Vanishing Order Research Articles

Bounding the order of vanishing of cuspidal newforms via the nth centered moments

Bounding the order of vanishing of cuspidal newforms via the nth centered moments

On the vanishing of Green’s function, desingularization and Carleman’s method

The subject of the present paper is the phenomenon of vanishing for the Green function of the operator − Δ + V -\Delta + V on R 3 \mathbb {R}^3 at the points where the potential V V has positive critical singularities. More precisely, under minimal assumptions on V V (i.e., the form-boundedness), an upper bound on the order of vanishing of the Green function is obtained. As a byproduct, the existing results on the strong unique continuation for eigenfunctions of − Δ + V -\Delta +V in dimension d = 3 d=3 are improved.

Validity of annealed approximation in a high-dimensional system

This study investigates the suitability of the annealed approximation in high-dimensional systems characterized by dense networks with quenched link disorder, employing models of coupled oscillators. We demonstrate that dynamic equations governing dense-network systems converge to those of the complete-graph version in the thermodynamic limit, where link disorder fluctuations vanish entirely. Consequently, the annealed-network systems, where fluctuations are attenuated, also exhibit the same dynamic behavior in the thermodynamic limit. However, a significant discrepancy arises in the incoherent (disordered) phase wherein the finite-size behavior becomes critical in determining the steady-state pattern. To explicitly elucidate this discrepancy, we focus on identical oscillators subject to competitive attractive and repulsive couplings. In the incoherent phase of dense networks, we observe the manifestation of random irregular states. In contrast, the annealed approximation yields a symmetric (regular) incoherent state where two oppositely coherent clusters of oscillators coexist, accompanied by the vanishing order parameter. Our findings imply that the annealed approximation should be employed with caution even in dense-network systems, particularly in the disordered phase.

Anomalous phase diagram of the elastic interface with nonlocal hydrodynamic interactions in the presence of quenched disorder.

We investigate the influence of quenched disorder on the steady states of driven systems of the elastic interface with nonlocal hydrodynamic interactions. The generalized elastic model (GEM), which has been used to characterize numerous physical systems such as polymers, membranes, single-file systems, rough interfaces, and fluctuating surfaces, is a standard approach to studying the dynamics of elastic interfaces with nonlocal hydrodynamic interactions. The criticality and phase transition of the quenched generalized elastic model are investigated numerically and the results are presented in a phase diagram spanned by two tuning parameters. We demonstrate that in the one-dimensional disordered driven GEM, three qualitatively different behavior regimes are possible with a proper specification of the order parameter (mean velocity) for this system. In the vanishing order parameter regime, the steady-state order parameter approaches zero in the thermodynamic limit. A system with a nonzero mean velocity can be in either the continuous regime, which is characterized by a second-order phase transition, or the discontinuous regime, which is characterized by a first-order phase transition. The focus of this research is to investigate the critical scaling features near the pinning-depinning threshold. The behavior of the quenched generalized elastic model at the critical depinning force is explored. Near the depinning threshold, the critical exponent is obtained numerically.

GENERALISED AUTOMORPHIC SHEAVES AND THE PROPORTIONALITY PRINCIPLE OF HIRZEBRUCH-MUMFORD

AbstractWe axiomatise the algebraic properties of toroidal compactifications of (mixed) Shimura varieties and their automorphic vector bundles. A notion of generalised automorphic sheaf is proposed which includes sheaves of (meromorphic) sections of automorphic vector bundles with prescribed vanishing and pole orders along strata in the compactification, and their quotients. These include, for instance, sheaves of Jacobi forms and weakly holomorphic modular forms. Using this machinery, we give a short and purely algebraic proof of the proportionality theorem of Hirzebruch and Mumford.

Tori over number fields and special values at $s=1$

We define a Weil-étale complex with compact support for duals (in the sense of the Bloch dualizing cycles complex \mathbb{Z}^c ) of a large class of \mathbb{Z} -constructible sheaves on an integral1-dimensional proper arithmetic scheme flat over Spec( \mathbb{Z} ). This complex can be thought of as computing Weil-étale homology. For those Z-constructible sheaves that are moreover tamely ramified, we define an “additive” complex which we think of as the Lie algebra of the dual of the \mathbb{Z} -constructible sheaf. The product of the determinants of the additive and Weil-étale complex is called the fundamental line. We prove a duality theorem which implies that the fundamental line has a natural trivialization, giving a multiplicative Euler characteristic. We attach a natural L -function to the dual of a \mathbb{Z} -constructible sheaf; up to a finite number of factors, this L -function is an Artin L -function at s = 1 . Our main theorem contains a vanishing order formula at s = 0 for the L -function and states that, in the tamely ramified case, the special value at s = 0 is given up to sign by the Euler characteristic. This generalizes the analytic class number formula for the special value at s = 1 of the Dedekind zeta function. In the function field case, this is a theorem of Geisser–Suzuki.

