IIndefinite Theta Series on Cones
Martin Westerholt-Raum Abstract:
We show that indefinite theta series on cones converge and provide an explicit modular completion.Our completion rests on a convolution of the Gaussian with a piecewise constant function supported on thecone. Our main innovation is to formulate this convolution in terms of euclidean geometry as opposed to hy-perbolic geometry. This change of perspective allows us to establish essential asymptotic estimates withoutfurther difficulty. indefinite lattices (cid:4) indefinite quadratic forms (cid:4) mock theta series (cid:4) modular completion A N indefinite theta series is the generating series associated with a vector norm on a suitable coneof an indefinite lattice. As the notion is so flexible, such generating series occur frequently in, e.g.number theory [Kit86; Sie51], infinite dimensional representation theory [BO09; KW01; KW14], enu-merative geometry [LZ15; OP06], combinatorics [BO06], and string theory [DMZ12; Gan16]. Indefinitetheta series are also connected directly to mock theta series. For decades, mathematicians sought toclarify the mystery around them, which emerged from Ramanujan’s Death Bed Letter and his LostNotebook [Ram00; Ram88]: What are the modular properties of mock theta series? Zwegers resolvedit in his pioneering thesis [Zwe02] by offering not one but three explanations, two of which are relatedto indefinite theta series. In the third section of [Zwe02], Zwegers relates forthright indefinite thetaseries and mock theta series. Section 1 of his thesis suggests to understand mock theta series in termsof Appell-Lerch sums, which are quotients of indefinite theta series on “cubical cones” by a usual thetaseries. Zwegers’s key discovery was a “modular completion” of indefinite theta series for lattices of sig-nature ( d + , 1) together with an explicit formula for its Fourier expansion. His piece is at the foundationof all applications that we mentioned.Zwegers’s assumption that a lattice has signature ( d + , 1) as opposed to the most general signature( d + , d − ) with d + , d − ≥ L in a real quadratic space V with quadratic form q of signature( d + , d − ) and bilinear form 〈 l , l (cid:48) 〉 = q ( l + l (cid:48) ) − q ( l ) − q ( l (cid:48) ), suitable cones C ⊆ V yield theta series θ L ,0 ( C ; τ , z ) = (cid:88) l ∈ L ∩ ( C − vy ) exp (cid:161) π i ( q ( l ) τ + 〈 z , l 〉 ) (cid:162) , τ = x + i y ∈ (cid:72) = { τ ∈ (cid:67) : Im( τ ) > z = u + i v ∈ V ( (cid:67) ) : = V ⊗ (cid:67) .The key questions are: (1) When does θ L ,0 ( C ) converge? (2) If it converges, what are its modularproperties? In other words, what is its modular completion? (3) If θ L ,0 ( C ) is not modular, what is the1-cocycle attached to it? We answer the first two questions in this work. The author was partially supported by Vetenskapsrådet Grant 2015-04139. – – ndefinite Theta Series on Cones M. Westerholt-Raum
Zwegers’s thesis treating the case of signatures ( d + , 1) was published in 2002. The existence of a mod-ular completion of indefinite theta series in general has been announced several times by Zagier andZwegers in talks over the past years. However, details or explicit formulas are not available. In 2012,the author [Wes15b] extended slightly the approach in Zwegers’s thesis to general lattices and cubi-cal cones whose faces meet mutually orthogonally. The orthogonality assumption, which is a majorobstacle in applications, was lifted for lattice signatures ( d + , 2) and some cubical cones by Alexan-drov, Banerjee, Manschot, and Pioline [Ale+16] in summer 2016. Shortly after [Ale+16] appeared asa preprint, Kudla [Kud16] illustrated how the modular completion of more specific indefinite thetaseries can be obtained from the theta kernels in [KM86; KM87]. Kudla remarked that the conditionsimposed on cones in [Ale+16] are stricter than necessary, and he treated additional cases. After thepresent manuscript appeared as a preprint, Nazaro˘glu [Naz16] extended considerations in [Ale+16] tocubical cones in arbitrary lattices, while still imposing rather strong assumptions on their faces.Kudla has announced a joint paper with J. Funke in which they generalize [Kud16] to the effect thatthey interpret θ L ,0 ( C ) in terms of intersection numbers of points with cubes in the Grassmannian ofmaximal negative definite subspaces of V . These cubes correspond to cubical cones in our terminol-ogy. Funke and Kudla’s ideas yield a modular completion of θ L ,0 ( C ) as an integral of the Kudla-Millsontheta kernel. If the cube that corresponds to C is compact, the integral can be evaluated and one ob-tains the Fourier expansion of the modular completion. In the non-compact case, the modular com-pletion is merely given as an integral. Using our results for cubical cones, one computes its Fourierexpansion as a uniform limit of Fourier expansions in the compact case.We discuss θ L ,0 ( C ) in more detail. Our