We give an explicit classification of the possible p-adic Galois representations that are attached to elliptic curves E with CM defined over ℚ(j(E)). More precisely, let K be an imaginary quadratic field, and let 𝒪K,f be an order in K of conductor f ≥ 1. Let E be an elliptic curve with CM by 𝒪K,f, such that E is defined by a model over ℚ(j(E)). Let p ≥ 2 be a prime, let Gℚ(j(E)) be the absolute Galois

We introduce a class of modules over Kac–Moody superalgebras; we call these modules snowflake modules. These modules are characterized by invariance property of their characters with respect to a certain subgroup of the Weyl group. Examples of snowflake modules appear as admissible modules in representation theory of affine vertex algebras and in the classification of bounded weight modules. Using

We use the arithmetic of ideals in orders to parametrize the roots μ(mod ⁡m) of the polynomial congruence F(μ) ≡ 0(mod ⁡m), F(X) ∈ ℤ[X] monic, irreducible and degree d. Our parametrization generalizes Gauss’s classic parametrization of the roots of quadratic congruences using binary quadratic forms, which had previously only been extended to the cubic polynomial F(X) = X3 − 2. We show that only a special

We prove that the normal bundle of a general Brill–Noether space curve of degree d and genus g ≥ 2 is stable if and only if (d,g)∉{(5,2),(6,4)}. When g ≤ 1 and the characteristic of the ground field is zero, it is classical that the normal bundle is strictly semistable. We show that this still holds in positive characteristic except when the characteristic is 2, the genus is 0 and the degree is even

The integral monodromy on the Milnor lattice of an isolated quasihomogeneous singularity is subject of an almost untouched conjecture of Orlik from 1972. We prove this conjecture for all iterated Thom–Sebastiani sums of chain type singularities and cycle type singularities. The main part of the paper is purely algebraic. It provides tools for dealing with sums and tensor products of ℤ-lattices with

Pour un morphisme T → S, plat et de présentation finie, l’enveloppe étale — un avatar du π0(T∕S) — peut ne pas exister ; par contre l’enveloppe étale séparée, i.e., celle qui est universelle pour les schémas étales et séparés sur S, existe dès que S est localement connexe. On la note T → πs(T∕S) ; c’est le quotient de T par la relation d’équivalence minimale à graphe ouvert et fermé dans T ×ST ; cette

We obtain the last of the standard Kuznetsov formulas for SL ⁡ (3, ℤ). In the previous paper, we were able to exploit the relationship between the positive-sign Bessel function and the Whittaker function to apply Wallach’s Whittaker expansion; now we demonstrate the expansion of functions into Bessel functions for all four signs, generalizing Wallach’s theorem for SL ⁡ (3). As applications, we again

Under relatively mild and natural conditions, we establish integral period relations for the (real or imaginary) quadratic base change of an elliptic cusp form. This answers a conjecture of Hida regarding the congruence ideal controlling the congruences between this base change and other eigenforms which are not base change. As a corollary, we establish the Bloch–Kato conjecture for adjoint modular

In this paper, we study the representations of the periplectic Lie superalgebra using the Duflo–Serganova functor. Given a simple 𝔭(n)-module L and a certain odd element x ∈ 𝔭(n) of rank 1, we give an explicit description of the composition factors of the 𝔭(n−1)-module DS ⁡ x(L), which is defined as the homology of the complex ΠM →xM →xΠM, where Π denotes the parity-change functor (−) ⊗ ℂ0|1. In

We define the covering gonality and separable covering gonality of varieties over arbitrary fields, generalizing the definition given by Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery for complex varieties. We show that, over an algebraically closed field, a smooth multidegree (d1,… ⁡,dk) complete intersection in ℙN has separable covering gonality at least d − N + 1, where d = d1 + ⋯ + dk. We also