Seshadri stratifications and standard monomial theory

We introduce the notion of a Seshadri stratification on an embedded projective variety. Such a structure enables us to construct a Newton-Okounkov simplicial complex and a flat degeneration of the projective variety into a union of toric varieties. We show that the Seshadri stratification provides a geometric setup for a standard monomial theory. In this framework, Lakshmibai-Seshadri paths for Schubert varieties get a geometric interpretation as successive vanishing orders of regular functions.

Borel’s rank theorem for Artin 𝐿-functions

Borel’s rank theorem identifies the ranks of algebraic K K -groups of the ring of integers of a number field with the orders of vanishing of the Dedekind zeta function attached to the field. Following the work of Gross, we establish a version of this theorem for Artin L L -functions by considering equivariant algebraic K K -groups of number fields with coefficients in rational Galois representations. This construction involves twisting algebraic K K -theory spectra with rational equivariant Moore spectra. We further discuss integral equivariant Moore spectra attached to Galois representations and their potential applications in L L -functions.

On the Hasse invariant of Hilbert modular varieties mod p

Let F be a totally real field and let S denote the geometric special fiber of a Hilbert modular variety associated to F, at a prime unramified in F. We show that the order of vanishing of the Hasse invariant on S is equal to the largest integer m such that the smallest piece of the conjugate filtration lies in the mth piece of the Hodge filtration. This result is a direct analogue of Ogus' on families of Calabi–Yau varieties in positive characteristic (see [15]). We also show that the order of vanishing at a point is the same as the codimension of the Ekedahl-Oort stratum containing it.

Space-like quantitative uniqueness for parabolic operators

We obtain sharp maximal vanishing order at a given time level for solutions to parabolic equations with a C1 potential V. Our main result Theorem 1.1 is a parabolic generalization of a well known result of Donnelly-Fefferman and Bakri. It also sharpens a previous result of Zhu that establishes similar vanishing order estimates which are instead averaged over time. The principal tool in our analysis is a new quantitative version of the well-known Escauriaza-Fernandez-Vessella type Carleman estimate that we establish in our setting.

Carleman estimates for sub-Laplacians on Carnot groups

In this note, we establish a new Carleman estimate with singular weights for the sub-Laplacian on a Carnot group mathbb G for functions satisfying the discrepancy assumption in (2.16) below. We use such an estimate to derive a sharp vanishing order estimate for solutions to stationary Schrödinger equations.

Perturbed eigenvalues of polyharmonic operators in domains with small holes

We study singular perturbations of eigenvalues of the polyharmonic operator on bounded domains under removal of small interior compact sets. We consider both homogeneous Dirichlet and Navier conditions on the external boundary, while we impose homogeneous Dirichlet conditions on the boundary of the removed set. To this aim, we develop a notion of capacity which is suitable for our higher-order context, and which permits to obtain a description of the asymptotic behaviour of perturbed simple eigenvalues in terms of a capacity of the removed set, in dependence of the respective normalized eigenfunction. Then, in the particular case of a subset which is scaling to a point, we apply a blow-up analysis to detect the precise convergence rate, which turns out to depend on the order of vanishing of the eigenfunction. In this respect, an important role is played by Hardy–Rellich inequalities in order to identify the appropriate functional space containing the limiting profile. Remarkably, for the biharmonic operator this turns out to be the same, regardless of the boundary conditions prescribed on the exterior boundary.

On analogues of Mazur-Tate type conjectures in the Rankin-Selberg setting

We study the Fitting ideals over the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of Selmer groups attached to the Rankin--Selberg convolution of two modular forms $f$ and $g$. Inspired by the Theta elements for modular forms defined by Mazur and Tate in ``Refined conjectures of the Birch and Swinnerton-Dyer type'', we define new Theta elements for Rankin--Selberg convolutions of $f$ and $g$ using Loeffler--Zerbes' geometric $p$-adic $L$-functions attached to $f$ and $g$. Under certain technical hypotheses, we generalize a recent work of Kim--Kurihara on elliptic curves to prove a result very close to the \emph{weak main conjecture} of Mazur and Tate for Rankin--Selberg convolutions. Special emphasis is given to the case where $f$ corresponds to an elliptic curve $E$ and $g$ to a two dimensional odd irreducible Artin representation $\rho$ with splitting field $F$. As an application, we give an upper bound of the dimension of the $\rho$-isotypic component of the Mordell-Weil group of $E$ over the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of $F$ in terms of the order of vanishing of our Theta elements.