focus is on polyhedral cones. We call a cone C ⊆ V poly-hedral, if its cross-section is a (possibly non-compact) polyhedron. Polyhedral cones are sufficientto cover all applications of indefinite theta series that are known to the author. Beyond this theycan be used to approximate any other cone. For instance, for Fano varieties the nef cone is poly-hedral, and the Morrison-Kawamata cone conjecture extends this in a suitable sense to further va-rieties. On the other end of the spectrum, the classical Appell-Lerch sum can be related to the lat-tice with Gram matrix (cid:161) (cid:162) , which has lattice signature (1, 1). In coordinates, the associated cone is C = { l ∈ (cid:82) : ( l , l ≥ ∨ ( l , l ≤ (cid:80) ( V ) of V is one-dimensional. Generalized Appell-Lerch sums can be ex-panded to indefinite theta series on cubical cones [Ale+16]. Figure 1: cross section of C ex To answer the first key question on indefinite theta series on cones,we note that if a polyhedral cone C contains negative vectors, then θ L ,0 ( C ) does not converge. Consider a polyhedral cone that containsonly non-negative vectors and whose isotropic parts are rational. Weshow in Corollary 4.1 on page 16 that θ L ,0 ( C ) converges to a meromor-phic function, whose poles are determined by the isotropic parts of C .Only cones with rational isotropic parts have so-far appeared in appli-cations. As for cases of non-rational isotropic parts of C , convergenceof θ L ,0 ( C ) is related to Roth’s theorem. These cases are not studied inthe present paper, while it might be interesting to examine them withan eye to arithmetic properties of the cone C itself. – – ndefinite Theta Series on Cones M. Westerholt-Raum
To illustrate our results, we inspect one tetrahedral case as a runningexample. It is among the simplest cases that are not covered by any prior work. L ex = (cid:90) , q ex ( l ) = ( l − l − l ), and C ex = (cid:110) l ∈ (cid:82) : ( l , l − l − l , 2 l + l − l ≥ ∨ ( l , l − l − l , 2 l + l − l ≤ (cid:111) .Notice that C ex contains the isotropic line spanned by ( ). It is bounded by the orthogonal comple-ments of ( ), ( ), and ( ). Its cross-section for fixed l including the circle of isotropic vectorsis displayed in Figure 1. The shaded area is the cross section of C ex and the displayed lines correspondto its faces.Recall that the major issue in applications with indefinite theta series is that they are in general notmodular. As a remedy Zwegers constructed for lattice signatures ( d + , 1) and polyhedral cones with twofaces (i.e. tetrahedral cones) a modular completion of a slight modification of θ L ,0 ( C ). To construct thismodification, one first observes that for every polyhedral cone, there is a polynomial sgn + C ( l ) in signfunctions l (cid:55)→ sgn 〈 l , w 〉 with w ∈ V such thatinterior( C ) ⊆ support(sgn + C ) ⊆ closure( C ) and sgn + C | interior( C ) = ± + C ( l ) as a linear combination of products of sign functions sgn E ( l ) : = (cid:81) w ∈ E sgn 〈 l , w 〉 with sets E ⊂ V of size at most d − . For the purpose of this paper, we say that such conesare determined by face indicators. We show in Section 2.1 that tetrahedral and cubical cones are de-termined by face indicators. In our running example, sgn + C ex ( l ) is explicitly given by (cid:161) + sgn 〈 l , ( ) 〉 sgn 〈 l , ( ) 〉 + sgn 〈 l , ( ) 〉 sgn 〈 l , ( ) 〉 + sgn 〈 l , ( ) 〉 sgn 〈 l , ( ) 〉 (cid:162) . (0.1)As a generalization of Zwegers’s theta series, we study the meromorphic function θ L ,0 (sgn + C ; τ , z ) = (cid:88) l ∈ L sgn + C (cid:161) (cid:112) y ( l + vy ) (cid:162) exp (cid:161) π i ( q ( l ) τ + 〈 z , l 〉 ) (cid:162) ,Comparing to the definition of θ L ,0 ( C ), the condition that l ∈ C − vy is replaced by sgn + C ( l + vy ) (cid:54)=
0. Thedifference θ L ,0 ( C ) − θ L ,0 (sgn + C ) can be expressed in terms of theta series for smaller lattices, if the facesof C span non-degenerate subspaces of V . In that case, the present paper yields a modular completionof θ L ,0 ( C ), which in general has mixed weight.Zwegers emphasized in many of his talks that the error function that appears in [Zwe02] should beviewed as a smoothened sign. Connecting to this idea, we obtain a smoothened variant (cid:100) sgn + C of sgn + C as a convolution with the Gaussian. The assumption that C is determined by its face indicators enters.Special cases of (cid:100) sgn + C appear in [Ale+16; Naz16], where they are called generalized error functions. Themain novelty in the present manuscript is that our description of (cid:100) sgn + C is based on euclidean geometryas opposed to hyperbolic geometry or contour integration. As a result, we can expose its asymptoticbehavior in a straightforward way. In particular, we find that the difference (cid:100) sgn + C − sgn + C decays square-exponentially on every closed cone that does not contain the faces of C . – – ndefinite Theta Series on Cones M. Westerholt-Raum
Theorem I.
