Reine Angew Math Research Articles

The Schur–Potapov Algorithm in the General Matrix Case and Its Application to the Matricial Schur Problem

This paper is a generalization of the topic handled in Bogner et al. (Oper Theory 1(1):55–95, 2007a, Oper Theory 1(2):235–278, 2007b) where the Schur–Potapov algorithm (SP-algorithm) was handled in the context of non-degenerate p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur sequences and non-degenerate p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur functions. In particular, the interplay between both types of algorithms was intensively studied there. This was itself a generalization of the classical Schur algorithm (Schur in J Reine Angew Math 148:122–145, 1918) to the non-degenerate matrix case. In treating the matrix case a result due to Potapov (Potapov in Trudy Moskov Mat Obšč 4:125–236, 1955) concerning particular linear fractional transformations of contractive p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} matrices was used. For this reason, the notation SP-algorithm was already chosen in Dubovoj et al. (Matricial version of the classical Schur problem, volume 129 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1992). We are going to introduce both types of SP-algorithms as well for arbitrary p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur sequences as for arbitrary p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur functions. Again we will intensively discuss the interplay between both types of algorithms. Applying the SP-algorithm, a complete treatment of the matricial Schur problem in the most general case is established. A one-step extension problem for finite p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur sequences is considered. Central p×q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${p\ imes q}$$\\end{document} Schur sequences are studied under the view of SP-parameters.

The Super Frobenius–Schur Indicator and Finite Group Gauge Theories on Pin$$^-$$ Surfaces

It is well-known that the value of the Frobenius–Schur indicator $$|G|^{-1} \sum _{g\in G} \chi (g^2)=\pm 1$$ of a real irreducible representation of a finite group G determines which of the two types of real representations it belongs to, i.e. whether it is strictly real or quaternionic. We study the extension to the case when a homomorphism $$\varphi :G\rightarrow \mathbb {Z}/2\mathbb {Z}$$ gives the group algebra $$\mathbb {C}[G]$$ the structure of a superalgebra. Namely, we construct of a super version of the Frobenius–Schur indicator whose value for a real irreducible super representation is an eighth root of unity, distinguishing which of the eight types of irreducible real super representations described in Wall (in J Reine Angew Math 213:187–199, 1963/64. https://doi.org/10.1515/crll.1964.213.187 ) it belongs to. We also discuss its significance in the context of two-dimensional finite-group gauge theories on pin $$^-$$ surfaces.

Intrinsic Flat Convergence of Points and Applications to Stability of the Positive Mass Theorem

We prove results on intrinsic flat convergence of points—a concept first explored by Sormani (Commun Anal Geom 26(6):1317–1373, 2018). In particular, we discuss compatibility with Gromov–Hausdorff convergence of points—a concept first described by Gromov (Inst Hautes Études Sci Publ Math 53:53–73, 1981). We apply these results to the problem of stability of the positive mass theorem in mathematical relativity. Specifically, we revisit the article (Huang et al. in J Reine Angew Math 727:269–299, 2017) on intrinsic flat stability for the case of graphical hypersurfaces of Euclidean space: We are able to fill in some details in the proofs of Theorems 1.4 and Lemma 5.1 of Huang et al. (2017) and strengthen some statements. Moreover, in light of an acknowledged error in the proof of Theorem 1.3 of Huang et al. (2017), we provide an alternative proof that extends recent work of Allen and Perales (Intrinsic flat stability of manifolds with boundary where volume converges and distance is bounded below, 2020. arXiv:2006.13030 ).

Dominant and global dimension of blocks of quantised Schur algebras

Group algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and S_q(n,r) with n geqslant r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59–85, 2021).

