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Convexity of multiplicities of filtrations on local rings Compos. Math. (IF 1.8) Pub Date : 2024-03-13 Harold Blum, Yuchen Liu, Lu Qi
We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction
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Discrepancy of rational points in simple algebraic groups Compos. Math. (IF 1.8) Pub Date : 2024-03-13 Alexander Gorodnik, Amos Nevo
The aim of the present paper is to derive effective discrepancy estimates for the distribution of rational points on general semisimple algebraic group varieties, in general families of subsets and at arbitrarily small scales. We establish mean-square, almost sure and uniform estimates for the discrepancy with explicit error bounds. We also prove an analogue of W. Schmidt's theorem, which establishes
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Constructing abelian varieties from rank 2 Galois representations Compos. Math. (IF 1.8) Pub Date : 2024-03-07 Raju Krishnamoorthy, Jinbang Yang, Kang Zuo
Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let ${\mathbb {L}}$ be a rank $2$, geometrically irreducible lisse $\overline {{\mathbb {Q}}}_\ell$-sheaf on $U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field $E\subset \overline {\mathbb {Q}}_{\ell }$, and has bad, infinite reduction at some
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On the Hasse principle for complete intersections Compos. Math. (IF 1.8) Pub Date : 2024-03-05 Matthew Northey, Pankaj Vishe
We prove the Hasse principle for a smooth projective variety $X\subset \mathbb {P}^{n-1}_\mathbb {Q}$ defined by a system of two cubic forms $F,G$ as long as $n\geq 39$. The main tool here is the development of a version of Kloosterman refinement for a smooth system of equations defined over $\mathbb {Q}$.
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There are at most finitely many singular moduli that are S-units Compos. Math. (IF 1.8) Pub Date : 2024-03-05 Sebastián Herrero, Ricardo Menares, Juan Rivera-Letelier
We show that for every finite set of prime numbers $S$, there are at most finitely many singular moduli that are $S$-units. The key new ingredient is that for every prime number $p$, singular moduli are $p$-adically disperse. We prove analogous results for the Weber modular functions, the $\lambda$-invariants and the McKay–Thompson series associated with the elements of the monster group. Finally,
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On straightening for Segal spaces Compos. Math. (IF 1.8) Pub Date : 2024-02-23 Joost Nuiten
The straightening–unstraightening correspondence of Grothendieck–Lurie provides an equivalence between cocartesian fibrations between $(\infty, 1)$-categories and diagrams of $(\infty, 1)$-categories. We provide an alternative proof of this correspondence, as well as an extension of straightening–unstraightening to all higher categorical dimensions. This is based on an explicit combinatorial result
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Modular forms of half-integral weight on exceptional groups Compos. Math. (IF 1.8) Pub Date : 2024-02-22 Spencer Leslie, Aaron Pollack
We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by ${\pm }1$. We analyze the minimal modular form $\Theta _{F_4}$ on the double cover of $F_4$, following Loke–Savin and Ginzburg. Using $\Theta _{F_4}$, we define a modular form
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Stratification of the transverse momentum map Compos. Math. (IF 1.8) Pub Date : 2024-02-12 Maarten Mol
Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that
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Most odd-degree binary forms fail to primitively represent a square Compos. Math. (IF 1.8) Pub Date : 2024-02-08 Ashvin A. Swaminathan
Let $F$ be a separable integral binary form of odd degree $N \geq 5$. A result of Darmon and Granville known as ‘Faltings plus epsilon’ implies that the degree-$N$ superelliptic equation $y^2 = F(x,z)$ has finitely many primitive integer solutions. In this paper, we consider the family $\mathscr {F}_N(f_0)$ of degree-$N$ superelliptic equations with fixed leading coefficient $f_0 \in \mathbb {Z} \smallsetminus
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Boundedness of the p-primary torsion of the Brauer group of an abelian variety Compos. Math. (IF 1.8) Pub Date : 2024-01-05 Marco D'Addezio
We prove that the $p^\infty$-torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic $p>0$ is bounded. This answers a (variant of a) question asked by Skorobogatov and Zarhin for abelian varieties. To do this, we prove a ‘flat Tate conjecture’ for divisors. We also study other geometric Galois-invariant $p^\infty$-torsion classes of the Brauer
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Isoparametric hypersurfaces in symmetric spaces of non-compact type and higher rank Compos. Math. (IF 1.8) Pub Date : 2024-01-05 Miguel Domínguez-Vázquez, Víctor Sanmartín-López
We construct inhomogeneous isoparametric families of hypersurfaces with non-austere focal set on each symmetric space of non-compact type and rank ${\geq }3$. If the rank is ${\geq }4$, there are infinitely many such examples. Our construction yields the first examples of isoparametric families on any Riemannian manifold known to have a non-austere focal set. They can be obtained from a new general
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On the Bezrukavnikov–Kaledin quantization of symplectic varieties in characteristic p Compos. Math. (IF 1.8) Pub Date : 2024-01-05 Ekaterina Bogdanova, Vadim Vologodsky
We prove that after inverting the Planck constant $h$, the Bezrukavnikov–Kaledin quantization $(X, {\mathcal {O}}_h)$ of symplectic variety $X$ in characteristic $p$ with $H^2(X, {\mathcal {O}}_X) =0$ is Morita equivalent to a certain central reduction of the algebra of differential operators on $X$.