Let L be an integral lattice. If C ⊆ V = L ⊗ (cid:82) is a non-degenerate, non-negative tetrahedralor cubical cone with rational isotropic edges, then (cid:98) θ L ,0 ( (cid:100) sgn + C ; τ , z ) = (cid:88) l ∈ L (cid:100) sgn + W (cid:161) (cid:112) y ( l + vy ) (cid:162) exp (cid:161) π i ( q ( l ) τ + 〈 z , l 〉 ) (cid:162) converges locally to a real-analytic function with meromorphic singularities and transforms like a Jacobiform of suitable level. For a discussion of each of the assumptions see the introduction above. Notice that we do not stateabsolute convergence, which in fact can fail on a certain exceptional set. As a consequence, the Fourierexpansion might exhibit “wall-crossing”. A more precise statement phrased in terms of vector-valuedJacobi forms is given in Theorem 4.2 on page 16.
Remark.
Bruinier’s and Funke’s ξ -operator [BF04] plays a decisive role in the modern theory of mocktheta series and harmonic weak Maaß forms, which emerged from Zwegers’s thesis [Zwe02] and fromBruinier’s and Funke’s treatment of theta lifts [BF04]. This role can be phrased more systematically interms of Harish-Chandra modules for real representation theory, as was done in [Sch11; Wes15a]. Adescription of the analytic properties of (cid:98) θ L ,0 along these lines can be inferred from [Wes15b] and thedefinition Equation (3.1).To illustrate Theorem I, we return to our running example. Given v = ( v v v ), write˜ v = ˜ v ( v ) = v · ( ) , ˜ v = ˜ v ( v ) = ( v + v − v ) · ( ) , ˜ v = ˜ v ( v ) = (2 v + v − v ) · ( ) .Then (cid:100) sgn + C ex ( v ) equals14 + + (cid:112) (cid:90) (cid:82) sgn(4 v (cid:48) + v (cid:48) )sgn(5 v (cid:48) + v (cid:48) )exp (cid:161) π (cid:161) v (cid:48) − ˜ v ) + v (cid:48) − ˜ v ) + v (cid:48) − ˜ v )( v (cid:48) − ˜ v ) (cid:162)(cid:162) dv (cid:48) dv (cid:48) + (cid:112) (cid:90) (cid:82) sgn( v (cid:48) + v (cid:48) )sgn(2 v (cid:48) + v (cid:48) )exp (cid:161) π (cid:161) ( v (cid:48) − ˜ v ) + v (cid:48) − ˜ v ) + v (cid:48) − ˜ v )( v (cid:48) − ˜ v ) (cid:162)(cid:162) dv (cid:48) dv (cid:48) .Notice that the second term in the expression (0.1) for sgn + C ex contributes to (cid:100) sgn + C ex ( v ) (definedin (3.9)), as opposed to a general real-analytic function. In this specific case, the reason is that d − is even and the span of ( ) and ( ) contains a totally isotropic subspace of dimension 1 = d − − (cid:100) sgn + C is studied in terms of an euclideangeometric decomposition in Section 3. In Section 4 we establish modularity of completed indefinitetheta series. Acknowledgment
Will be entered after the referees’ comments are received. – – ndefinite Theta Series on Cones M. Westerholt-Raum