Harmonic mappings between singular metric spaces

In this paper, we survey the existence, uniqueness and interior regularity of solutions to the Dirichlet problem associated with various energy functionals in the setting of mappings between singular metric spaces. Based on known ideas and techniques, we separate the necessary analytical assumptions to axiomatizing the theory in the singular setting. More precisely, (1) we extend the existence result of Guo and Wenger (Comm Anal Geom 28(1):89–112, 2020) for solutions to the Dirichlet problem of Korevaar–Schoen energy functional to more general energy functionals in purely singular setting. (2) When Y has non-positive curvature in the sense of Alexandrov (NPC), we show that the ideas of Jost (Calc Var Partial Differ Equ 5(1):1–19, 1997) and Lin (Analysis on singular spaces, collection of papers on geometry, analysis and mathematical physics, World Science Publishers, River Edge, pp 114–126, 1997) can be adapted to the purely singular setting to yield local Hölder continuity of solutions of the Dirichlet problem of Korevaar–Schoen and Kuwae–Shioya. (3) We extend the Liouville theorem of Sturm (J Reine Angew Math 456:173–196, 1994) for harmonic functions to harmonic mappings between singular metric spaces. (4) We extend the theorem of Mayer (Comm Anal Geom 6:199–253, 1998) on the existence of the harmonic mapping flow and solve the corresponding initial boundary value problem. Combing these known ideas, with the more or less standard techniques from analysis on metric spaces based on upper gradients, leads to new results when we consider harmonic mappings from \({{\,\mathrm{RCD}\,}}(K,N)\) spaces into NPC spaces. Similar results for the Dirichlet problem associated with the Kuwae–Shioya energy functional and the upper gradient functional are also derived.

Regularity for almost-minimizers of variable coefficient Bernoulli-type functionals

In David et al. (Adv Math 350:1109–1192, 2019) and David and Toro (Regularity of almost minimizers with free boundary. Calculus of variations and PDEs, 2020), the authors studied almost minimizers for functionals of the type first studied by Alt and Caffarelli (J Reine Angew Math 325:105–144, 1981) and Alt et al. (Trans Am Math Soc 282:431–461, 1984). In this paper we study the regularity of almost minimizers to energy functionals with variable coefficients (as opposed to Alt and Caffarelli, J Reine Angew Math 325:105–144, 1981; Alt et al., Trans Am Math Soc 282:431–461, 1984; David et al., Adv Math 350:1109–1192, 2019; David and Toro, Regularity of almost minimizers with free boundary. Calculus of variations and PDEs, 2020) which deal only with the “Laplacian” setting). We prove Lipschitz regularity up to, and across, the free boundary, fully generalizing the results of David and Toro (Regularity of almost minimizers with free boundary. Calculus of variations and PDEs, 2020) to the variable coefficient setting.

Vector bundles on rational homogeneous spaces.

We consider a uniform r-bundle E on a complex rational homogeneous space X and show that if E is poly-uniform with respect to all the special families of lines and the rank r is less than or equal to some number that depends only on X, then E is either a direct sum of line bundles or unstable with respect to some numerical class of a line. So we partially answer a problem posted by Muñoz et al. (Eur J Math 6:430–452, 2020). In particular, if X is a generalized Grassmannian {mathcal {G}} and the rank r is less than or equal to some number that depends only on X, then E splits as a direct sum of line bundles. So we improve the main theorem of Muñoz et al. (J Reine Angew Math (Crelles J) 664:141–162, 2012, Theorem 3.1) when X is a generalized Grassmannian. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert–Mülich–Barth theorem on rational homogeneous spaces.

