We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the differential graded (DG) algebra $\operatorname{RHom}^{\bullet }(F,F)$ is formal for any sheaf $F$ polystable with respect to an ample line bundle. Our main tool is the uniqueness of the DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.

We prove a dynamical version of the Mordell–Lang conjecture in the context of Drinfeld modules. We use analytic methods similar to those employed by Skolem, Chabauty, and Coleman for studying diophantine equations.

Using a recent computation of the rational minus part of $SH(k)$ by Ananyevskiy, Levine and Panin, a theorem of Cisinski and Déglise and a version of the Röndigs and Østvær theorem, rational stable motivic homotopy theory over an infinite perfect field of characteristic different from 2 is recovered in this paper from finite Milnor–Witt correspondences in the sense of Calmès and Fasel.

Let D≡ 7 mod 8 be a positive squarefree integer, and let h$_D$ be the ideal class number of E$_D$=Q($\sqrt{−D}$). Let d≡1 mod 4 be a squarefree integer relatively prime to D. Then for any integer k[ges ]0 there is a constant M=M(k), independent of the pair (D,D), such that if (−1)$^k$=sign (d), (2k+1,h$_D$)=1, and$\sqrt{D}$>(12/π)d$^2$(log|d+M(k)), then the central L-value L(k+1, χ$^2k+1$$_D, d$>0. Furthermore, for k[les ]1, we can take M(k)=0. Finally, if D=p is a prime, and d>0, then the associated elliptic curve A(p)$^d$ has Mordell–Weil rank 0 (over its definition field) when $\sqrt{D}$>(12/π)d$^2$ log d.

The moduli space of n-pointed stable curves of genus g is stratified by the topological type of the curves being parameterized: the closure of the locus of curves with k nodes has codimension k. The one-dimensional components of this stratification are smooth rational curves called F-curves. These are believed to determine all ample divisors. F-conjecture 

A divisor on  is ample if and only if it positively intersects theF-curves.

We study the motive of the moduli spaces of rank two vector bundles on a curve. In the smooth case we obtain the Hodge numbers, intermediate Jacobians and number of points over a finite field as corollaries. In the singular case our computations yield the Poincaré–Hodge polynomial of Seshadri's smooth model.

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $\bar{\unicode[STIX]{x1D70C}}$ be a continuous, absolutely irreducible representation of its absolute Galois group with values in a finite field of characteristic $p$. We prove that the Galois representations that become crystalline of a fixed regular weight after an abelian extension are Zariski-dense in the generic fiber of the universal deformation ring of $\bar{\unicode[STIX]{x1D70C}}$. In fact we deduce this from a similar density result for the space of trianguline representations. This uses an embedding of eigenvarieties for unitary groups into the spaces of trianguline representations as well as the corresponding density claim for eigenvarieties as a global input.

A logarithmic 1-form on $\mathbb C\mathbb P^n$ can be written as

\omega=\biggl(\prod_0^m F_j\biggr)\sum_0^m \lambda_i\frac{dF_i}{F_i}=\lambda_0 \widehat F_0 \,dF_0+\cdots+\lambda_m \widehat F_m \,dF_m

with $\widehat F_i=(\prod_0^m F_j)/F_i$ for some homogeneous polynomials Fi of degree di and constants $\lambda_i\in{\mathbb C}^\star$ such that $\sum\lambda_id_i=0$. For general $F_i,\lambda_i$, the singularities of $\omega$ consist of a schematic union of the codimension 2 subvarieties Fi = Fj = 0 together with, possibly, finitely many isolated points. This is the case when all Fi are smooth and in general position. In this situation, we give a formula which prescribes the number of isolated singularities.

Given a curve of genus 3 with an unramified double cover, we give an explicit description of the associated Prym variety. We also describe how an unramified double cover of a non-hyperelliptic genus 3 curve can be mapped into the Jacobian of a curve of genus 2 over its field of definition and how this can be used to perform Chabauty- and Brauer–Manin-type calculations for curves of genus 5 with an fixed-point-free involution. As an application, we determine the rational points on a smooth plane quartic and give examples of curves of genus 3 and 5 violating the Hasse principle. The methods are, in principle, applicable to any genus 3 curve with a double cover. We also show how these constructions can be used to design smooth plane quartics with specific arithmetic properties. As an example, we give a smooth plane quartic with all 28 bitangents defined over . By specialization, this also gives examples over .

In this paper, we study the structure of the local components of the (shallow, i.e. without $U_{p}$) Hecke algebras acting on the space of modular forms modulo $p$ of level $1$, and relate them to pseudo-deformation rings. In many cases, we prove that those local components are regular complete local algebras of dimension $2$, generalizing a recent result of Nicolas and Serre for the case $p=2$.

Let $Y$ be a compact Kähler normal space and let $\unicode[STIX]{x1D6FC}\in H_{\mathit{BC}}^{1,1}(Y)$ be a Kähler class. We study metric properties of the space ${\mathcal{H}}_{\unicode[STIX]{x1D6FC}}$ of Kähler metrics in $\unicode[STIX]{x1D6FC}$ using Mabuchi geodesics. We extend several results of Calabi, Chen, and Darvas, previously established when the underlying space is smooth. As an application, we analytically characterize the existence of Kähler–Einstein metrics on $\mathbb{Q}$-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.