We let V be a quadratic space over (cid:81) with quadratic form q and scalar product 〈 · , · 〉 of signature ( d + , d − ). Its realification is V ( (cid:82) ) : = V ⊗ (cid:82) and its complexification V ( (cid:67) ) = V ⊗ (cid:67) . Theset of vectors v ∈ V ( (cid:82) ) with q ( v ) > q ( v ) < V ( (cid:82) ) + and V ( (cid:82) ) − , respectively. Wewrite V ( (cid:82) ) for the set of isotropic vectors in V ( (cid:82) ). Given a lattice L in V its dual with respect to thescalar product is denoted by L ∨ .We frequently use the projectivization (cid:80) ( V ( (cid:82) )) : = ( V ( (cid:82) ) \ {0})/ (cid:82) of V ( (cid:82) ). The class in (cid:80) ( V ( (cid:82) )) thatis represented by v is denoted by [ v ]. Given S ⊆ V ( (cid:82) ), we write [ S ] : = {[ v ] : v ∈ S \ {0}} for the set ofprojective vectors that are represented by some element in S .The orthogonal complement with respect to 〈 · , · 〉 in V ( (cid:82) ) of a vector w or a set W is denoted by w ⊥ and W ⊥ , respectively. We write W ⊥ W (cid:48) to say that the elements of two set W and W (cid:48) are mutuallyorthogonal. Given a definite subspace U and a vector v , we write v U for its orthogonal projectiononto U . Similarly, if W is a set with definite span, we write v W for the orthogonal projection ontospan W . Let (cid:72) = { τ ∈ (cid:67) : Im τ > (cid:72) J ( V ) = (cid:72) × V ( (cid:67) ) (1.1)be the Poincaré upper half-plane and the Jacobi upper half-space attached to V . We write elementsof (cid:72) J ( V ) as pairs ( τ , z ), where the imaginary part of τ is generally denoted by y , and the one of z by v .Note that in Section 3, we use the letter v for elements of V ( (cid:82) ), but the Jacobi upper half-space will notoccur there. We have an action of SL ( (cid:82) ) on (cid:72) defined by g τ = a τ + bc τ + d , g = (cid:161) a bc d (cid:162) ∈ SL ( (cid:82) ), (1.2)where the displayed notation for elements of SL ( (cid:82) ) is used here and throughout the paper. There is aunique, non-split extension Mp ( (cid:82) ) of SL ( (cid:82) ) by (cid:90) /2 (cid:90) , which is called its metaplectic cover. We writeMp ( (cid:90) ) for the preimage of SL ( (cid:90) ) in Mp ( (cid:82) ). Elements of Mp ( (cid:82) ) can be identified with pairs ( g , ω ),where g ∈ SL ( (cid:82) ) and ω : (cid:72) → (cid:67) is a holomorphic square root of τ (cid:55)→ c τ + d . The action of SL ( (cid:82) ) on (cid:72) gives rise to one of Mp ( (cid:82) ) by the projection ( g , ω ) (cid:55)→ g .Fix the Heisenberg group 0 → (cid:82) → H( V ) → V ( (cid:82) ) → λ , µ , κ ) · ( λ , µ , κ ) = (cid:161) λ + λ , µ + µ , κ + κ + 〈 λ , µ 〉 − 〈 λ , µ 〉 (cid:162) (1.3)for λ , λ , µ , µ ∈ V ( (cid:82) ) and κ , κ ∈ (cid:82) . It carries a right action of SL ( (cid:82) ) and Mp ( (cid:82) ) by means of thestandard representation of SL ( (cid:82) ) on the second factor of V ( (cid:82) ) ∼= V ( (cid:82) ) ⊗ (cid:82) .The real Jacobi group G J ( V ) attached to V is the extension0 −→ H( V ) −→ G J ( V ) −→ Mp ( (cid:82) ) −→ – – ndefinite Theta Series on Cones M. Westerholt-Raum with multiplication (( g , ω ), h ) · (( g , ω ), h ) = (cid:161) ( g , ω )( g , ω ), h g + h (cid:162) . (1.4)The full Jacobi Group Γ J ( L ) is the discrete subgroup of G J ( V ) generated by Mp ( (cid:90) ) and L ⊂ H( V ).We usually denoted its elements by γ J , identifying them with ( γ , λ , µ , κ ) ( γ ∈ Mp ( (cid:90) ), λ , µ ∈ L , κ ∈ (cid:90) )according to our needs. Assume that L is integral. The Weil representation ρ L of L is a representa-tion of Mp ( (cid:90) ) with representation space V ( ρ L ) : = span (cid:67) { e l : l ∈ L ∨ / L }. (1.5)On the generators T : = (cid:161) (cid:161) (cid:162) , 1 (cid:162) and S : = (cid:161) (cid:161) −