Using twisted sheaves, formal-local methods, and elementary transformations we show that separated algebraic spaces which are constructed as pushouts or contractions (of curves) have enough Azumaya algebras. This implies (1) Under mild hypothesis, every cohomological Brauer class is representable by an Azumaya algebra away from a closed subset of codimension ≥ 3, generalizing an early result of Grothendieck

We construct a descent-of-scalars criterion for K-linear abelian categories. Using advances in the Langlands correspondence due to Abe, we build a correspondence between certain rank 2 local systems and certain Barsotti–Tate groups on complete curves over a finite field. We conjecture that such Barsotti–Tate groups “come from” a family of fake elliptic curves. As an application of these ideas, we provide

Let K be a finite extension of ℚp. The field of norms of a strictly APF extension K∞∕K is a local field of characteristic p equipped with an action of Gal ⁡ (K∞∕K). When can we lift this action to characteristic zero, along with a compatible Frobenius map? In this article, we explain what we mean by lifting the field of norms, explain its relevance to the theory of (φ,Γ)-modules, and show that under

Let K be a characteristic zero algebraic function field with a valuation ν. Let L be a finite extension of K and ω be an extension of ν to L. We establish that the valuation ring V ω of ω is essentially finitely generated over the valuation ring V ν of ν if and only if the initial index 𝜖(ω|ν) is equal to the ramification index e(ω|ν) of the extension. This gives a positive answer, for characteristic

We introduce the use of p-descent techniques for elliptic surfaces over a perfect field of characteristic not 2 or 3. Under mild hypotheses, we obtain an upper bound for the rank of a nonconstant elliptic surface. When p = 2, this bound is an arithmetic refinement of a well-known geometric bound for the rank deduced from Igusa’s inequality. This answers a question raised by Ulmer. We give some applications

For a regular scheme and a prime number p, we define the FW-cotangent bundle as a vector bundle on the closed subscheme defined by p = 0, under a certain finiteness condition. For a constructible complex on the étale site of the scheme, we introduce the condition to be microsupported on a closed conical subset in the FW-cotangent bundle. At the end of the article, we compute the singular supports in

T. Dupuy, E. Katz, J. Rabinoff, and D. Zureick-Brown introduced the module of total p-differentials for a ring over ℤ∕p2ℤ. We study the same construction for a ring over ℤ(p) and prove a regularity criterion. For a local ring, the tensor product with the residue field is constructed in a different way by O. Gabber and L. Ramero. In another article we use the sheaf of FW-differentials to define the

Let k be an algebraically closed field of characteristic p and X the projective line over k with three points removed. We investigate which finite groups G can arise as the monodromy group of finite étale covers of X that are tamely ramified over the three removed points. This provides new information about the tame fundamental group of the projective line. In particular, we show that for each prime

Let U be a regular connected affine semilocal scheme over a field k. Let G be a reductive group scheme over U. Assuming that G has an appropriate parabolic subgroup scheme, we prove the following statement. Given an affine k-scheme W, a principal G-bundle over W ×kU is trivial if it is trivial over the generic fiber of the projection W ×kU → U. We also simplify the proof of the Grothendieck–Serre conjecture:

Noncommutative hypersurfaces, in particular, noncommutative quadric hypersurfaces are major objects of study in noncommutative algebraic geometry. In the commutative case, Knörrer’s periodicity theorem is a powerful tool to study Cohen–Macaulay representation theory since it reduces the number of variables in computing the stable category CM ⁡ ¯(A) of maximal Cohen–Macaulay modules over a hypersurface

We prove that, in certain situations, intersection numbers on formal schemes that come in profinite families vary locally constantly in the parameter. To this end, we define the product S × M of a profinite set S with a locally noetherian formal scheme M and study intersections thereon. Our application is to the arithmetic fundamental lemma of W. Zhang where the result helps to overcome a restriction

We compute the relative orbifold Gromov–Witten invariants of [ℂ2∕ℤn+1] × ℙ1 with respect to vertical fibers. Via a vanishing property of the Hurwitz–Hodge bundle, 2-point rubber invariants are calculated explicitly using Pixton’s formula for the double ramification cycle, and the orbifold quantum Riemann–Roch. As a result parallel to its crepant resolution counterpart for 𝒜n, the GW/DT/Hilb/Sym correspondence

