Reine Angew Math Research Articles

Lattices in the cohomology of Shimura curves

We prove the main conjectures of Breuil (J Reine Angew Math, 2012) (including a generalisation from the principal series to the cuspidal case) and Dembele (J Reine Angew Math, 2012), subject to a mild global hypothesis that we make in order to apply certain $$R=\mathbb {T}$$ theorems. More precisely, we prove a multiplicity one result for the mod $$p$$ cohomology of a Shimura curve at Iwahori level, and we show that certain apparently globally defined lattices in the cohomology of Shimura curves are determined by the corresponding local $$p$$ -adic Galois representations. We also indicate a new proof of the Buzzard–Diamond–Jarvis conjecture in generic cases. Our main tools are the geometric Breuil–Mezard philosophy developed in Emerton and Gee (J Inst Math Jussieu, 2012), and a new and more functorial perspective on the Taylor–Wiles–Kisin patching method. Along the way, we determine the tamely potentially Barsotti–Tate deformation rings of generic two-dimensional mod $$p$$ representations, generalising a result of Breuil and Mezard (Bull Soc Math de France, 2012) in the principal series case.

Around the Van Daele–Schmüdgen Theorem

For a bounded non-negative self-adjoint operator acting in a complex, infinite-dimensional, separable Hilbert space \({\mathcal{H}}\) and possessing a dense range \({\mathcal{R}}\) we propose a new approach to characterisation of phenomenon concerning the existence of subspaces \({\mathfrak{M}\subset \mathcal{H}}\) such that \({\mathfrak{M}\cap\mathcal{R}=\mathfrak{M}^\perp\cap\mathcal{R}=\{0\}}\). We show how the existence of such subspaces leads to various pathological properties of unbounded self-adjoint operators related to von Neumann theorems (J Reine Angew Math 161:208–236, 1929; Math Ann 102:49–131, 1929; Math Ann 102:370–427, 1929). We revise the von Neumann–Van Daele-Schmüdgen assertions (J Reine Angew Math 161:208–236, 1929; J Oper Theory 11:379–393, 1984; Can J Math 36:1245–1250, 1982) to refine them. We also develop a new systematic approach, which allows to construct for any unbounded densely defined symmetric/self-adjoint operator T infinitely many pairs \({\langle T_1 , T_2 \rangle}\) of its closed densely defined restrictions \({T_k\subset T}\) such that \({{\rm dom}(T^* T_{k})=\{0\} (\Rightarrow{\rm dom} T_{k}^2=\{0\})\, k=1,2}\) and \({{\rm dom} T_1\cap{\rm dom} T_2=\{0\}, {\rm dom} T_1\dot+{\rm dom} T_2={\rm dom}T}\).

A Few Endpoint Geodesic Restriction Estimates for Eigenfunctions

We prove a couple of new endpoint geodesic restriction estimates for eigenfunctions. In the case of general 3-dimensional compact manifolds, after a TT* argument, simply by using the L 2-boundedness of the Hilbert transform on \({\mathbb{R}}\) , we are able to improve the corresponding L 2-restriction bounds of Burq, Gérard and Tzvetkov (Duke Math J 138:445–486, 2007) and Hu (Forum Math 6:1021–1052, 2009). Also, in the case of 2-dimensional compact manifolds with nonpositive curvature, we obtain improved L 4-estimates for restrictions to geodesics, which, by Hölder’s inequality and interpolation, implies improved L p-bounds for all exponents p ≥ 2. We do this by using oscillatory integral theorems of Hörmander (Ark Mat 11:1–11, 1973), Greenleaf and Seeger (J Reine Angew Math 455:35–56, 1994) and Phong and Stein (Int Math Res Notices 4:49–60, 1991), along with a simple geometric lemma (Lemma 3.2) about properties of the mixed-Hessian of the Riemannian distance function restricted to pairs of geodesics in Riemannian surfaces. We are also able to get further improvements beyond our new results in three dimensions under the assumption of constant nonpositive curvature by exploiting the fact that, in this case, there are many totally geodesic submanifolds.