On the non-critical exceptional zeros of Katz p-adic L-functions for CM fields

The primary goal of this article is to study p-adic Beilinson conjectures in the presence of exceptional zeros for Artin motives over CM fields. In more precise terms, we address a question raised by Hida and Tilouine on the order of vanishing of Katz p-adic L-functions associated to CM fields, by means of a leading term formula we prove in terms of Rubin–Stark elements. In the particular case when the CM field in question is imaginary quadratic, our leading term formula and its consequences we record herein are unconditional.

Enhanced dissipation and Hörmander's hypoellipticity

We examine the phenomenon of enhanced dissipation from the perspective of Hörmander's classical theory of second order hypoelliptic operators [33]. Consider a passive scalar in a shear flow, whose evolution is described by the advection–diffusion equation∂tf+b(y)∂xf−νΔf=0 on T×(0,1)×R+ with periodic, Dirichlet, or Neumann conditions in y. We demonstrate that decay is enhanced on the timescale T∼ν−(N+1)/(N+3), where N is the maximal order of vanishing of the derivative b′(y) of the shear profile and N=0 for monotone shear flows. In the periodic setting, we recover the known timescale of Bedrossian and Coti Zelati [8]. Our results are new in the presence of boundaries.

On order of vanishing of characteristic elements

AbstractLetpbe a fixed odd prime. LetEbe an elliptic curve defined over a number fieldFwith either good ordinary reduction or multiplicative reduction at each prime ofFabovep. We shall study the characteristic element of the Selmer group ofEover ap-adic Lie extension. In particular, we relate the order of vanishing of these characteristic elements evaluated at Artin representations to the Selmer coranks and their twists in the intermediate subextensions of thep-adic Lie extension.

The nodal set of solutions to some nonlocal sublinear problems

We study the nodal set of stationary solutions to equations of the form (-Delta )^s u = lambda _+ (u_+)^{q-1} - lambda _- (u_-)^{q-1}quad text {in }B_1, where lambda _+,lambda _->0, q in [1,2), and u_+ and u_- are respectively the positive and negative part of u. This collection of nonlinearities includes the unstable two-phase membrane problem q=1 as well as sublinear equations for 1<q<2. We initially prove the validity of the strong unique continuation property and the finiteness of the vanishing order, in order to implement a blow-up analysis of the nodal set. As in the local case s=1, we prove that the admissible vanishing orders can not exceed the critical value k_q= 2s/(2- q). Moreover, we study the regularity of the nodal set and we prove a stratification result. Ultimately, for those parameters such that k_q< 1, we prove a remarkable difference with the local case: solutions can only vanish with order k_q and the problem admits one dimensional solutions. Our approach is based on the validity of either a family of Almgren-type or a 2-parameter family of Weiss-type monotonicity formulas, according to the vanishing order of the solution.

The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds

We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold Sigma with Betti number b_1, the order of vanishing of the Ruelle zeta function at zero equals 4-b_1, while in the hyperbolic case it is equal to 4-2b_1. This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle SSigma with harmonic 1-forms on Sigma .

Goss polynomials, q-adic expansions, and Sheats compositions

The zeroes of Goss polynomials Gk,Λ(X) for Λ=A:=Fq[T] and similar lattices Λ are studied. Generically, the zero distribution follows a simple pattern governed by the q-adic expansion of k−1. However, if q=pf with f≥2 is a proper power of the prime p, irregularities of the q-adic sum-of-digits function may lead to deviations from this pattern, to irregular zeroes, and an abundance of trivial zeroes of Gk,Λ compared to the generic formula. These phenomena are related to properties of the Sheats compositions of natural numbers n divisible by q−1. Among other things, we give a necessary and sufficient condition for the existence of irregular zeroes of Gk,A and a formula for the vanishing order of Gk,A at x=0.

Orders of vanishing and U(1) charges in F-theory

Many interesting questions about F-theory models, including several concerning the F-theory swampland, involve massless matter charged under U(1) gauge symmetries. It is therefore important to better understand the geometric properties of F-theory models realizing various U(1) charges. We propose that, for F-theory models described by elliptic fibrations in Weierstrass form, the U(1) charge of light matter is encoded in the orders of vanishing of the section components corresponding to the U(1) gauge symmetry. We give specific equations relating the U(1) charges to the orders of vanishing that seem to hold for both U(1)-charged singlets and for matter additionally charged under a simply-laced nonabelian gauge algebra. Our formulas correctly describe properties of F-theory models in the prior literature, and we give an argument that they should describe the orders of vanishing for arbitrarily high U(1) charges. They also resemble formulas for the p-adic valuations of elliptic divisibility sequences developed by Stange [1]. These proposals could serve as a U(1) analogue of the Katz-Vafa method, allowing one to determine U(1) charges without resolution. Additionally, they predict geometric information about F-theory models with general U(1) charges, which may be useful for exploring the F-theory landscape and swampland.