box ^{r})$ of powers of the Laplacian $\Box$ on $\mathbb{C}^{n}$. The connection is made through the construction of a highest-weight representation of $\mathfrak{g}$ via the ring of differential operators ${\mathcal{D}}(X)$ on the singular scheme $X=(\mathtt{F}^{r}=0)\subset \mathbb{C}^{n}$, for $\mathtt{F}=\sum _{j=1}^{n}X_{i}^{2}\in \mathbb{C}[X_{1},\ldots ,X_{n}]$. In particular, we prove that $U(\mathfrak{g})/K_{r}\cong \mathscr{S}(\Box ^{r})\cong {\mathcal{D}}(X)$ for a certain primitive ideal $K_{r}$. Interestingly, if (and only if) $n$ is even with $r\geqslant n/2$, then both $\mathscr{S}(\Box ^{r})$ and its natural module ${\mathcal{A}}=\mathbb{C}[\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n},\ldots ,\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n}]/(\Box ^{r})$ have a finite-dimensional factor. The same holds for the ${\mathcal{D}}(X)$-module ${\mathcal{O}}(X)$. We also study higher-dimensional analogues $M_{r}=\{x\in A:\Box ^{r}(x)=0\}$ of the module of harmonic elements in $A=\mathbb{C}[X_{1},\ldots ,X_{n}]$ and of the space of ‘harmonic densities’. In both cases we obtain a minimal $\mathfrak{g}$-representation that is closely related to the $\mathfrak{g}$-modules ${\mathcal{O}}(X)$ and ${\mathcal{A}}$. Essentially all these results have real analogues, with the Laplacian replaced by the d’Alembertian $\Box _{p}$ on the pseudo-Euclidean space $\mathbb{R}^{p,q}$ and with $\mathfrak{g}$ replaced by the real Lie algebra $\mathfrak{so}(p+1,q+1)$.

To a polynomial map f : Cn + 1 → C one can attach a monodromy transformation on the complex cohomology of its generic fiber that reflects its asymptotic behaviour. In this paper this transformation is determined for a class of generic polynomials in terms of data attached to projective hypersurfaces with isolated singularities.

We consider deformations of singular Lagrangian varieties in symplectic manifolds. We prove that a Lagrangian deformation of a Lagrangian complete intersection is analytically rigid provided that this is the case infinitesimally. This result is given as a consequence of the coherence of the direct image sheaves of relative infinitesimal Lagrangian deformations.

We prove that algebraic solutions of Garnier systems in the irregular case are of two types. The classical ones come from isomonodromic deformations of linear equations with diagonal or dihedral differential Galois group; we give a complete list in the rank-2 case (two indeterminates). The pull-back ones come from deformations of coverings over a fixed degenerate hypergeometric equation; we provide a complete list when the differential Galois group is $\text{SL}_{2}(\mathbb{C})$. As a byproduct, we obtain a complete list of algebraic solutions for the rank-2 irregular Garnier systems.

This note proves Cellini’s conjecture that, in a Coxeter system (W,S) with reflections T, the T-increasing paths in W are self-avoiding. Here, a T-increasing path is a sequence v,t1v,…,tn⋯t1v in W with ti∈T and t1≺⋯≺tn in a reflection order ⪯ of T.

To a symmetric, relatively ample line bundle on an Abelian scheme one can associate a linear combination of the determinant bundle and the relative canonical bundle, which is a torsion element in the Picard group of the base. We improve the bound on the order of this element found by Faltings and Chai. In particular, we obtain an optimal bound when the degree of the line bundle d is odd and the set of residue characteristics of the base does not intersect the set of primes p dividing d, such that p$\equiv -1$ mod(4) and p$\le 2$g$-1$, where g is the relative dimension of the Abelian scheme. Also, we show that in some cases these torsion elements generate the entire torsion subgroup in the Picard group of the corresponding moduli stack.

In this paper we study the subgroup of the Picard group of Voevodsky’s category of geometric motives $\operatorname{DM}_{\text{gm}}(k;\mathbb{Z}/2)$ generated by the reduced motives of affine quadrics. Our main tools here are the functors of Bachmann [On the invertibility of motives of affine quadrics, Doc. Math. 22 (2017), 363–395], but we also provide an alternative method. We show that the group in question can be described in terms of indecomposable direct summands in the motives of projective quadrics over $k$. In particular, we describe all the relations among the reduced motives of affine quadrics. We also extend the criterion of motivic equivalence of projective quadrics.

We consider the Dirichlet series Z(P,A;s) = [sum ]$_m∈ A ∩ Z^n$P$^-s$(m) (s ∈ C) where P ∈ R[X$_1$, …, X$_n$] and A is an open semi-algebraic subset of R$^n$. We will say that Z(P,A;s) exists if this multiple series is absolutely convergent. In this paper we study the existence and several properties of meromorphic continuations of such series, under certain assumption on P and A. As an application, we show the existence of a finite asymptotic expansion of the counting function with support in A: N$_p$(A,t):= [sharp ] m ∈ A ∩ Z$^n$ | P(m) [les ] t} when t → +∞.

Estimates are established for the number of integers of size N, in intervals of size $N^\theta$, that fail to admit a representation as the sum of s cubes (s = 5, 6). Thereby it is shown that almost all such integers are represented in the proposed manner. When s = 5 one may take $\theta = 10/21$, and when s = 6 one may take any $\theta >17/63$. Similar such conclusions are also established for the related problem associated with the expected asymptotic formula.

We undertake a systematic study of Lipschitz normally embedded normal complex surface germs. We prove, in particular, that the topological type of such a germ determines the combinatorics of its minimal resolution which factors through the blowup of its maximal ideal and through its Nash transform, as well as the polar curve and the discriminant curve of a generic plane projection, thus generalizing results of Spivakovsky and Bondil that were known for minimal surface singularities. In an appendix, we give a new example of a Lipschitz normally embedded surface singularity.

We establish a Harder–Narasimhan formalism for modifications of $G$-bundles on the Fargues–Fontaine curve. The semi-stable stratum of the associated stratification of the ${B^+_{{\rm dR}}}$-Grassmannian coincides with the variant of the weakly admissible locus defined by Viehmann, and its classical points agree with those of the basic Newton stratum. When restricted to minuscule affine Schubert cells, the stratification corresponds to the Harder–Narasimhan stratification of Dat, Orlik and Rapoport. We also study basic geometric properties of the strata, and the relation to the Hodge–Newton decomposition.