DG-resolutions of NC-smooth thickenings and NC-Fourier–Mukai transforms

We give a construction of NC-smooth thickenings [a notion defined by Kapranov (J Reine Angew Math 505:73–118, 1998)] of a smooth variety equipped with a torsion free connection. We show that a twisted version of this construction realizes all NC-smooth thickenings as \(0\)th cohomology of a differential graded sheaf of algebras, similarly to Fedosov’s construction in (J Differ Geom 40:213–238, 1994). We use this dg resolution to construct and study sheaves on NC-smooth thickenings. In particular, we construct an NC version of the Fourier–Mukai transform from coherent sheaves on a (commutative) curve to perfect complexes on the canonical NC-smooth thickening of its Jacobian. We also define and study analytic NC-manifolds. We prove NC-versions of some of GAGA theorems, and give a \(C^\infty \)-construction of analytic NC-thickenings that can be used in particular for Kähler manifolds with constant holomorphic sectional curvature. Finally, we describe an analytic NC-thickening of the Poincaré line bundle for the Jacobian of a curve, and the corresponding Fourier–Mukai functor, in terms of \(A_\infty \)-structures.

A tableau formula for eta polynomials

We use the Pieri and Giambelli formulas of Buch et al. (Invent Math 178:345–405, 2009; J Reine Angew, 2013) and the calculus of raising operators developed in Buch et al. (A Giambelli formula for isotropic Grassmannians, arXiv:0811.2781, 2008) and Tamvakis (J Reine Angew Math 652, 207–244, 2011) to prove a tableau formula for the eta polynomials of Buch et al. (J Reine Angew, 2013) and the Stanley symmetric functions which correspond to Grassmannian elements of the Weyl group $$\widetilde{W}_n$$ of type $$\text {D}_n$$ . We define the skew elements of $$\widetilde{W}_n$$ and exhibit a bijection between the set of reduced words for any skew $$w\in \widetilde{W}_n$$ and a set of certain standard typed tableaux on a skew shape $$\lambda /\mu $$ associated to $$w$$ .

Loewner Theory in annulus II: Loewner chains

Loewner Theory, based on dynamical viewpoint, proved itself to be a powerful tool in Complex Analysis and its applications. Recently Bracci et al. (J Reine Angew Math to appear, arXiv:0807.1594; Math Ann 344:947–962, 2009) and Contreras et al. (Revista Matematica Iberoamericana 26:975–1012, 2010) have proposed a new approach bringing together all the variants of the (deterministic) Loewner Evolution in a simply connected reference domain. This paper is devoted to the construction of a general version of Loewner Theory for the annulus launched in Contreras et al. (Trans Amer Math Soc to appear, arXiv:1011.4253). We introduce the general notion of a Loewner chain over a system of annuli and obtain a 1-to-1 correspondence between Loewner chains and evolution families in the doubly connected setting similar to that in the Loewner Theory for the unit disk. Furthermore, we establish a conformal classification of Loewner chains via the corresponding evolution families and via semicomplete weak holomorphic vector fields. Finally, we extend the explicit characterization of the semicomplete weak holomorphic vector fields obtained in Contreras et al. (Trans Amer Math Soc to appear, arXiv:1011.4253) to the general case.