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Sixfolds of generalized Kummer type and K3 surfaces Compos. Math. (IF 1.8) Pub Date : 2024-01-05 Salvatore Floccari
We prove that any hyper-Kähler sixfold $K$ of generalized Kummer type has a naturally associated manifold $Y_K$ of $\mathrm {K}3^{[3]}$ type. It is obtained as crepant resolution of the quotient of $K$ by a group of symplectic involutions acting trivially on its second cohomology. When $K$ is projective, the variety $Y_K$ is birational to a moduli space of stable sheaves on a uniquely determined projective
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Dynamics on ℙ1: preperiodic points and pairwise stability Compos. Math. (IF 1.8) Pub Date : 2024-01-05 Laura DeMarco, Niki Myrto Mavraki
DeMarco, Krieger, and Ye conjectured that there is a uniform bound B, depending only on the degree d, so that any pair of holomorphic maps $f, g :{\mathbb {P}}^1\to {\mathbb {P}}^1$ with degree $d$ will either share all of their preperiodic points or have at most $B$ in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, $\mathrm {Rat}_d \times
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The Grothendieck–Serre conjecture over valuation rings Compos. Math. (IF 1.8) Pub Date : 2024-01-05 Ning Guo
In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme $G$ over a valuation ring $V$ with fraction field $K$, a $G$-torsor over $V$ is trivial if it is trivial over $K$. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought
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Families of rational curves on holomorphic symplectic varieties and applications to 0-cycles Compos. Math. (IF 1.8) Pub Date : 2023-12-18 François Charles, Giovanni Mongardi, Gianluca Pacienza
We study families of rational curves on irreducible holomorphic symplectic varieties. We give a necessary and sufficient condition for a sufficiently ample linear system on a holomorphic symplectic variety of $K3^{[n]}$-type to contain a uniruled divisor covered by rational curves of primitive class. In particular, for any fixed $n$, we show that there are only finitely many polarization types of holomorphic
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Higher semiadditive algebraic K-theory and redshift Compos. Math. (IF 1.8) Pub Date : 2023-12-15 Shay Ben-Moshe, Tomer M. Schlank
We define higher semiadditive algebraic K-theory, a variant of algebraic K-theory that takes into account higher semiadditive structure, as enjoyed for example by the $\mathrm {K}(n)$- and $\mathrm {T}(n)$-local categories. We prove that it satisfies a form of the redshift conjecture. Namely, that if $R$ is a ring spectrum of height $\leq n$, then its semiadditive K-theory is of height $\leq n+1$.
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Subsets of without L-shaped configurations Compos. Math. (IF 1.8) Pub Date : 2023-12-04 Sarah Peluse
Fix a prime $p\geq 11$. We show that there exists a positive integer $m$ such that any subset of $\mathbb {F}_p^n\times \mathbb {F}_p^n$ containing no nontrivial configurations of the form $(x,y)$, $(x,y+z)$, $(x,y+2z)$, $(x+z,y)$ must have density $\ll 1/\log _{m}{n}$, where $\log _{m}$ denotes the $m$-fold iterated logarithm. This gives the first reasonable bound in the multidimensional Szemerédi
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Drinfeld's lemma for F-isocrystals, II: Tannakian approach Compos. Math. (IF 1.8) Pub Date : 2023-12-01 Kiran S. Kedlaya, Daxin Xu
We prove a Tannakian form of Drinfeld's lemma for isocrystals on a variety over a finite field, equipped with actions of partial Frobenius operators. This provides an intermediate step towards transferring V. Lafforgue's work on the Langlands correspondence over function fields from $\ell$-adic to $p$-adic coefficients. We also discuss a motivic variant and a local variant of Drinfeld's lemma.