Shin and Templier studied families of automorphic representations with local restrictions: roughly, Archimedean components contained in a fixed L-packet of discrete series and non-Archimedean components ramified only up to a fixed level. They computed limiting statistics of local components as either the weight of the L-packet or level went to infinity. We extend their weight-aspect results to families

We fill a gap in the literature by computing the Chow ring of the classifying space of the unitary group of a hermitian form on a finite-dimensional vector space over a separable quadratic field extension.

We show that the complete symmetric polynomials are dual generators of compressed artinian Gorenstein algebras satisfying the strong Lefschetz property. This is the first example of an explicit dual form with these properties. For complete symmetric forms of any degree in any number of variables, we provide an upper bound for the Waring rank by establishing an explicit power sum decomposition. Moreover

We initiate a systematic study of the cohomology of cluster varieties. We introduce the Louise property for cluster algebras that holds for all acyclic cluster algebras, and for most cluster algebras arising from marked surfaces. For cluster varieties satisfying the Louise property and of full rank, we show that the cohomology satisfies the curious Lefschetz property of Hausel and Rodriguez-Villegas

In this note, we consider special algebraic cycles on the Shimura variety S associated to a quadratic space V over a totally real field F, |F : ℚ| = d, of signature ((m,2)d+ ,(m + 2,0)d−d+ ),1 ≤ d+ < d. For each n, 1 ≤ n ≤ m, there are special cycles Z(T) in S of codimension nd+, indexed by totally positive semidefinite matrices with coefficients in the ring of integers OF. The generating series for

We study model-theoretic properties of algebraic differential equations of order two defined over constant differential fields. In particular, we show that the set of solutions of a “general” differential equation of order two and of degree d ≥ 3 in a differentially closed field is strongly minimal and disintegrated (in other words, is strongly minimal with trivial forking geometry). We also give two

Using a local monomialization result of Knaf and Kuhlmann, which was generalized by Cutkosky, we prove that the valuation ring of an Abhyankar valuation of a function field over an F-finite ground field of prime characteristic is Frobenius split. We show that a Frobenius splitting of a sufficiently well-behaved center lifts to a Frobenius splitting of the valuation ring. We also investigate properties

Malle proposed a conjecture for counting the number of G-extensions L∕K with discriminant bounded above by X, denoted N(K,G;X), where G is a fixed transitive subgroup G ⊂ Sn and X tends towards infinity. We introduce a refinement of Malle’s conjecture, if G is a group with a nontrivial Galois action then we consider the set of crossed homomorphisms in Z1(K,G) (or equivalently 1-coclasses in H1(K,G))

The purpose of this article is to compare several versions of bivariant algebraic cobordism constructed previously by the author and others. In particular, we show that a simple construction based on the universal precobordism theory of Annala and Yokura agrees with the more complicated theory of bivariant derived algebraic cobordism constructed earlier by the author, and that both of these theories

Let ℓ≠3 be a prime. We show that there are only finitely many cyclic number fields F of degree ℓ for which the unit equation λ + μ = 1,λ,μ ∈𝒪F× has solutions. Our result is effective. For example, we deduce that the only cyclic quintic number field for which the unit equation has solutions is ℚ(ζ11)+.

We investigate integrality and divisibility properties of Fourier coefficients of meromorphic modular forms of weight $2k$ associated to positive definite integral binary quadratic forms. For example, we show that if there are no non-trivial cusp forms of weight $2k$, then the $n$-th coefficients of these meromorphic modular forms are divisible by $n^{k-1}$ for every natural number $n$. Moreover, we

We study short sums of algebraic trace functions via the $q$-analogue of van der Corput method, and develop methods of arithmetic exponent pairs that coincide with the classical case while the moduli has sufficiently good factorizations. As an application, we prove a quadratic analogue of Brun-Titchmarsh theorem on average, bounding the number of primes $p\leqslant X$ with $p^2+1\equiv0\pmod q$. The

Deformations of ordinary varieties of K3 type can be described in terms of displays by recent work of Langer-Zink. We extend this to the general (non-ordinary) case using displays with $G$-structure for a reductive group $G$. As a basis we suggest a modified definition of the tensor category of displays and variants which is similar to the Frobenius gauges of Fontaine-Jannsen.

