arXiv subjects

Stephen Kudla

Publications and source records attributed to Stephen Kudla.

17 recordsLinked to original sources

arXiv2021

The case of an N-gon

We construct the indefinite theta series attached to N-gons in the symmetric space of an indefinite inner product space of signature (m-2,2) following the suggestions of section C in the recent paper of Alexandrov, Banerjee, Manschot, and Pioline. We prove the termwise absolute convergence of the holomorphic mock modular part of these series and also obtain an interpretation of the coefficients of this part as linking numbers. Thus we prove the convergence conjecture of ABMP provided none of the vectors in the collection CC={C_1,..., C_N} is a null vector. The use of linking numbers and a homotopy argument eliminates the need for an explicit parametrization of a surface S spanning the N-gon that was used in an essential way in our previous work. We indicate how our method could be carried over to a more general situation for signature (m-q,q) where higher homotopy groups are now involved. In the last section, we apply the method to the case of a dodecahedral cell in the symmetric space of a quadratic form of signature (m-3,3).

math.NT↗

arXiv2020

On the p-adic uniformization of unitary Shimura curves

We prove $p$-adic uniformization for Shimura curves attached to the group of unitary similitudes of certain binary skew hermitian spaces $V$ with respect to an arbitrary CM field $K$ with maximal totally real subfield $F$. For a place $v|p$ of $F$ that is not split in $K$ and for which $V_v$ is anisotropic, let $\nu$ be an extension of $v$ to the reflex field $E$. We define an integral model of the corresponding Shimura curve over ${\rm Spec}\, O_{E, (\nu)}$ by means of a moduli problem for abelian schemes with suitable polarization and level structure prime to $p$. The formulation of the moduli problem involves a Kottwitz condition, an Eisenstein condition, and an adjusted invariant. The first two conditions are conditions on the Lie algebra of the abelian varieties; the last condition is a condition on the Riemann form of the polarization. The uniformization of the formal completion of this model along its special fiber is given in terms of the formal Drinfeld upper half plane for $F_v$. The proof relies on the construction of the contracting functor which relates a relative Rapoport-Zink space for strict formal $O_{F_v}$-modules with a Rapoport-Zink space of $p$-divisible groups which arise from the moduli problem, where the $O_{F_v}$-action is usually not strict when $F_v\ne \mathbb {Q}_p$. Our main tool is the theory of displays, in particular the Ahsendorf functor.

math.AG↗

arXiv2020

On the subring of special cycles

For a totally real field F of degree d>1 and a quadratic space V of signature (m,2)^{d_+} x (m+2,0)^{d-d_+} with associated Shimura variety Sh(V), we consider the subring of cohomology generated by the classes of weighted special cycles. We assume that d_+<d. We take the quotient SC(V) of this ring by the radical of the restriction of the intersection pairing to it. We show that the inner products of classes in SC(V) are determined by Fourier coefficients of pullbacks of Hilbert-Siegel Eisenstein series of genus m to products of smaller Siegel spaces and that the products of classes in SC(V) are determined by Fourier coefficients of pullbacks to triple products of smaller Siegel spaces. As a consequence, we show that, for quadratic spaces V and V' over F that are isomorphic at all finite places, but with no restriction on d_+(V) and d_+(V') other than the necessary condition that they have the same parity, the special cycles rings SC(V) and SC(V') are isometrically isomorphic. This is a consequence of the Siegel-Weil formula and the matching principle. Finally, we give a combinatorial construction of a ring SC(V_+) associated to a totally positive definite quadratic space V_+ of dimension m+2 over F and show that the comparison isomorphism extends to this case.

math.NT↗

arXiv2019

Remarks on generating series for special cycles

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\le d_+<d. For each n, 1\le n\le m, there are special cycles Z(T) in S, of codimension nd_+, indexed by totally positive semi-definite matrices with coefficients in the ring of integers O_F. The generating series for the classes of these cycles in the cohomology group H^{2nd_+}(S) are Hilbert-Siegel modular forms of parallel weight m/2+1. One can form analogous generating series for the classes of the special cycles in the Chow group CH^{nd_+}(S). For d_+=1 and n=1, the modularity of these series was proved by Yuan-Zhang-Zhang. In this note we prove the following: Assume the Bloch-Beilinson conjecture on the injectivity of Abel-Jacobi maps. Then the Chow group valued generating series for special cycles of codimension nd_+ on S is modular for all n with 1\le n\le m.