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Finite orbits for large groups of automorphisms of projective surfaces Compos. Math. (IF 1.8) Pub Date : 2023-11-30 Serge Cantat, Romain Dujardin
We study finite orbits of non-elementary groups of automorphisms of compact projective surfaces. We prove that if the surface and the group are defined over a number field $\mathbf {k}$ and the group contains parabolic elements, then the set of finite orbits is not Zariski dense, except in certain very rigid situations, known as Kummer examples. Related results are also established when $\mathbf {k}
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Slopes in eigenvarieties for definite unitary groups Compos. Math. (IF 1.8) Pub Date : 2023-11-22 Lynnelle Ye
We generalize bounds of Liu–Wan–Xiao for slopes in eigencurves for definite unitary groups of rank $2$ to slopes in eigenvarieties for definite unitary groups of any rank. We show that for a definite unitary group of rank $n$, the Newton polygon of the characteristic power series of the $U_p$ Hecke operator has exact growth rate $x^{1+2/{n(n-1)}}$, times a constant proportional to the distance of the
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The tamely ramified geometric quantitative minimal ramification problem Compos. Math. (IF 1.8) Pub Date : 2023-11-09 Mark Shusterman
We prove a large finite field version of the Boston–Markin conjecture on counting Galois extensions of the rational function field with a given Galois group and the smallest possible number of ramified primes. Our proof involves a study of structure groups of (direct products of) racks.
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Approximation and homotopy in regulous geometry Compos. Math. (IF 1.8) Pub Date : 2023-11-09 Wojciech Kucharz
Let $X$, $Y$ be nonsingular real algebraic sets. A map $\varphi \colon X \to Y$ is said to be $k$-regulous, where $k$ is a nonnegative integer, if it is of class $\mathcal {C}^k$ and the restriction of $\varphi$ to some Zariski open dense subset of $X$ is a regular map. Assuming that $Y$ is uniformly rational, and $k \geq 1$, we prove that a $\mathcal {C}^{\infty }$ map $f \colon X \to Y$ can be approximated
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On equivariant topological modular forms Compos. Math. (IF 1.8) Pub Date : 2023-11-06 David Gepner, Lennart Meier
Following ideas of Lurie, we give a general construction of equivariant elliptic cohomology without restriction to characteristic zero. Specializing to the universal elliptic curve we obtain, in particular, equivariant spectra of topological modular forms. We compute the fixed points of these spectra for the circle group and more generally for tori.
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Locality of relative symplectic cohomology for complete embeddings Compos. Math. (IF 1.8) Pub Date : 2023-10-10 Yoel Groman, Umut Varolgunes
A complete embedding is a symplectic embedding $\iota :Y\to M$ of a geometrically bounded symplectic manifold $Y$ into another geometrically bounded symplectic manifold $M$ of the same dimension. When $Y$ satisfies an additional finiteness hypothesis, we prove that the truncated relative symplectic cohomology of a compact subset $K$ inside $Y$ is naturally isomorphic to that of its image $\iota (K)$
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Generic Torelli and local Schottky theorems for Jacobian elliptic surfaces Compos. Math. (IF 1.8) Pub Date : 2023-10-06 N. I. Shepherd-Barron
Suppose that $f:X\to C$ is a general Jacobian elliptic surface over ${\mathbb {C}}$ of irregularity $q$ and positive geometric genus $h$. Assume that $10 h>12(q-1)$, that $h>0$ and let $\overline {\mathcal {E}\ell \ell }$ denote the stack of generalized elliptic curves. (1) The moduli stack $\mathcal {JE}$ of such surfaces is smooth at the point $X$ and its tangent space $T$ there is naturally a direct
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Lagrangian configurations and Hamiltonian maps Compos. Math. (IF 1.8) Pub Date : 2023-09-18 Leonid Polterovich, Egor Shelukhin
We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofer's metric, prove constraints on Lagrangian packing, find instances of Lagrangian Poincaré recurrence, and present a new hierarchy of normal
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Perfect points of abelian varieties Compos. Math. (IF 1.8) Pub Date : 2023-09-08 Emiliano Ambrosi