Strongly Semihereditary Rings and Rings with Dimension

The existence of a well-behaved dimension of a finite von Neumann algebra (see Luck, J Reine Angew Math 495:135–162, 1998) has lead to the study of such a dimension of finite Baer *-rings (see Vas, J Algebra 289(2):614–639, 2005) that satisfy certain *-ring axioms (used in Berberian, 1972). This dimension is closely related to the equivalence relation $ {\sim^{\raisebox{-.1ex}[0pc][0pc]{\scriptsize{*}}}}$ on projections defined by $p{\sim^{\raisebox{-.1ex}[0pc][0pc]{\scriptsize{*}}}} q$ iff p = xx * and q = x * x for some x. However, the equivalence ${\sim^{{\raisebox{.3ex}[0pc][0pc]{\scriptsize\em{a}}}}}$ on projections (or, in general, idempotents) defined by $p{\sim^{{\raisebox{.3ex}[0pc][0pc]{\scriptsize\em{a}}}}} q$ iff p = xy and q = yx for some x and y, can also be relevant. There were attempts to unify the two approaches (see Berberian, preprint, 1988)). In this work, our agenda is three-fold: (1) We study assumptions on a ring with involution that guarantee the existence of a well-behaved dimension defined for any general equivalence relation on projections ~. (2) By interpreting ~ as ${\sim^{{\raisebox{.3ex}[0pc][0pc]{\scriptsize\em{a}}}}},$ we prove the existence of a well-behaved dimension of strongly semihereditary *-rings with positive definite involution. This class is wider than the class of finite Baer *-rings with dimension considered in the past: it includes some non Rickart *-rings. Moreover, none of the *-ring axioms from Berberian (1972) and Vas (J Algebra 289(2):614–639, 2005) are assumed. (3) As the first corollary of (2), we obtain dimension of noetherian Leavitt path algebras over positive definite fields. Secondly, we obtain dimension of a Baer *-ring R satisfying the first seven axioms from Vas (J Algebra 289(2):614–639, 2005) (in particular, dimension of finite AW *-algebras). Assuming the eight axiom as well, R has dimension for ${\sim^{\raisebox{-.1ex}[0pc][0pc]{\scriptsize{*}}}}$ also and the two dimensions coincide. While establishing (2), we obtain some additional results for a right strongly semihereditary ring R: we prove that every finitely generated R-module M splits as a direct sum of a finitely generated projective module and a singular module; we describe right strongly semihereditary rings in terms of relations between their maximal and total rings of quotients; and we characterize extending Leavitt path algebras over finite graphs.

The canonical shrinking soliton associated to a Ricci flow

To every Ricci flow on a manifold \({\mathcal{M}}\) over a time interval \({I\subset\mathbb{R}_-}\), we associate a shrinking Ricci soliton on the space–time \({\mathcal{M}\times I}\). We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric construction was discovered by consideration of the theory of optimal transportation, and in particular the results of the second author Topping (J Reine Angew Math 636:93–122, 2009), and McCann and the second author (Am J Math 132:711–730, 2010); we briefly survey the link between these subjects.

Deconstruction of a Dirichlet–Nazimow formula

Dirichlet (J Reine Angew Math 24, 1842) announced a generalization of his class number formula for real quadratic fields to biquadratic fields containing \({\mathbb{Q}(i)}\), by replacing the circular trigonometric functions with certain elliptic trigonometric functions. Subsequently, Nazimow (Ann Sci Ecole Norm Sup (3) 5:147–176, 1888) published the corresponding formula. Here we analyze the Dirichlet–Nazimow formula and update its proof in the context of Class Field Theory.

The Representations of Cyclotomic BMW Algebras, II

In this paper, we give a recursive formula to compute the Gram determinant associated to each cell module of the cyclotomic BMW algebras over an integral domain. As a by-product, we determine explicitly when is semisimple over a field. This generalizes our previous result on Birman-Murakami-Wenzl algebras in Rui and Si (J Reine Angew Math 631:153–180, 2009).

Stickelberger ideals and Fitting ideals of class groups for abelian number fields

In this paper, we determine completely the initial Fitting ideal of the minus part of the ideal class group of an abelian number field over Q up to the 2-component. This answers an open question of Mazur and Wiles (Invent Math 76:179–330, 1984) up to the 2-component, and proves Conjecture 0.1 in Kurihara (J Reine Angew Math 561:39–86, 2003). We also study Brumer’s conjecture and prove a stronger version for a CM-field, assuming certain conditions, in particular on the Galois group.