Bergman–Weil expansion for holomorphic functions

Using a modified Cauchy–Weil representation formula in a Weil polyhedron $${\varvec{D}}_f\subset U\subset \mathbb {C}^n$$ , we prove a generalized version of Lagrange interpolation formula (at any order) with respect to a discrete set defined by $$V_{{\varvec{D}}_f}(f):=\{f_1=\cdots = f_m =0\}\, \cap {\varvec{D}}_f$$ , when $$m>n$$ and $$\{f_1,\ldots ,f_m\}$$ is minimal as a defining system. Thus the set $$V_{{\varvec{D}}_f}(f)$$ fails to be a complete intersection. We present our result as an averaged version of the classic Lagrange interpolation formula in the case $$m=n$$ . We invoke to that purpose Crofton’s formula, which plays a key role in the construction of Vogel generalized cycles as proposed in Andersson et al. (J Reine Angew Math 728: 105–136, 2017; Math Ann, 2020. https://doi.org/10.1007/s00208-020-01973-y). This leads us naturally to the construction of Bochner–Martinelli kernels. We also introduce $$f^{-1}(\{0\})$$ -Lagrange interpolators (at any order) subordinate to the choice of a smooth hermitian metric on the trivial m-bundle $$\mathbb {C}_U^m=U\times \mathbb {C}^m$$ , while the mapping $$f = (f_1,\ldots ,f_m)$$ is considered as its section.

Diagonal restrictions of p-adic Eisenstein families

We compute the diagonal restriction of the first derivative with respect to the weight of a p-adic family of Hilbert modular Eisenstein series attached to a general (odd) character of the narrow class group of a real quadratic field, and express the Fourier coefficients of its ordinary projection in terms of the values of a distinguished rigid analytic cocycle in the sense of Darmon and Vonk (Duke Math J, to appear, 2020) at appropriate real quadratic points of Drinfeld’s p-adic upper half-plane. This can be viewed as the p-adic counterpart of a seminal calculation of Gross and Zagier (J Reine Angew Math 355:191–220, 1985, §7) which arose in their “analytic proof” of the factorisation of differences of singular moduli, and whose inspiration can be traced to Siegel’s proof of the rationality of the values at negative integers of the Dedekind zeta function of a totally real field. Our main identity enriches the dictionary between the classical theory of complex multiplication and its extension to real quadratic fields based on RM values of rigid meromorphic cocycles, and leads to an expression for the p-adic logarithms of Gross–Stark units and Stark–Heegner points in terms of the first derivatives of certain twisted Rankin triple product p-adic L-functions.

A note on presentations of supersingular representations of $${\text {GL}}_2(F)$$

We prove that any smooth irreducible supersingular representation with central character of $${\text {GL}}_2(F)$$ is never of finite presentation when F is a finite field extension of $$\mathbb {Q}_p$$ such that $$F\ne \mathbb {Q}_p$$ , extending a result of Schraen in (J Reine Angew Math (Crelle’s J) 2015(704):187–208, 2015) for quadratic extensions.

Commutative Algebras Generated by Toeplitz Operators on the Unit Sphere

The classical result by Brown and Halmos (J Reine Angew Math 213:8–102, 1964) implies that there is no nontrivial commutative $$C^*$$ -algebra generated by Toeplitz operators acting on the Hardy space $$H^2(S^1)$$ , while there are only two commutative Banach algebras. One of them is generated by Toeplitz operators with analytic symbols, and the other one is generated by Toeplitz operators with conjugate analytic symbols. At the same time there are many nontrivial commutative $$C^*$$ and Banach algebras generated by Toeplitz operators acting on the Bergman spaces. In the paper we show that the situation on the multidimensional Hardy space $$H^2(S^{2n-1})$$ is drastically different from the one on $$H^2(S^1)$$ . We represent the Hardy space $$H^2(S^{2n-1})$$ as a direct sum of weighted Bergman spaces over $$\mathbb {B}^{n-1}$$ , and use the already known results for the Bergman space operators to describe a variety of nontrivial commutative $$C^*$$ and Banach algebras generated by Toeplitz operators acting on the multidimensional Hardy space $$H^2(S^{2n-1})$$ .