Let $p$ be a prime number, $k$ a finite field of characteristic $p>0$ and $K/k$ a finitely generated extension of fields. Let $A$ be a $K$-abelian variety such that all the isogeny factors are neither isotrivial nor of $p$-rank zero. We give a necessary and sufficient condition for the finite generation of $A(K^{\mathrm {perf}})$ in terms of the action of $\mathrm {End}(A)\otimes \mathbb {Q}_p$ on
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Conformal blocks for Galois covers of algebraic curves Compos. Math. (IF 1.8) Pub Date : 2023-08-29 Jiuzu Hong, Shrawan Kumar
We study the spaces of twisted conformal blocks attached to a $\Gamma$-curve $\Sigma$ with marked $\Gamma$-orbits and an action of $\Gamma$ on a simple Lie algebra $\mathfrak {g}$, where $\Gamma$ is a finite group. We prove that if $\Gamma$ stabilizes a Borel subalgebra of $\mathfrak {g}$, then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf
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Subgraph distributions in dense random regular graphs Compos. Math. (IF 1.8) Pub Date : 2023-08-25 Ashwin Sah, Mehtaab Sawhney
Given a connected graph $H$ which is not a star, we show that the number of copies of $H$ in a dense uniformly random regular graph is asymptotically Gaussian, which was not known even for $H$ being a triangle. This addresses a question of McKay from the 2010 International Congress of Mathematicians. In fact, we prove that the behavior of the variance of the number of copies of $H$ depends in a delicate
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Realization of GKM fibrations and new examples of Hamiltonian non-Kähler actions Compos. Math. (IF 1.8) Pub Date : 2023-08-24 Oliver Goertsches, Panagiotis Konstantis, Leopold Zoller
We classify fibrations of abstract $3$-regular GKM graphs over $2$-regular ones, and show that all fibrations satisfying the known necessary conditions for realizability are, in fact, realized as the projectivization of equivariant complex rank-$2$ vector bundles over quasitoric $4$-manifolds or $S^4$. We investigate the existence of invariant (stable) almost complex, symplectic, and Kähler structures
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Hecke category actions via Smith–Treumann theory Compos. Math. (IF 1.8) Pub Date : 2023-08-23 J. Ciappara
Let $\textbf {G}$ be a simply connected semisimple algebraic group over a field of characteristic greater than the Coxeter number. We construct a monoidal action of the diagrammatic Hecke category on the principal block $\operatorname {Rep}_0(\textbf {G})$ of $\operatorname {Rep}(\textbf {G})$ by wall-crossing functors. This action was conjectured to exist by Riche and Williamson. Our method uses constructible
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Connected components of affine Deligne–Lusztig varieties for unramified groups Compos. Math. (IF 1.8) Pub Date : 2023-08-17 Sian Nie
For an unramified reductive group, we determine the connected components of affine Deligne–Lusztig varieties in the affine flag variety. Based on work of Hamacher, Kim, and Zhou, this result allows us to verify, in the unramified group case, the He–Rapoport axioms, the almost product structure of Newton strata, and the precise description of isogeny classes predicted by the Langlands–Rapoport conjecture
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Measure equivalence rigidity via s-malleable deformations Compos. Math. (IF 1.8) Pub Date : 2023-08-14 Daniel Drimbe
We single out a large class of groups ${\rm {\boldsymbol {\mathscr {M}}}}$ for which the following unique prime factorization result holds: if $\Gamma _1,\ldots,\Gamma _n\in {\rm {\boldsymbol {\mathscr {M}}}}$ and $\Gamma _1\times \cdots \times \Gamma _n$ is measure equivalent to a product $\Lambda _1\times \cdots \times \Lambda _m$ of infinite icc groups, then $n \ge m$, and if $n = m$, then, after
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Projective varieties with nef tangent bundle in positive characteristic Compos. Math. (IF 1.8) Pub Date : 2023-08-03 Akihiro Kanemitsu, Kiwamu Watanabe
Let $X$ be a smooth projective variety defined over an algebraically closed field of positive characteristic $p$ whose tangent bundle is nef. We prove that $X$ admits a smooth morphism $X \to M$ such that the fibers are Fano varieties with nef tangent bundle and $T_M$ is numerically flat. We also prove that extremal contractions exist as smooth morphisms. As an application, we prove that, if the Frobenius