math.NT↗

arXiv2017

Theta integrals and generalized error functions, II

Theta series for indefinite quadratic lattices were introduced by Zwegers, for signature (m-1,1), Alexandrov, Banerjee, Manschot and Pioline, for signature (m-2,2), and Nazaroglu, for signature (m-q,q). These series are modular modular completions, defined by means of generalized error functions, of certain non-modular holomorphic generating series associated to lattice vectors in positive cones. We show that these modular forms arise as integrals of the theta forms, defined in work of Millson and the second author, over certain singular q-cubes. We also give an explicit formula for the integrals of such forms over singular q-simplices. The sign function occurring in the holomorphic generating series arises as in intersection number of the singular q-cube or q-simplex with a totally geodesic subsymmetric space of codimension q. The cubical case for q=2 was treated in [11].

math.NT↗

arXiv2017

Trapped imbalanced fermionic superfluids in one dimension: A variational approach

We propose and analyze a variational wave function for a population-imbalanced one-dimensional Fermi gas that allows for Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type pairing correlations among the two fermion species, while also accounting for the harmonic confining potential. In the strongly interacting regime, we find large spatial oscillations of the order parameter, indicative of an FFLO state. The obtained density profiles versus imbalance are consistent with recent experimental results as well as with theoretical calculations based on combining Bethe ansatz with the local density approximation. Although we find no signature of the FFLO state in the densities of the two fermion species, we show that the oscillations of the order parameter appear in density-density correlations, both in-situ and after free expansion. Furthermore, above a critical polarization, the value of which depends on the interaction, we find the unpaired Fermi-gas state to be energetically more favorable.

cond-mat.quant-gas↗

arXiv2016

A classification of harmonic Maass forms

We give a classification of the Harish-Chandra modules generated by the pullback to $\text{SL}_2(\mathbb R)$ of harmonic Maass forms for congruence subgroups of $\text{SL}_2(\mathbb Z)$ with exponential growth allowed at the cusps. We assume that the weight is integral but include vector-valued forms. Due to the weak growth condition, these modules need not be irreducible. Elementary Lie algebra considerations imply that there are 9 possibilities, and we show, by giving explicit examples, that all of them arise from harmonic Maass forms. Finally, we briefly discuss the case of forms that are not harmonic but rather are annihilated by a power of the Laplacian, where much more complicated Harish-Chandra modules can arise.

math.NT↗

arXiv2016

Theta integrals and generalized error functions

In a recent preprint, arXiv:1606.05495v1, Alexandrov, Banerjee, Manschot and Pioline introduced generalized error functions and used them to construct indefinite theta series associated to quadratic lattices L of signature (n-2,2). These series are generalizations of those constructed by Zwegers for lattices of signature (n-1,1) and are shown to be `modular completions' of certain nice $q$-series. In this paper, we show that the ABMP-indefinite theta series for signature (n-2,2) can also be obtained as integrals of the form valued theta series introduced in joint work with J. Millson in 1986. Given two pairs {C1,C2} and {C2,C2'} of negative vectors in the real quadratic space V obtained from L, we suppose that these vectors determine 4 distinct oriented negative 2-planes {C1,C2},{C1,C2'},{C1',C2'},{C1',C2} lying in the same component of the space D of oriented negative 2 -planes in V. These 2-planes determined a surface S in D and the non-holomorphic modular form obtained by integrating the KM-theta series over S is show to coincide with the ABMP-indefinite theta series. Moreover, the associated q-series in interpreted as the generating series for the intersection numbers of S with the codimension 2 subspaces D_x of D defined by positive lattice vectors.

math.NT↗

arXiv2015

Pairing correlations in a trapped one-dimensional Fermi gas

We use a BCS-type variational wavefunction to study attractively-interacting quasi one-dimensional (1D) fermionic atomic gases, motivated by cold-atom experiments that access the 1D regime using an anisotropic harmonic trapping potential (with trapping frequencies $ω_x = ω_y \gg ω_z$) that confines the gas to a cigar-shaped geometry. To handle the presence of the trap along the $z$-direction, we construct our variational wavefunction from the harmonic oscillator Hermite functions that are the eigenstates of the single-particle problem. Using an analytic determination of the effective interaction among harmonic oscillator states along with a numerical solution of the resulting variational equations, we make specific experimental predictions for how pairing correlations would be revealed in experimental probes like the local density and the momentum correlation function.

cond-mat.quant-gas↗

arXiv2014

A note about special cycles on moduli spaces of K3 surfaces

We describe the application of the results of Kudla-Millson on the modularity of generating series for cohomology classes of special cycles to the case of lattice polarized K3 surfaces. In this case, the special cycles can be interpreted as higher Noether-Lefschetz loci. These generating series can be paired with the cohomology classes of complete subvarieties of the moduli space to give classical Siegel modular forms with higher Noether-Lefschetz numbers as Fourier coefficients. Examples of such complete families associated to quadratic spaces over totally real number fields are constructed. A more explicit and concrete construction of such families and the resulting modular forms would be of interest.

