Showing papers in "Algebra & Number Theory in 2014"

A generalized Bogomolov–Gieseker inequality for the three-dimensional projective space

Abstract: A generalized Bogomolov–Gieseker inequality for tilt-stable complexes on a smooth projective threefold was conjectured by Bayer, Toda, and the author. We show that such inequality holds true in general if it holds true when the polarization is sufficiently small. As an application, we prove it for the three-dimensional projective space.

Intermediate co-$t$-structures, two-term silting objects, $\tau$-tilting modules, and torsion classes

Abstract: If (A,B) and (A′,B′) are co-t-structures of a triangulated category, then (A′,B′) is called intermediate if A⊆A′⊆ΣA. Our main results show that intermediate co-t-structures are in bijection with two-term silting subcategories, and also with support τ-tilting subcategories under some assumptions. We also show that support τ-tilting subcategories are in bijection with certain finitely generated torsion classes. These results generalise work by Adachi, Iyama, and Reiten.

Monodromy and local-global compatibility for l = p

Abstract: We strengthen the compatibility between local and global Langlands correspondences for GLn when n is even and l=p. Let L be a CM field and Π a cuspidal automorphic representation of GLn(AL) which is conjugate self-dual and regular algebraic. In this case, there is an l-adic Galois representation associated to Π, which is known to be compatible with local Langlands in almost all cases when l=p by recent work of Barnet-Lamb, Gee, Geraghty and Taylor. The compatibility was proved only up to semisimplification unless Π has Shin-regular weight. We extend the compatibility to Frobenius semisimplification in all cases by identifying the monodromy operator on the global side. To achieve this, we derive a generalization of Mokrane’s weight spectral sequence for log crystalline cohomology.

New equidistribution estimates of Zhang type

Abstract: We prove distribution estimates for primes in arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese remainder theorem conditions, obtaining an exponent of distribution 1/2 + 7/300.

Yangians and quantizations of slices in the affine Grassmannian

Abstract: We study quantizations of transverse slices to Schubert varieties in the affine Grassmannian. The quantization is constructed using quantum groups called shifted Yangians — these are subalgebras of the Yangian we introduce which generalize the Brundan-Kleshchev shifted Yangian to arbitrary type. Building on ideas of Gerasimov-Kharchev-Lebedev-Oblezin, we prove that a quotient of the shifted Yangian quantizes a scheme supported on the transverse slices, and we formulate a conjectural description of the defining ideal of these slices which implies that the scheme is reduced. This conjecture also implies the conjectural quantization of the Zastava spaces for PGLn of Finkelberg-Rybnykov.

Explicit Gross–Zagier and Waldspurger formulae

Abstract: We give an explicit Gross–Zagier formula which relates the height of an explicitly constructed Heegner point to the derivative central value of a Rankin L-series. An explicit form of the Waldspurger formula is also given.

Relations between Dieudonné displays and crystalline Dieudonné theory

Abstract: We discuss the relation between crystalline Dieudonne theory and Dieudonne displays of p-divisible groups. The theory of Dieudonne displays is extended to the prime 2 without restriction, which implies that the classification of finite locally free group schemes by Breuil–Kisin modules holds for the prime 2 as well.

K3 surfaces and equations for Hilbert modular surfaces

Abstract: We outline a method to compute rational models for the Hilbert modular surfaces Y−(D), which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in ℚ(D), via moduli spaces of elliptic K3 surfaces with a Shioda–Inose structure. In particular, we compute equations for all thirty fundamental discriminants D with 1

Tropical independence I: Shapes of divisors and a proof of the Gieseker–Petri theorem

Abstract: We develop a framework to apply tropical and nonarchimedean analytic methods to multiplication maps for linear series on algebraic curves, studying degenerations of these multiplications maps when the special fiber is not of compact type. As an application, we give a new proof of the GiesekerPetri Theorem, including an explicit tropical criterion for a curve over a valued field to be Gieseker-Petri general.

On the supersingular locus of the GU(2,2) Shimura variety

Abstract: We describe the supersingular locus of a GU(2,2) Shimura variety at a prime inert in the corresponding quadratic imaginary field.