Given an action of a reductive group on a normal variety, we describe all invariant open subsets admitting a good quotient with a quasiprojective or a divisorial quotient space. We obtain several new Hilbert–Mumford type theorems, and we extend a projectivity criterion of Białynicki-Birula and Święcicka for varieties with semisimple group action from the smooth to the singular case.

For a proper, smooth scheme $X$ over a $p$-adic field $K$, we show that any proper, flat, semistable ${\mathcal{O}}_{K}$-model ${\mathcal{X}}$ of $X$ whose logarithmic de Rham cohomology is torsion free determines the same ${\mathcal{O}}_{K}$-lattice inside $H_{\text{dR}}^{i}(X/K)$ and, moreover, that this lattice is functorial in $X$. For this, we extend the results of Bhatt–Morrow–Scholze on the construction and the analysis of an $A_{\text{inf}}$-valued cohomology theory of $p$-adic formal, proper, smooth ${\mathcal{O}}_{\overline{K}}$-schemes $\mathfrak{X}$ to the semistable case. The relation of the $A_{\text{inf}}$-cohomology to the $p$-adic étale and the logarithmic crystalline cohomologies allows us to reprove the semistable conjecture of Fontaine–Jannsen.

We give a construction of the moduli space of stable maps to the classifying stack Bμr of a cyclic group by a sequence of rth root constructions on . We prove a closed formula for the total Chern class of μr-eigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus-zero Gromov–Witten theory of stacks of the form [ℂN/μr]. We deduce linear recursions for genus-zero Gromov–Witten invariants.

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums $\text{Kl}_{p}(a)$, as $a$ varies over $\mathbf{F}_{p}^{\times }$ and as $p$ tends to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.

We investigate exponential sums over those numbers ${\leqslant}x$ all of whose prime factors are ${\leqslant}y$. We prove fairly good minor arc estimates, valid whenever $\log ^{3}x\leqslant y\leqslant x^{1/3}$. Then we prove sharp upper bounds for the $p$th moment of (possibly weighted) sums, for any real $p>2$ and $\log ^{C(p)}x\leqslant y\leqslant x$. Our proof develops an argument of Bourgain, showing that this can succeed without strong major arc information, and roughly speaking it would give sharp moment bounds and restriction estimates for any set sufficiently factorable relative to its density. By combining our bounds with major arc estimates of Drappeau, we obtain an asymptotic for the number of solutions of $a+b=c$ in $y$-smooth integers less than $x$ whenever $\log ^{C}x\leqslant y\leqslant x$. Previously this was only known assuming the generalised Riemann hypothesis. Combining them with transference machinery of Green, we prove Roth’s theorem for subsets of the $y$-smooth numbers whenever $\log ^{C}x\leqslant y\leqslant x$. This provides a deterministic set, of size ${\approx}x^{1-c}$, inside which Roth’s theorem holds.

For the dual pair Sp(n)×O(m) with m≤n, we prove an identity between a special value of a certain Eisenstein series and the regularized integral of a theta function. The proof uses the functional equation of the Eisenstein series and the regularized Siegel–Weil formula for Sp(n)×O(2n+2−m). Analogous results for unitary and orthogonal groups are included.

Let be a classical Lie superalgebra and let ℱ be the category of finite-dimensional -supermodules which are completely reducible over the reductive Lie algebra . In [B. D. Boe, J. R. Kujawa and D. K. Nakano, Complexity and module varieties for classical Lie superalgebras, Int. Math. Res. Not. IMRN (2011), 696–724], we demonstrated that for any module M in ℱ the rate of growth of the minimal projective resolution (i.e. the complexity of M) is bounded by the dimension of . In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra . In both cases we show that the complexity is related to the atypicality of the block containing the module.

If $E$ is an elliptic curve over ${\bb Q}$, then let $E(D)$ denote the $D$-quadratic twist of $E$. It is conjectured that there are infinitely many primes $p$ for which $E(p)$ has rank 0, and that there are infinitely many primes $l$ for which $E(l)$ has positive rank. For some special curves $E$ we show that there is a set $S$ of primes $p$ with density $\frac{1}{3}$; for which if $D = \Pi_{pj}$ is a squarefree integer where $pj \in S$, then $E(D)$ has rank 0. In particular $E(p)$ has rank 0 for every $p \in S$. As an example let $E_1$ denote the curve \[ E_1: y^2 = x^3 + 44x^2 -19360x + 1682384. \]Then its associated set of primes $S_1$ consists of the prime 11 and the primes $p$ for which the order of the reduction of $X0(11)$ modulo $p$ is odd. To obtain the general result we show for primes $p \in S$ that the rational factor of $L(E(p),1)$ is nonzero which implies that $E(p)$ has rank 0. These special values are related to surjective ${\bb Z}/2{\bb Z}$ Galois representations that are attached to modular forms. Another example of this result is given, and we conclude with some remarks regarding the existence of positive rank prime twists via polynomial identities.

For each non-negative integer n we define the nth Nash blowup of an algebraic variety, and call them all higher Nash blowups. When n=1, it coincides with the classical Nash blowup. We study higher Nash blowups of curves in detail and prove that any curve in characteristic zero can be desingularized by its nth Nash blowup with n large enough. Moreover, we completely determine for which n the nth Nash blowup of an analytically irreducible curve singularity in characteristic zero is normal, in terms of the associated numerical monoid.