Harmonic conjugates on Bergman spaces induced by doubling weights

A radial weight $$\omega $$ belongs to the class $$\widehat{\mathcal {D}}$$ if there exists $$C=C(\omega )\ge 1$$ such that $$\int _r^1 \omega (s)\,ds\le C\int _{\frac{1+r}{2}}^1\omega (s)\,ds$$ for all $$0\le r<1$$ . Write $$\omega \in \check{\mathcal {D}}$$ if there exist constants $$K=K(\omega )>1$$ and $$C=C(\omega )>1$$ such that $${\widehat{\omega }}(r)\ge C{\widehat{\omega }}\left( 1-\frac{1-r}{K}\right) $$ for all $$0\le r<1$$ . These classes of radial weights arise naturally in the operator theory of Bergman spaces induced by radial weights (Pelaez and Rattya in Bergman projection induced by radial weight, 2019. arXiv:1902.09837 ). Classical results by Hardy and Littlewood (J Reine Angew Math 167:405–423, 1932) and Shields and Williams (Mich Math J 29(1):3–25, 1982) show that the weighted Bergman space of harmonic functions is not closed by harmonic conjugation if $${\omega \in \widehat{\mathcal {D}}\setminus \check{\mathcal {D}}}$$ and $$0<p\le 1$$ . In this paper we establish sharp estimates for the norm of the analytic Bergman space $$A^p_\omega $$ , with $${\omega \in \widehat{\mathcal {D}}\setminus \check{\mathcal {D}}}$$ and $$0<p<\infty $$ , in terms of quantities depending on the real part of the function. It is also shown that these quantities result equivalent norms for certain classes of radial weights.

Generalized Heegner cycles on Mumford curves

We study generalised Heegner cycles, originally introduced by Bertolini–Darmon–Prasanna for modular curves in Bertolini et al. (Duke Math J 162(6):1033–1148, 2013), in the context of Mumford curves. The main result of this paper relates generalized Heegner cycles with the two variable anticyclotomic p-adic L-function attached to a Coleman family $$f_\infty $$ and an imaginary quadratic field K, constructed in Bertolini and Darmon (Invent Math 168(2):371–431, 2007) and Seveso (J Reine Angew Math 686:111–148, 2014). While in Bertolini and Darmon (Invent Math 168(2):371–431, 2007) and Seveso (J Reine Angew Math 686:111–148, 2014) only the restriction to the central critical line of this 2 variable p-adic L-function is considered, our generalised Heegner cycles allow us to study the restriction of this function to non-central critical lines. The main result expresses the derivative along the weight variable of this anticyclotomic p-adic L-function restricted to non necessarily central critical lines as a combination of the image of generalized Heegner cycles under a p-adic Abel–Jacobi map. In studying generalised Heegner cycles in the context of Mumford curves, we also obtain an extension of a result of Masdeu (Compos Math 148(4):1003–1032, 2012) for the (one variable) anticyclotomic p-adic L-function of a modular form f and K at non-central critical integers.

Modularity of Galois traces of ray class invariants

After Zagier’s significant work (in: Bogomolov and Katzarkov (eds) Motives, polylogarithms and hodge theory, part I, International Press, Somerville, 2002) on traces of singular moduli, Bruinier and Funke (J Reine Angew Math 594:1–33, 2006) generalized his result to the traces of singular values of modular functions on modular curves of arbitrary genus. In class field theory, the extended ring class field is a generalization of the ray class field over an imaginary quadratic field. By using Shimura’s reciprocity law, we construct primitive generators of the extended ring class fields by using Siegel functions of arbitrary level $$N\ge 2$$ and identify their Galois traces with Fourier coefficients of weight 3/2 harmonic weak Maass forms. This would extend the results of Jeon et al. (Math Ann 353:37–63, 2012) and Jung et al. (Modularity of Galois traces of Weber’s resolvents, under revision).

On the Minimality of Keplerian Arcs with Fixed Negative Energy

We revisit a classical result by Jacobi (J Reine Angew Math 17:68–82, 1837) on the local minimality, as critical points of the corresponding energy functional, of fixed-energy solutions of the Kepler equation joining two distinct points with the same distance from the origin. Our proof relies on the Morse index theorem, together with a characterization of the conjugate points as points of geodesic bifurcation.