Decompositions of commutative monoid congruences and binomial ideals

Abstract: Primary decomposition of commutative monoid congruences is insensitive to certain features of primary decomposition in commutative rings. These features are captured by the more refined theory of mesoprimary decomposition of congruences, introduced here complete with witnesses and associated prime objects. The combinatorial theory of mesoprimary decomposition lifts to arbitrary binomial ideals in monoid algebras. The resulting binomial mesoprimary decomposition is a new type of intersection decomposition for binomial ideals that enjoys computational efficiency and independence from ground field hypotheses. Binomial primary decompositions are easily recovered from mesoprimary decomposition.

Parameterizing tropical curves, I: Curves of genus zero and one

Abstract: In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. Our methods focus on describing curves by parameterizations, not by their defining equations; we give parameterizations by rational functions in the genus-zero case and by nonarchimedean elliptic functions in the genus-one case. For genus-zero curves, those graphs which can be lifted can be characterized in a completely combinatorial manner. For genus-one curves, we show that certain conditions identified by Mikhalkin are sufficient and we also identify a new necessary condition.

Local cohomology with support in generic determinantal ideals

Abstract: For positive integers m n p, we compute the GLm GLn-equivariant description of the local cohomology modules of the polynomial ring SD Sym.C m C n / with support in the ideal of p p minors of the generic m n matrix. Our techniques allow us to explicitly compute all the modules Ext S .S=Ix; S/, for x a partition and Ix the ideal generated by the irreducible subrepresentation of S indexed by x. In particular we determine the regularity of the ideals Ix, and we deduce that the only ones admitting a linear free resolution are the powers of the ideal of maximal minors of the generic matrix, as well as the products between such powers and the maximal ideal of S.

On direct images of pluricanonical bundles

Abstract: We show that techniques inspired by Kollar and Viehweg’s study of weak positivity, combined with vanishing for log-canonical pairs, lead to new generation and vanishing results for direct images of pluricanonical bundles. We formulate the strongest such results as Fujita conjecture-type statements, which are then shown to govern a range of fundamental properties of direct images of pluricanonical and pluriadjoint line bundles, like effective vanishing theorems, weak positivity, or generic vanishing.

The radius of a subcategory of modules

Abstract: We introduce a new invariant for subcategories X of finitely generated modules over a local ring R which we call the radius of X. We show that if R is a complete intersection and X is resolving, then finiteness of the radius forces X to contain only maximal Cohen‐Macaulay modules. We also show that the category of maximal Cohen‐Macaulay modules has finite radius when R is a Cohen‐Macaulay complete local ring with perfect coefficient field. We link the radius to many well-studied notions such as the dimension of the stable category of maximal Cohen‐Macaulay modules, finite/countable Cohen‐Macaulay representation type and the uniform Auslander condition.

Multivariate Apéry numbers and supercongruences of rational functions

Abstract: One of the many remarkable properties of the Apery numbers A(n), introduced in Apery’s proof of the irrationality of ζ(3), is that they satisfy the two-term supercongruences A(pm) ≡ A(pr−1m) (mod p) for primes p > 5. Similar congruences are conjectured to hold for all Apery-like sequences. We provide a fresh perspective on the supercongruences satisfied by the Apery numbers by showing that they extend to all Taylor coefficients A(n1, n2, n3, n4) of the rational function 1 (1 − x1 − x2)(1 − x3 − x4) − x1x2x3x4 . The Apery numbers are the diagonal coefficients of this function, which is simpler than previously known rational functions with this property. Our main result offers analogous results for an infinite family of sequences, indexed by partitions λ, which also includes the Franel and Yang–Zudilin numbers as well as the Apery numbers corresponding to ζ(2). Using the example of the Almkvist–Zudilin numbers, we further indicate evidence of multivariate supercongruences for other Apery-like sequences.

On the Picard number of K3 surfaces over number fields

Abstract: We discuss some aspects of the behavior of specialization at a finite place of Neron–Severi groups of K3 surfaces over number fields. We give optimal lower bounds for the Picard number of such specializations, thus answering a question of Elsenhans and Jahnel. As a consequence of these results, we show that it is possible to compute explicitly the Picard number of any given K3 surface over a number field.