math.AG↗

arXiv2014

Another product for a Borcherds form

In his celebrated 1998 Inventiones paper, Borcherds constructed meromorphic automorphic forms Psi(F) for arithmetic subgroups associated to even integral lattices M of signature (n,2). The input to his construction is a vector valued weakly holomorphic modular form F of weight 1 - n/2, and the resulting Borcherds form has an explicit divisor on the arithmetic quotient X = Gamma_M\ D. Most remarkably, in the neighborhood of each cusp (= rational point boundary component), there is a beautiful product formula for Psi(F), reminiscent of the classical product formula for the Dedekind eta-function. In this paper, we describe an analogous product formula for Psi(F) in the neighborhood of each 1-dimensional rational boundary component. This formula, which, like that of Borcherds, is obtained through the calculation of a regularized theta integral, reveals the behavior of Psi(F) on a (partial) smooth compactification of X. Information about Fourier-Jacobi coefficients is added to this revised version.

math.AG↗

arXiv2013

On occult period maps

We consider the "occult" period maps into ball quotients which exist for the moduli spaces of cubic surfaces, cubic threefolds, non-hyperelliptic curves of genus three and four. These were constructed in the work of Allcock/Carlson/Toledo, Looijenga/Swierstra, and Kondo. We interpret these maps as morphisms into moduli spaces of polarized abelian varieties of Picard type, and show that these morphisms, whose initial construction is transcendental, are defined over the natural field of definition of the spaces involved. This paper is extracted from section 15 of our paper arXiv:0912.3758, and differs from it only in some points of exposition.

math.AG↗

arXiv2013

New cases of p-adic uniformization

We prove a Cherednik style $p$-adic uniformization theorem for Shimura varieties associated to certain groups of unitary similitudes of size two over totally real fields. Our basic tool is the alternative modular interpretation of the Drinfeld $p$-adic halfplane of our earlier paper (arXiv 1108.5713)

math.AG↗

arXiv2013

An alternative description of the Drinfeld p-adic half-plane

We show that the Deligne formal model of the Drinfeld p-adic halfplane relative to a non-archimedean local field F represents a moduli problem of polarized O_F-modules with an action of the ring of integers O_E in a quadratic extension E of F. The proof proceeds by establishing a comparison isomorphism with the Drinfeld moduli problem. This isomorphism reflects the accidental isomorphism of SL_2(F) and SU(C)(F) for a two-dimensional split hermitian space C for E/F.

math.NT↗

arXiv2012

Special cycles on unitary Shimura varieties II: global theory

We introduce moduli spaces of abelian varieties which are arithmetic models of Shimura varieties attached to unitary groups of signature (n-1, 1). We define arithmetic cycles on these models and study their intersection behaviour. In particular, in the non-degenerate case, we prove a relation between their intersection numbers and Fourier coefficients of the derivative at s=0 of a certain incoherent Eisenstein series for the group U(n, n). This is done by relating the arithmetic cycles to their formal counterpart from Part I via non-archimedean uniformization, and by relating the Fourier coefficients to the derivatives of representation densities of hermitian forms. The result then follows from the main theorem of Part I and a counting argument.

math.AG↗

arXiv2011

On the pullback of an arithmetic theta function

In this paper, we consider the relation between the simplest types of arithmetic theta series, those associated to the cycles on the moduli space $\Cal C$ of elliptic curves with CM by the ring of integers $\OK$ in an imaginary quadratic field $\kay$, on the one hand, and those associated to cycles on the arithmetic surface $\M$ parametrizing 2-dimensional abelian varieties with an action of the maximal order $O_B$ in an indefinite quaternion algebra $B$ over $\Q$, on the other. We show that the arithmetic degree of the pullback to $Cal C$ of the arithmetic theta function of weight 3/2 valued in $\hat CH^1(\M)$ can be expressed as a linear combination of arithmetic theta functions of weight 1 for $\Cal C$ and unary theta series. This identity can be viewed as an arithmetic seesaw identity. In addition, we show that the arithmetic theta series of weight 1 coincide with the central derivative of certain incoherent Eisenstein series for SL(2)/Q, generalizing earlier joint work with M. Rapoport for the case of a prime discriminant.

math.NT↗

arXiv2011

Special cycles on unitary Shimura varieties I. unramified local theory

The supersingular locus in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n-1) over Q is uniformized by a formal scheme \Cal N. In the case when p is inert, we define special cycles Z(x) in \Cal N, associated to a collection x of m `special homomorphisms' with fundamental matrix T in Herm_m(OK). When m=n and T is nonsingular, we show that the cycle Z(x) is a union of components of the Ekedahl-Oort stratification, and we give a necessary and sufficient conditions, in terms of T, for Z(x) to be irreducible. When Z(x) is zero dimensional -- in which case it reduces to a single point -- we determine the length of the corresponding local ring by using a variant of the theory of quasi-canonical liftings. We show that this length coincides with the derivative of a representation density for hermitian forms.

math.AG↗