11 0 (cid:162) , (cid:112) τ (cid:162) of Mp ( (cid:90) ), where (cid:112) τ is the principal branch ofthe holomorphic square root, it is given by ρ L ( T ) e l : = exp (cid:161) π i q ( l ) (cid:162) e l and ρ L ( S ) e l : = σ ( L ) (cid:112) L ∨ / L (cid:88) l (cid:48) ∈ L ∨ / L exp (cid:161) − π i 〈 l , l (cid:48) 〉 (cid:162) e l (cid:48) ,where σ ( L ) : = (cid:112) L ∨ / L (cid:88) l ∈ L ∨ / L exp (cid:161) − π i q ( l ) (cid:162) .The Weil representation extends to a representation of Γ J ( L ) via the projection Γ J ( L ) (cid:16) Mp ( (cid:90) ). Assume that L is integral. Let φ : (cid:72) J ( V ) → V ( ρ L ) be a real-analytic function withpossible singularities. We call φ a real-analytic, weak Jacobi form of weight k , index L , and type ρ L iffor γ J ∈ Γ J ( L ) we have φ (cid:161) a τ + bc τ + d , z + λτ + µ c τ + d (cid:162) = ρ L ( γ J ) ω ( τ ) k exp (cid:161) − π i (cid:161) − c q ( z + λτ + µ ) c τ + d + q ( λ ) τ + 〈 z , λ 〉 (cid:162)(cid:162) φ ( τ , z ).Observe that is suffices to verify that φ ( τ + z ) = ρ L ( T ) φ ( τ , z ), φ ( τ , z + = φ ( τ , z ), and φ (cid:161) − τ , z τ (cid:162) = ρ L ( S ) (cid:112) τ k exp (cid:161) π i q ( z )/ τ (cid:162) φ ( τ , z ),in order to check the transformation behavior of φ . We write E = 〈 v , ∂ v 〉 and ∆ = 〈 ∂ v , ∂ v 〉 for the Euler and Laplace dif-ferential operators. Note that ∂ v must be viewed as an element of the dual V ∨ of V . In other word,in coordinates v i of v , we have E = (cid:80) i v i ∂ v i and ∆ = (cid:80) i , j ( m − ) i , j ∂ v i ∂ v j , where m is the Gram matrixassociated with q .Vignéras proved a stronger version of the following theorem. It will serve as a major tool for ourmodularity theorem. We reformulate it slightly, to include the vector valued case, which allows us towork with the full modular group throughout. – – ndefinite Theta Series on Cones M. Westerholt-Raum
Theorem 1.1 (cf. Vignéras [Vig77]).
Assume that q is non-degenerate of signature ( d + , d − ) . Given afunction h : V ( (cid:82) ) → (cid:67) such that v (cid:55)→ h ( v ) exp( − π q ( v )) is a Schwartz function and such that we have ∆ h /4 π = ( E − k ) h (1.6) for some integer k , then the theta series θ L ( h ; τ , z ) : = y − k (cid:88) l ∈ L ∨ e l h (cid:161) (cid:112) y ( l + vy ) (cid:162) exp (cid:161) π i ( q ( l ) τ + 〈 z , l 〉 ) (cid:162) converges locally absolutely and uniformly, and is a real-analytic Jacobi form of weight k + d + + d − , in-dex L, and type ρ L . Fix a negative definite subspace U ⊆ V ( (cid:82) ), and let B D (0) ⊂ U be the ballof radius D with respect to − q . To examine the difference of our approximation (cid:100) sgn E and the sign-product sgn E in Section 3, we need the follow estimate. Lemma 1.2.