We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane $\mathbb{H}$. The angles of lattice points arising from the orbit of the modular group $PSL_{2}(\mathbb{Z})$, and lying on hyperbolic circles, are shown to be equidistributed for generic radii. However, the angles fail to equidistribute on a thin set of exceptional radii, even in the presence of growing

We establish the analytic Hasse principle for Diophantine systems consisting of one diagonal form of degree $k$ and one general form of degree $d$, where $d$ is smaller than $k$. By employing a hybrid method that combines ideas from the study of general forms with techniques adapted to the diagonal case, we are able to obtain bounds that grow exponentially in $d$ but only quadratically in $k$, reflecting

We formulate and prove an analog of Poonen's finite-field Bertini theorem with Taylor conditions that holds in the Grothendieck ring of varieties. This gives a broad generalization of the work of Vakil-Wood, who treated the case of smooth hypersurface sections. In fact, our techniques give analogs in motivic statistics of all known results in arithmetic statistics that have been proven using Poonen's

We study Tamagawa numbers and other invariants (especially Tate–Shafarevich sets) attached to commutative and pseudoreductive groups over global function fields. In particular, we prove a simple formula for Tamagawa numbers of commutative groups and pseudoreductive groups. We also show that the Tamagawa numbers and Tate–Shafarevich sets of such groups are invariant under inner twist, as well as proving

We give a new characterisation of elliptic curves of Shimura type in terms of commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of global minimal models for such curves, generalising a result of Gross. Along the way we also prove a handful of small but new results regarding elliptic curves with

Roth’s theorem is extended to finitely generated field extensions of ℚ, using Moriwaki’s theory of heights.

We consider the question of simplicity of a ℂ-algebra R under the action of its ring of differential operators DR∕ℂ. We give examples to show that even when R is Gorenstein and has rational singularities, R need not be a simple DR∕ℂ-module; for example, this is the case when R is the homogeneous coordinate ring of a smooth cubic surface. Our examples are homogeneous coordinate rings of smooth Fano

In this paper, we develop the theory of flag manifolds over a semifield for any Kac–Moody root datum. We show that a flag manifold over a semifield admits a natural action of the monoid over that semifield associated with the Kac–Moody datum and admits a cellular decomposition. This extends the previous work of Lusztig, Postnikov, Rietsch, and others on the totally nonnegative flag manifolds (of finite

We prove a conjecture of Lehmann and Tanimoto about the behaviour of the Fujita invariant (or a-constant appearing in Manin’s conjecture) under pull-back to generically finite covers. As a consequence we obtain results about geometric consistency of Manin’s conjecture.

We prove that the ring of Siegel modular forms of weight divisible by g + n + 1 is isomorphic to the ring of (log) pluricanonical forms on the n-fold Kuga family of abelian varieties and certain compactifications of it, for every arithmetic group for a symplectic form of rank 2g > 2. We also give applications to the Kodaira dimension of the Kuga variety. In most cases, the Kuga variety has canonical

We show that all of the nice behavior for Tamagawa numbers, Tate–Shafarevich sets, and other arithmetic invariants of pseudoreductive groups over global function fields, proved in another work, fails in general for noncommutative unipotent groups. We also give some positive results which show that Tamagawa numbers do exhibit some reasonable behavior for arbitrary connected linear algebraic groups over