An isoperimetric inequality related to a Bernoulli problem

Given a bounded domain Ω we look at the minimal parameter Λ(Ω) for which a Bernoulli free boundary value problem for the p-Laplacian has a solution minimising an energy functional. We show that amongst all domains of equal volume Λ(Ω) is minimal for the ball. Moreover, we show that the inequality is sharp with essentially only the ball minimising Λ(Ω). This resolves a problem related to a question asked in Flucher et al. (Reine Angew Math 486:165–204, 1997).

Metrics with conical singularities on the sphere and sharp extensions of the theorems of Landau and Schottky

An explicit formula for the generalized hyperbolic metric on the thrice-punctured sphere $${\mathbb {P} \backslash \{z_1, z_2, z_3\}}$$ with singularities of order α j ≤ 1 at z j is obtained in all possible cases α 1 + α 2 + α 3 > 2. The existence and uniqueness of such a metric was proved long time ago by Picard (J Reine Angew Math 130:243–258, 1905) and Heins (Nagoya Math J 21:1–60, 1962), while explicit formulas for the cases α 1 = α 2 = 1 were given earlier by Agard (Ann Acad Sci Fenn Ser A I 413, 1968) and recently by Anderson et al. (Math Z, to appear, doi: 10.1007/s00209-009-0560-5 ). We also establish precise and explicit lower bounds for the generalized hyperbolic metric. This extends work of Hempel (J Lond Math Soc II Ser 20:435–445, 1979) and Minda (Complex Var 8:129–144, 1987). As applications, sharp versions of Landau- and Schottky-type theorems for meromorphic functions are obtained.

Isolated Singularities of the 1D Complex Viscous Burgers Equation

The Cauchy problem for the 1D real-valued viscous Burgers equation ut+uux = uxx is globally well posed (Hopf in Commun Pure Appl Math 3:201–230, 1950). For complex-valued solutions finite time blow-up is possible from smooth compactly supported initial data, see Polacik and Sverak (J Reine Angew Math 616:205–217, 2008). It is also proved in Polacik and Sverak (J Reine Angew Math 616:205–217, 2008) that the singularities for the complex-valued solutions are isolated if they are not present in the initial data. In this paper we study the singularities in more detail. In particular, we classify the possible blow-up rates and blow-up profiles. It turns out that all singularities are of type II and that the blow-up profiles are regular steady state solutions of the equation.

Bounds on smooth matrix coefficients on L 2-spaces

Let G be a semisimple Lie group with a finite number of connected components and a finite center. Let K be a maximal compact subgroup. Let X be a smooth G-space equipped with a G-invariant measure. In this paper, we give upper bounds for K-finite and $${\mathfrak k}$$ -smooth matrix coefficients of the regular representation L 2(X) under an assumption about $${{\rm supp}(L^2(X)) \cap \hat G_K}$$ . Furthermore, we show that this bound holds for unitary representations that are weakly contained in L 2(X). Our result generalizes a result of Cowling–Haagerup–Howe (J Reine Angew Math 387:97–110, 1988). As an example, we discuss the matrix coefficients of the O(p, q) representation $${L^2(\mathbb{R}^{p+q})}$$ .

Symmetry and monotonicity of least energy solutions

We give a simple proof of the fact that for a large class of quasilinear elliptic equations and systems the solutions that minimize the corresponding energy in the set of all solutions are radially symmetric. We require just continuous nonlinearities and no cooperative conditions for systems. Thus, in particular, our results cannot be obtained by using the moving planes method. In the case of scalar equations, we also prove that any least energy solution has a constant sign and is monotone with respect to the radial variable. Our proofs rely on results in Brothers and Ziemer (J Reine Angew Math 384:153–179, 1988) and Maris (Arch Ration Mech Anal, 192:311–330, 2009) and answer questions from Brezis and Lieb (Comm Math Phys 96:97–113, 1984) and Lions (Ann Inst H Poincare Anal Non Lineaire 1:223–283, 1984).