Tetrahedral elliptic curves and the local-global principle for isogenies

Abstract: We study the failure of a local-global principle for the existence of l-isogenies for elliptic curves over number fields K. Sutherland has shown that over ℚ there is just one failure, which occurs for l=7 and a unique j-invariant, and has given a classification of such failures when K does not contain the quadratic subfield of the l-th cyclotomic field. In this paper we provide a classification of failures for number fields which do contain this quadratic field, and we find a new “exceptional” source of such failures arising from the exceptional subgroups of PGL2(Fl). By constructing models of two modular curves, Xs(5) and XS4(13), we find two new families of elliptic curves for which the principle fails, and we show that, for quadratic fields, there can be no other exceptional failures.

Bounded gaps between primes with a given primitive root

Abstract: Fix an integer g≠−1 that is not a perfect square. In 1927, Artin conjectured that there are infinitely many primes for which g is a primitive root. Forty years later, Hooley showed that Artin’s conjecture follows from the generalized Riemann hypothesis (GRH). We inject Hooley’s analysis into the Maynard–Tao work on bounded gaps between primes. This leads to the following GRH-conditional result: Fix an integer m≥2. If q1

Compatibility between Satake and Bernstein isomorphisms in characteristic p

Abstract: We study the center of the pro-p Iwahori-Hecke ring H of a connected split p-adic reductive group G. For k an algebraically closed field with characteristic p, we prove that the center of the k-algebra H_k:= H\otimes_Z k contains an affine semigroup algebra which is naturally isomorphic to the Hecke algebra attached to any irreducible smooth k-representation of a given hyperspecial maximal compact subgroup of G. This isomorphism is obtained using the inverse Satake isomorphism constructed in arXiv:1207.5557. We apply this to classify the simple supersingular H_k-modules, study the supersingular block in the category of finite length H_k-modules, and relate the latter to supersingular representations of G.

Polynomial bounds for Arakelov invariants of Belyi curves

Abstract: We explicitly bound the Faltings height of a curve over Q polynomially in its Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings’ delta invariant and the self-intersection of the dualising sheaf. Our results allow us to explicitly bound these Arakelov invariants for modular curves, Hurwitz curves and Fermat curves in terms of their genus. Moreover, as an application, we show that the Couveignes‐Edixhoven‐Bruin algorithm to compute coefficients of modular forms for congruence subgroups of SL2.Z/ runs in polynomial time under the Riemann hypothesis for -functions of number fields. This was known before only for certain congruence subgroups. Finally, we use our results to prove a conjecture of Edixhoven, de Jong and Schepers on the Faltings height of a cover of P 1 with fixed branch locus.

Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras

Abstract: The question of whether or not a Hopf algebra $H$ is faithfully flat over a Hopf subalgebra $A$ has received positive answers in several particular cases: when $H$ (or more generally, just $A$) is commutative, or cocommutative, or pointed, or when $K$ contains the coradical of $H$. We prove the statement in the title, adding the class of cosemisimple Hopf algebras to those known to be faithfully flat over all Hopf subalgebras. We also show that the third term of the resulting "exact sequence" $A\to H\to C$ is always a cosemisimple coalgebra, and that the expectation $H\to A$ is positive when $H$ is a CQG algebra.

Sato-Tate distributions of twists of y^2=x^5-x and y^2=x^6+1

Abstract: We determine the limiting distribution of the normalized Euler factors of an abelian surface A defined over a number field k when A is isogenous to the square of an elliptic curve defined over k with complex multiplication. As an application, we prove the Sato-Tate Conjecture for Jacobians of Q-twists of the curves y^2=x^5-x and y^2=x^6+1, which give rise to 18 of the 34 possibilities for the Sato-Tate group of an abelian surface defined over Q. With twists of these two curves one encounters, in fact, all of the 18 possibilities for the Sato-Tate group of an abelian surface that is isogenous to the square of an elliptic curve with complex multiplication. Key to these results is the twisting Sato-Tate group of a curve, which we introduce in order to study the effect of twisting on the Sato-Tate group of its Jacobian.

McKay natural correspondences on characters

Abstract: Let G be a finite group, let p be an odd prime, and let P∈ Sylp(G). If NG(P)=PCG(P), then there is a canonical correspondence between the irreducible complex characters of G of degree not divisible by p belonging to the principal block of G and the linear characters of P. As a consequence, we give a characterization of finite groups that possess a self-normalizing Sylow p-subgroup or a p-decomposable Sylow normalizer.