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Obstruction theory and the level n elliptic genus Compos. Math. (IF 1.8) Pub Date : 2023-08-03 Andrew Senger
Given a height at most two Landweber exact $\mathbb {E}_\infty$-ring $E$ whose homotopy is concentrated in even degrees, we show that any complex orientation of $E$ which satisfies the Ando criterion admits a unique lift to an $\mathbb {E}_\infty$-complex orientation $\mathrm {MU} \to E$. As a consequence, we give a short proof that the level $n$ elliptic genus lifts uniquely to an $\mathbb {E}_\infty$-complex
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Twisted GGP problems and conjectures Compos. Math. (IF 1.8) Pub Date : 2023-07-31 Wee Teck Gan, Benedict H. Gross, Dipendra Prasad
In a series of three earlier papers, we considered a family of restriction problems for classical groups (over local and global fields) and proposed precise answers to these problems using the local and global Langlands correspondence. These restriction problems were formulated in terms of a pair $W \subset V$ of orthogonal, Hermitian, symplectic, or skew-Hermitian spaces. In this paper, we consider
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Sutured instanton homology and Heegaard diagrams Compos. Math. (IF 1.8) Pub Date : 2023-07-28 John A. Baldwin, Zhenkun Li, Fan Ye
Suppose $\mathcal {H}$ is an admissible Heegaard diagram for a balanced sutured manifold $(M,\gamma )$. We prove that the number of generators of the associated sutured Heegaard Floer complex is an upper bound on the dimension of the sutured instanton homology $\mathit {SHI}(M,\gamma )$. It follows, in particular, that strong L-spaces are instanton L-spaces.
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On the strict Picard spectrum of commutative ring spectra Compos. Math. (IF 1.8) Pub Date : 2023-07-28 Shachar Carmeli
We compute the connective spectra of maps from $\mathbb {Z}$ to the Picard spectra of the spherical Witt vectors associated with perfect rings of characteristic $p$. As an application, we determine the connective spectrum of maps from $\mathbb {Z}$ to the Picard spectrum of the sphere spectrum.
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On weighted-blowup formulae of genus zero orbifold Gromov–Witten invariants Compos. Math. (IF 1.8) Pub Date : 2023-07-19 Bohui Chen, Cheng-Yong Du
In this paper, we provide a new approach to prove some weighted-blowup formulae for genus zero orbifold Gromov–Witten invariants. As a consequence, we show the invariance of symplectically rational connectedness with respect to weighted-blowup along positive centers. Furthermore, we use this method to give a new proof to the genus zero relative-orbifold correspondence of Gromov–Witten invariants.
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G-valued crystalline deformation rings in the Fontaine–Laffaille range Compos. Math. (IF 1.8) Pub Date : 2023-07-17 Jeremy Booher, Brandon Levin
Let $G$ be a split reductive group over the ring of integers in a $p$-adic field with residue field $\mathbf {F}$. Fix a representation $\overline {\rho }$ of the absolute Galois group of an unramified extension of $\mathbf {Q}_p$, valued in $G(\mathbf {F})$. We study the crystalline deformation ring for $\overline {\rho }$ with a fixed $p$-adic Hodge type that satisfies an analog of the Fontaine–Laffaille
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Floor diagrams and enumerative invariants of line bundles over an elliptic curve Compos. Math. (IF 1.8) Pub Date : 2023-07-07 Thomas Blomme
We use the tropical geometry approach to compute absolute and relative enumerative invariants of complex surfaces which are $\mathbb {C} P^1$-bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be refined by the standard Block–Göttsche refined multiplicity to give tropical refined invariants. We then give a concrete algorithm using floor diagrams to
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On the u∞-torsion submodule of prismatic cohomology Compos. Math. (IF 1.8) Pub Date : 2023-06-30 Shizhang Li, Tong Liu
We investigate the maximal finite length submodule of the Breuil–Kisin prismatic cohomology of a smooth proper formal scheme over a $p$-adic ring of integers. This submodule governs pathology phenomena in integral $p$-adic cohomology theories. Geometric applications include a control, in low degrees and mild ramifications, of (1) the discrepancy between two naturally associated Albanese varieties in