Stability of quadratic modules

A finitely generated quadratic module or preordering in the real polynomial ring is called stable, if it admits a certain degree bound on the sums of squares in the representation of polynomials. Stability, first defined explicitly in Powers and Scheiderer (Adv Geom 1, 71–88, 2001), is a very useful property. It often implies that the quadratic module is closed; furthermore, it helps settling the Moment Problem, solves the Membership Problem for quadratic modules and allows applications of methods from optimization to represent nonnegative polynomials. We provide sufficient conditions for finitely generated quadratic modules in real polynomial rings of several variables to be stable. These conditions can be checked easily. For a certain class of semi-algebraic sets, we obtain that the nonexistence of bounded polynomials implies stability of every corresponding quadratic module. As stability often implies the non-solvability of the Moment Problem, this complements the result from Schmüdgen (J Reine Angew Math 558, 225–234, 2003), which uses bounded polynomials to check the solvability of the Moment Problem by dimensional induction. We also use stability to generalize a result on the Invariant Moment Problem from Cimpric et al. (Trans Am Math Soc, to appear).

Existence of traveling waves for integral recursions with nonmonotone growth functions

A class of integral recursion models for the growth and spread of a synchronized single-species population is studied. It is well known that if there is no overcompensation in the fecundity function, the recursion has an asymptotic spreading speed c*, and that this speed can be characterized as the speed of the slowest non-constant traveling wave solution. A class of integral recursions with overcompensation which still have asymptotic spreading speeds can be found by using the ideas introduced by Thieme (J Reine Angew Math 306:94-121, 1979) for the study of space-time integral equation models for epidemics. The present work gives a large subclass of these models with overcompensation for which the spreading speed can still be characterized as the slowest speed of a non-constant traveling wave. To illustrate our results, we numerically simulate a series of traveling waves. The simulations indicate that, depending on the properties of the fecundity function, the tails of the waves may approach the carrying capacity monotonically, may approach the carrying capacity in an oscillatory manner, or may oscillate continually about the carrying capacity, with its values bounded above and below by computable positive numbers.

Determination of simple CM abelian surfaces defined over $${\mathbb{Q}}$$

It is known that in the moduli space \({\mathcal{A}_1}\) of elliptic curves, there exist precisely nine \({\mathbb{Q}}\)-rational points represented by an elliptic curve with complex multiplication by the maximal order of an imaginary quadratic field. In Murabayashi and Umegaki (J Algebra 235:267–274, 2001) and Umegaki [Determination of all \({\mathbb{Q}}\)-rational CM-points in the moduli spaces of polarized abelian surfaces, Analytic number theory (Beijng/Kyoto, 1999). Dev. Math., vol 6. Kluwer, Dordrecht, pp 349–357, 2002] we determined all \({\mathbb{Q}}\)-rational points in \({\mathcal{A}_2(d)}\) (the moduli space of d-polarized abelian surfaces) represented by a d-polarized abelian surface whose endomorphism ring is isomorphic to the maximal order of a quartic CM-field by using the result in Murabayashi (J Reine Angew Math 470:1–26, 1996). In this paper, we prove that polarized abelian surfaces corresponding to these \({\mathbb{Q}}\)-rational CM points have a \({\mathbb{Q}}\)-rational model by constructing certain Hecke characters.

On syzygies of non-complete embedding of projective varieties

Let X be a non-degenerate, not necessarily linearly normal projective variety in \(\mathbb{P}^r\). Recently the generalization of property Np to non-linearly normal projective varieties have been considered and its algebraic and geometric properties are studied extensively. One of the generalizations is the property Nd,p for the saturated ideal IX (Eisenbud et al. in Compos Math 141: 1460–1478, 2005) and the other is the property \(N_p^S\) for the graded module of the twisted global sections of \(\mathcal{O}_X(1)\) (Kwak and Park in J Reine Angew Math 582: 87–105, 2005). In this paper, we are interested in the algebraic and geometric meaning of properties \(N_p^S\) for every p ≥ 0 and the syzygetic behaviors of isomorphic projections and hyperplane sections of a given variety with property \(N_p^S\).