The homotopy category of injectives

Abstract: Krause studied the homotopy category K(InjA) of complexes of injectives in a locally noetherian Grothendieck abelian category A. Because A is assumed locally noetherian, we know that arbitrary direct sums of injectives are injective, and hence, the category K(InjA) has coproducts. It turns out that K(InjA) is compactly generated, and Krause studies the relation between the compact objects in K(InjA), the derived category D(A), and the category Kac(InjA) of acyclic objects in K(InjA). We wish to understand what happens in the nonnoetherian case, and this paper begins the study. We prove that, for an arbitrary Grothendieck abelian category A, the category K(InjA) has coproducts and is μ-compactly generated for some sufficiently large μ. The existence of coproducts follows easily from a result of Krause: the point is that the natural inclusion of K(InjA) into K(A) has a left adjoint and the existence of coproducts is a formal corollary. But in order to prove anything about these coproducts, for example the μ-compact generation, we need to have a handle on this adjoint. Also interesting is the counterexample at the end of the article: we produce a locally noetherian Grothendieck abelian category in which products of acyclic complexes need not be acyclic. It follows that D(A) is not compactly generated. I believe this is the first known example of such a thing.

Lefschetz theorem for abelian fundamental group with modulus

Abstract: We prove a Lefschetz hypersurface theorem for abelian fundamental groups allowing wild ramification along some divisor. In fact, we show that isomorphism holds if the degree of the hypersurface is large relative to the ramification along the divisor.

The algebraic dynamics of generic endomorphisms of ℙn

Abstract: We investigate some general questions in algebraic dynamics in the case of generic endomorphisms of projective spaces over a field of characteristic zero. The main results that we prove are that a generic endomorphism has no nontrivial preperiodic subvarieties, any infinite set of preperiodic points is Zariski-dense and any infinite subset of a single orbit is also Zariski-dense, thereby verifying the dynamical “Manin–Mumford” conjecture of Zhang and the dynamical “Mordell–Lang” conjecture of Denis and Ghioca and Tucker in this case.

Quantum Matrices by Paths

Abstract: We study, from a combinatorial viewpoint, the quantized coordinate ring of mxn matrices over an infinite field K (also called quantum matrices) and its torus-invariant prime ideals. The first part of this paper shows that this algebra, traditionally defined by generators and relations, can be seen as subalgebra of a quantum torus by using paths in a certain directed graph. Roughly speaking, we view each generator of quantum matrices as a sum over paths in the graph, each path being assigned an element of the quantum torus. The quantum matrices relations then arise naturally by considering intersecting paths. This viewpoint is closely related to Cauchon's deleting-derivations algorithm. The second part of this paper is to apply the paths viewpoint to the theory of torus-invariant prime ideals of quantum matrices. We prove a conjecture of Goodearl and Lenagan that all such prime ideals, when the quantum parameter q is a non-root of unity, have generating sets consisting of quantum minors. Previously, this result was known to hold only for char(K)=0 and q transcendental over Q. Our strategy is to show that the quantum minors in a given torus-invariant ideal form a Grobner basis.

On the number of cubic orders of bounded discriminant having automorphism group $C_3$, and related problems

Abstract: For a binary quadratic form Q, we consider the action of SOQ on a 2-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and of a coregular space whose underlying group is not semisimple. We show that the nondegenerate integer orbits of this representation are in natural bijection with orders in cubic fields having a fixed “lattice shape”. Moreover, this correspondence is discriminant-preserving: the value of the invariant polynomial of an element in this representation agrees with the discriminant of the corresponding cubic order. We use this interpretation of the integral orbits to solve three classical-style counting problems related to cubic orders and fields. First, we give an asymptotic formula for the number of cubic orders having bounded discriminant and nontrivial automorphism group. More generally, we give an asymptotic formula for the number of cubic orders that have bounded discriminant and any given lattice shape (i.e., reduced trace form, up to scaling). Via a sieve, we also count cubic fields of bounded discriminant whose rings of integers have a given lattice shape. We find, in particular, that among cubic orders (resp. fields) having lattice shape of given discriminant D, the shape is equidistributed in the class group ClD of binary quadratic forms of discriminant D. As a by-product, we also obtain an asymptotic formula for the number of cubic fields of bounded discriminant having any given quadratic resolvent field.