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Kudla–Rapoport conjecture for Krämer models Compos. Math. (IF 1.8) Pub Date : 2023-06-29 Qiao He, Yousheng Shi, Tonghai Yang
In this paper, we propose a modified Kudla–Rapoport conjecture for the Krämer model of unitary Rapoport–Zink space at a ramified prime, which is a precise identity relating intersection numbers of special cycles to derivatives of Hermitian local density polynomials. We also introduce the notion of special difference cycles, which has surprisingly simple description. Combining this with induction formulas
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Weight filtrations on Selmer schemes and the effective Chabauty–Kim method Compos. Math. (IF 1.8) Pub Date : 2023-06-20 L. Alexander Betts
We develop an effective version of the Chabauty–Kim method which gives explicit upper bounds on the number of $S$-integral points on a hyperbolic curve in terms of dimensions of certain Bloch–Kato Selmer groups. Using this, we give a new ‘motivic’ proof that the number of solutions to the $S$-unit equation is bounded uniformly in terms of $\#S$.
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Relative regular Riemann–Hilbert correspondence II Compos. Math. (IF 1.8) Pub Date : 2023-06-15 Luisa Fiorot, Teresa Monteiro Fernandes, Claude Sabbah
We develop the theory of relative regular holonomic $\mathcal {D}$-modules with a smooth complex manifold $S$ of arbitrary dimension as parameter space, together with their main functorial properties. In particular, we establish in this general setting the relative Riemann–Hilbert correspondence proved in a previous work in the one-dimensional case.
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Refined unramified cohomology of schemes Compos. Math. (IF 1.8) Pub Date : 2023-06-15 Stefan Schreieder
We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This generalizes to cycles of arbitrary codimension previous results of Bloch–Ogus, Colliot-Thélène–Voisin, Kahn, Voisin, and Ma. We combine our approach with the Bloch–Kato conjecture, proven by Voevodsky, to show that on a smooth complex
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Generic differential equations are strongly minimal Compos. Math. (IF 1.8) Pub Date : 2023-06-08 Matthew DeVilbiss, James Freitag
In this paper we develop a new technique for showing that a nonlinear algebraic differential equation is strongly minimal based on the recently developed notion of the degree of non-minimality of Freitag and Moosa. Our techniques are sufficient to show that generic order $h$ differential equations with non-constant coefficients are strongly minimal, answering a question of Poizat (1980).
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q-stability conditions on Calabi–Yau-𝕏 categories Compos. Math. (IF 1.8) Pub Date : 2023-06-07 Akishi Ikeda, Yu Qiu
We introduce $q$-stability conditions $(\sigma,s)$ on Calabi–Yau-$\mathbb {X}$ categories $\mathcal {D}_\mathbb {X}$, where $\sigma$ is a stability condition on $\mathcal {D}_\mathbb {X}$ and $s$ a complex number. We prove the corresponding deformation theorem, that $\operatorname {QStab}_s\mathcal {D}_\mathbb {X}$ is a complex manifold of dimension $n$ for fixed $s$, where $n$ is the rank of the Grothendieck
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Isometric actions on Lp-spaces: dependence on the value of p Compos. Math. (IF 1.8) Pub Date : 2023-05-26 Amine Marrakchi, Mikael de la Salle
Answering a question by Chatterji–Druţu–Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty ]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($0< p<\infty$) with unbounded orbits if and only if $p \geq p_G$. A similar result holds for the existence of proper continuous affine isometric actions on $L_p$ spaces
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The Balmer spectrum of certain Deligne–Mumford stacks Compos. Math. (IF 1.8) Pub Date : 2023-05-24 Eike Lau
We consider a Deligne–Mumford stack $X$ which is the quotient of an affine scheme $\operatorname {Spec}A$ by the action of a finite group $G$ and show that the Balmer spectrum of the tensor triangulated category of perfect complexes on $X$ is homeomorphic to the space of homogeneous prime ideals in the group cohomology ring $H^*(G,A)$.