Let U be a quadratic space with quadratic form q that is negative definite. If dv (cid:48) is themeasure associated to − q, then for every D ≥ , we have (cid:90) U \ B D (0) exp(4 π q ( v (cid:48) )) dv (cid:48) (cid:46) (cid:90) U exp(4 π q ( v (cid:48) )) dv (cid:48) (cid:191) D d − exp( − π D ) .Proof. Fix coordinates v , . . . , v d of U . Let S d − ( r ) be the ( d − r . Reducing ourselvesby a transformation in GL( U ) to the case q ( v ) = v + · · · + v n , we obtain (cid:90) U \ B D (0) exp(4 π q ( v (cid:48) )) dv (cid:48) = (cid:90) ∞ D exp( − π r ) vol( S d − ( r )) dr = π d − Γ ( d − ) (cid:90) ∞ D r d − exp( − π r ) dr = − d (cid:112) π Γ ( d − ) (cid:90) ∞ π D r d − exp( − r ) dr = − d (cid:112) π Γ ( d − ) Γ ( d , 4 π D ) (cid:191) D d − exp( − π D ). Given C ⊆ (cid:80) ( V ( (cid:82) )), we call C ( C ) : = (cid:169) v ∈ V ( (cid:82) ) : [ v ] ∈ C (cid:170) the cone associated to C . We say that C ( C ) is positive, if C ⊂ [ V ( (cid:82) ) + ]. If C ⊆ [ V ( (cid:82) ) \ V ( (cid:82) ) − ], we say thatit is non-negative.The following theorem is the most elementary convergence theorem for theta series in this paper.We state it separately in order to recurse to it in the proof of the more subtle Theorem 2.7. Theorem 2.1.
Given a closed set C ⊆ [ V ( (cid:82) ) + ] , the theta series θ L ( C ( C )) converges locally absolutely anduniformly. – – ndefinite Theta Series on Cones M. Westerholt-Raum
Proof.
Let (cid:107) · (cid:107) be an auxiliary euclidean norm on V ( (cid:82) ). Since C is closed and (cid:80) ( V ( (cid:82) )) is compact thereis a minimum m = min (cid:169) q ( v ) : v ∈ V ( (cid:82) ), (cid:107) v (cid:107) =
1, [ v ] ∈ C (cid:170) ,which is positive. We set (cid:101) q ( v ) = m (cid:107) v (cid:107) , which is a positive definite minorant to q on C ( C ). That is, wehave q ( v ) > m (cid:107) v (cid:107) for all v ∈ C ( C ). In particular, we have (cid:175)(cid:175)(cid:175) θ L ( C ( C ); τ , z ) (cid:175)(cid:175)(cid:175) ≤ (cid:88) l ∈ L ∩ ( C ( C ) − vy ) exp( − π y q ( l + vy )) ≤ (cid:88) l ∈ ( L + vy ) ∩ C ( C ) exp( − π y (cid:101) q ( l )) ≤ (cid:88) v ∈ L + vy exp( − π y (cid:101) q ( l )).The right hand side is a theta series associated with a positive definite lattice and therefore converges. If C ⊆ (cid:80) ( V ( (cid:82) )) is given as C ( W ) : = (cid:169) [ v ] ∈ (cid:80) ( V ( (cid:82) )) : ∀ w ∈ W : 〈 v , w 〉 ≥ (cid:170) for a finite set W ⊂ V ( (cid:82) ), we call C ( W ) : = C ( C ( W )) the polyhedral cone with walls W . A polyhedralcone is said to be non-degenerate if it has inner points. Throughout this paper we call a cone C ( W )with W = + d − a tetrahedral cone and the corresponding W a tetrahedral set of walls. A set of walls W is called cubical, if W allows for a partition S into d − pairs such thatinterior( C ( W )) = (cid:169) [ v ] ∈ (cid:80) ( V ( (cid:82) )) : ∀ ( w , w (cid:48) ) ∈ S : 〈 v , w 〉 = 〈 v , w (cid:48) 〉 (cid:170) .The associated cones are called cubical. Lemma 2.2.