Let E∕ℚ be an elliptic curve of conductor N, let p > 3 be a prime where E has good ordinary reduction, and let K be an imaginary quadratic field satisfying the Heegner hypothesis. In 1987, Perrin-Riou formulated an Iwasawa main conjecture for the Tate–Shafarevich group of E over the anticyclotomic ℤp-extension of K in terms of Heegner points. In this paper, we give a proof of Perrin-Riou’s conjecture

Nous construisons un modèle local pour les schémas de Hilbert–Siegel de niveau Γ1(p), lorsque p est non-ramifié dans le corps totalement réel. Notre outil clé est une variante du complexe de Lie anneau-équivariant défini par Illusie. We construct a local model for Hilbert–Siegel moduli schemes with Γ1(p)-level structures, when p is unramified in the totally real field. Our key tool is a variant of

Let K be an algebraically closed field of characteristic zero, let G be a finitely generated subgroup of the multiplicative group of K, and let X be a quasiprojective variety defined over K. We consider K-valued sequences of the form an := f(φn(x0)), where φ : X −−→ X and f : X −−→ ℙ1 are rational maps defined over K and x0 ∈ X is a point whose forward orbit avoids the indeterminacy loci of φ and f

We prove that a certain genuine Hecke algebra ℋ on the nonlinear double cover of a simple, simply laced, simply connected, Chevalley group G over ℚ2 admits a Bernstein presentation. This presentation has two consequences. First, the Bernstein component containing the genuine unramified principal series is equivalent to ℋ-mod. Second, ℋ is isomorphic to the Iwahori–Hecke algebra of the linear group G∕Z2

Let G be a finite primitive permutation group on a set Ω with point stabiliser H. Recall that a subset of Ω is a base for G if its pointwise stabiliser is trivial. We define the base size of G, denoted b(G,H), to be the minimal size of a base for G. Determining the base size of a group is a fundamental problem in permutation group theory, with a long history stretching back to the 19th century. Here

We study fields of values of the restriction to Sylow subgroups of irreducible characters of the normalizers of Sylow subgroups in symmetric and alternating groups. As an application, we show that these classes of groups admit a McKay bijection that respects fields of values and restriction to Sylow subgroups.

Let K be a number field, and let E∕K be an elliptic curve over K. The Mordell–Weil theorem asserts that the K-rational points E(K) of E form a finitely generated abelian group. In this work, we complete the classification of the finite groups which appear as the torsion subgroup of E(K) for K a cubic number field. To do so, we determine the cubic points on the modular curves X1(N) for N = 21,22,24

We establish the Sato–Tate equidistribution of Hecke eigenvalues of the family of Hecke–Maass cusp forms on SL(n, ℤ)∖SL(n, ℝ)∕SO(n). As part of the proof, we establish a uniform upper-bound for spherical functions on semisimple Lie groups which is of independent interest. For each of the principal, symmetric square and exterior square L-functions, we deduce the level distribution with restricted support

Let C∕K be a smooth plane quartic over a discrete valuation field. We characterize the type of reduction (i.e., smooth plane quartic, hyperelliptic genus 3 curve or bad) over K in terms of the existence of a special plane quartic model and, over K¯, in terms of the valuations of certain algebraic invariants of C when the characteristic of the residue field is not 2,3,5 or 7. On the way, we gather several

We study certain mod p differential operators that act on automorphic forms over Shimura varieties of type A or C. We show that, over the ordinary locus, these operators agree with the mod p reduction of the p-adic theta operators previously studied by some of the authors. In the characteristic 0, p-adic case, there is an obstruction that makes it impossible to extend the theta operators to the whole

We prove that if X is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring QH(X) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring H∗(X) that multiplies with nonnegative structure constants. This implies that the (three point, genus zero) Gromov–Witten invariants of X are uniquely determined by Witten’s presentation of QH(X)

We prove a purely local form of a result of Saito and Yatagawa. They proved that the characteristic cycle of a constructible étale sheaf is determined by wild ramification of the sheaf along the boundary of a compactification. But they had to consider ramification at all the points of the compactification. We give a pointwise result, that is, we prove that the characteristic cycle of a constructible