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Hofer's geometry and topological entropy Compos. Math. (IF 1.8) Pub Date : 2023-05-22 Arnon Chor, Matthias Meiwes
In this paper we study persistence features of topological entropy and periodic orbit growth of Hamiltonian diffeomorphisms on surfaces with respect to Hofer's metric. We exhibit stability of these dynamical quantities in a rather strong sense for a specific family of maps studied by Polterovich and Shelukhin. A crucial ingredient comes from enhancement of lower bounds for the topological entropy and
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p-adic Eichler–Shimura maps for the modular curve Compos. Math. (IF 1.8) Pub Date : 2023-05-22 Juan Esteban Rodríguez Camargo
We give a new proof of Faltings's $p$-adic Eichler–Shimura decomposition of the modular curves via Bernstein–Gelfand–Gelfand (BGG) methods and the Hodge–Tate period map. The key property is the relation between the Tate module and the Faltings extension, which was used in the original proof. Then we construct overconvergent Eichler–Shimura maps for the modular curves providing ‘the second half’ of
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Integral Fourier transforms and the integral Hodge conjecture for one-cycles on abelian varieties Compos. Math. (IF 1.8) Pub Date : 2023-05-15 Thorsten Beckmann, Olivier de Gaay Fortman
We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, the Jacobian of a smooth projective curve over the complex numbers satisfies the integral Hodge conjecture for one-cycles. The main ingredient is a lift of the Fourier transform to integral Chow groups. Similarly, we prove the integral Tate conjecture
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Foliated affine and projective structures Compos. Math. (IF 1.8) Pub Date : 2023-05-15 Bertrand Deroin, Adolfo Guillot
We formalize the concepts of holomorphic affine and projective structures along the leaves of holomorphic foliations by curves on complex manifolds. We show that many foliations admit such structures, we provide local normal forms for them at singular points of the foliation, and we prove some index formulae in the case where the ambient manifold is compact. As a consequence of these, we establish
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Quadratic Chabauty for modular curves: algorithms and examples Compos. Math. (IF 1.8) Pub Date : 2023-05-15 Jennifer S. Balakrishnan, Netan Dogra, J. Steffen Müller, Jan Tuitman, Jan Vonk
We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus $g>1$ whose Jacobians have Mordell–Weil rank $g$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We
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On the derived category of the Iwahori–Hecke algebra Compos. Math. (IF 1.8) Pub Date : 2023-05-04 Eugen Hellmann
We state a conjecture that relates the derived category of smooth representations of a $p$-adic split reductive group with the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We investigate the conjecture in the case of the principal block of ${\rm GL}_n$ by showing that the functor should be given by the derived tensor product with the family of representations interpolating
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Arakelov–Milnor inequalities and maximal variations of Hodge structure Compos. Math. (IF 1.8) Pub Date : 2023-04-25 Olivier Biquard, Brian Collier, Oscar García-Prada, Domingo Toledo
In this paper we study the $\mathbb {C}^*$-fixed points in moduli spaces of Higgs bundles over a compact Riemann surface for a complex semisimple Lie group and its real forms. These fixed points are called Hodge bundles and correspond to complex variations of Hodge structure. We introduce a topological invariant for Hodge bundles that generalizes the Toledo invariant appearing for Hermitian Lie groups
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Integration questions in separably good characteristics Compos. Math. (IF 1.8) Pub Date : 2023-04-24 Marion Jeannin
Let $G$ be a reductive group over an algebraically closed field $k$ of separably good characteristic $p>0$ for $G$. Under these assumptions, a Springer isomorphism $\phi : \mathcal {N}_{\mathrm {red}}(\mathfrak {g}) \rightarrow \mathcal {V}_{\mathrm {red}}(G)$ from the nilpotent scheme of $\mathfrak {g}$ to the unipotent scheme of $G$ always exists and allows one to integrate any $p$-nilpotent element
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Local terms for transversal intersections Compos. Math. (IF 1.8) Pub Date : 2023-04-20 Yakov Varshavsky
The goal of this note is to show that in the case of ‘transversal intersections’ the ‘true local terms’ appearing in the Lefschetz trace formula are equal to the ‘naive local terms’. To prove the result, we extend the strategy used in our previous work, where the case of contracting correspondences is treated. Our new ingredients are the observation of Verdier that specialization of an étale sheaf