Galois descent for higher Brauer groups

For X a smooth projective variety over a field k, we consider the problem of Galois descent for higher Brauer groups. More precisely, we extend a finiteness result of Colliot-Thelene and Skorobogatov (J Reine Angew Math 682:141–165, 2013) to higher Brauer groups.

The Kepler problem: the energy point, the Levi-Civita, the Burdet and the KS regularizations via the primigenial sphere

In our unitary description (Vivarelli in Meccanica 50:915–925, 2015) of the Kepler problem (obtained via the introduction of a simple structure, the primigenial sphere $$S_{p^{-1}}$$), we have shown that this sphere encompasses, in a sort of inbred order of its elements, several fundamental elements of the Kepler problem. In this paper, we show that also the mechanical energy of an elliptic Kepler orbit is an element embedded in the sphere through a peculiar point, the energy point $$P^*$$. We show that this point in its circular motion on the sphere has a velocity which is strictly linked to so-called Sundman–Levi-Civita regularizing time transformation (Levi-Civita in Opere matematiche, 1973). Moreover in this spherical scenario, we reconsider both the two regularizations of the Kepler problem, namely the Bohlin–Burdet (Burdet in Z Angew Math Phys 18:434–438, 1967) and the Kustaanheimo and Stiefel (KS) regularizations (J Reine Angew Math 218:204–219, 1965): we present a geometrical interpretation of the first one, and we show an explicit link between their regularizing fundamental equations.

Extremal Kähler Poincaré Type Metrics on Toric Varieties

We develop a general theory for the existence of extremal Kahler metrics of Poincare type in the sense of Auvray (J Reine Angew Math 722:1–64, 2017), defined on the complement of a torus invariant divisor of a smooth compact toric variety. In the case when the divisor is smooth, we obtain a list of necessary conditions which must be satisfied for such a metric to exist. Using the explicit methods of Apostolov et al. (Ann Sci Ecole Norm Supp (4) 48:1075–1112, 2015; J Reine Angew Math 721:109–147, 2016, https://doi.org/10.1515/crelle-2014-0060 ) together with the computational approach of Sektnan (N Y J Math 24:317–354, 2018), we show that on a Hirzebruch complex surface the necessary conditions are also sufficient. In particular, on such a complex surface the complement of the infinity section admits an extremal Kahler metric of Poincare type, whereas the complement of a fibre fixed by the torus action admits a complete ambitoric extremal Kahler metric which is not of Poincare type.

Congruences for modular forms and generalized Frobenius partitions

The partition function is known to exhibit beautiful congruences that are often proved using the theory of modular forms. In this paper, we study the extent to which these congruence results apply to the generalized Frobenius partitions defined by Andrews (Mem Am Math Soc 49(301):iv+44, 1984). In particular, we prove that there are infinitely many congruences for $$c\phi _k(n)$$ modulo $$\ell ,$$ where $$\gcd (\ell ,6k)=1,$$ and we also prove results on the parity of $$c\phi _k(n).$$ Along the way, we prove results regarding the parity of coefficients of weakly holomorphic modular forms which generalize work of Ono (J Reine Angew Math 472:1–15, 1996).

On the problem of differentiation of hyperelliptic functions

We describe a construction that leads to an explicit solution of the problem of differentiation of hyperelliptic functions. A classical genus $$g=1$$ example of such a solution is the result of Frobenius and Stickelberger (J Reine Angew Math 92:311–337, 1882). Our method follows the works Buchstaber (Proc Steklov Inst Math 294:176–200, 2016) and Bunkova (Eur J Math 4(1):93–112, 2018) that led to constructions of explicit solutions of the problem for genus $$g=2$$ and $$g=3$$ .