Given a set of walls W such that C ( W ) is non-degenerate and non-negative. Then for everyE ⊆ W with E > d − we have V ( (cid:82) ) ∩ C ( W ) ∩ E ⊥ = (cid:59) .Proof. The case of d − = d − >
0. Suppose that V ( (cid:82) ) ∩ C ( W ) ∩ E ⊥ wasnot empty and contained a vector e . Choose a positive definite subspace V (cid:48) ⊂ V of dimension d + − e . Replacing V by the orthogonal complement of V (cid:48) , we can and will reduce ourconsiderations to the case that d + = v + ∈ V . The cross section A ∩ V ( (cid:82) ) of the cone V ( (cid:82) ) with respect to the affinesubspace A = { v ∈ V : 〈 v , v + 〉 =
1} generates V ( (cid:82) ) . Since C ( W ) is positive, the analogue holds for itscross section A ∩ C ( W ). In particular, we may assume that e ∈ A .Since d + = d − ≥
1, and C ( W ) is polyhedral and non-negative, the cross section A ∩ C ( W ) is acompact Euclidean polyhedron in A . There is a tessellation of A ∩ C ( W ) by tetrahedrons. Replacing A ∩ C ( W ) with a tetrahedron containing e , we can thus focus on the case of tetrahedral cones. Inparticular, we can work with the stronger assumption that e ∈ V ( (cid:82) ) ∩ W ⊥ .Since C ( W ) has inner points, there is v ∈ V ( (cid:82) ) with 〈 v , w 〉 > w ∈ W . There exists some (cid:101) v (cid:54)∈ e ⊥ ,and by adding a sufficiently small multiple of it to v , we can achieve that 〈 v , w 〉 > w ∈ W and 〈 v , e 〉 (cid:54)=
0. Then for all t ∈ (cid:82) , we have v + t e ∈ C ( W ), since e ∈ W ⊥ . On the other hand, we find that q ( v + t e ) = q ( v ) + t 〈 v , e 〉 + t q ( e ) = q ( v ) + t 〈 v , e 〉 .For some t , we have q ( v + t e ) < C ( W ) is positive. – – ndefinite Theta Series on Cones M. Westerholt-Raum
For a tetrahedral set W , we call E ( W ) = (cid:169) E ⊆ W : E = d − (cid:170) the set of its edges. The orthogonalcomplements of the E ∈ E ( W ) yield the edges, i.e. the lowest dimensional faces, of the cone C ( W ). Wecall E ( W ) : = (cid:169) E ∈ E ( W ) : E ⊥ ∩ V ( (cid:82) ) (cid:54)= {0} (cid:170) the set of isotropic edges of W . In the next two Lemmas, we split up C ( W ) into a closed positive setand neighborhoods of isotropic edges. Lemma 2.3.
For every polyhedral cone C ( W ) that is non-degenerate and non-negative, we have C ( W ) ∩ V ( (cid:82) ) = (cid:91) E ∈ E ( W ) E ⊥ = (cid:91) E ∈ E ( W ) (cid:169) v ∈ V ( (cid:82) ) : ∀ w ∈ E : 〈 v , w 〉 = (cid:170) .Proof. As in the proof of Lemma 2.2, it suffices to consider the case of tetrahedral cones and d + = v ∈ C ( W ) ∩ V ( (cid:82) ) and let E be a maximal subset of W such that v ∈ E ⊥ . We have E (cid:54)= W byLemma 2.2. Suppose that E < d − . Then E ⊥ contains at least one vector v (cid:48) with q ( v (cid:48) ) <
0. Observethat 〈 v , w 〉 > w ∈ W \ E . For t ∈ (cid:82) with | t | sufficiently small, we therefore have v + t v (cid:48) ∈ C ( W ).On the other hand, q ( v + t v (cid:48) ) = q ( v ) + t 〈 v , v (cid:48) 〉 + t q ( v (cid:48) ) = t 〈 v , v (cid:48) 〉 + t q ( v (cid:48) ).If t (cid:54)=
0, sgn t (cid:54)= sgn 〈 v , v (cid:48) 〉 , and | t | is small enough, then this is negative, contradicting positivity of C ( W ).Therefore E = d − as claimed.Given E ∈ E ( W ) fix some auxiliary euclidean norm (cid:107) · (cid:107) on V ( (cid:82) ). With this norm and for (cid:178) >
0, let E ( (cid:178) ) : = E ( (cid:107) · (cid:107) , (cid:178) ) : = (cid:169) [ v ] ∈ (cid:80) ( V ( (cid:82) )) : (cid:107) v (cid:107) = ∀ w ∈ E : 0 ≤ 〈 v , w 〉 < (cid:178) (cid:170) . (2.1)Notice that we suppress the norm from notation, since for any other choice (cid:107) · (cid:107) (cid:48) we have E ( (cid:107) · (cid:107) (cid:48) , (cid:178) (cid:48) ) ⊆ E ( (cid:107) · (cid:107) , (cid:178) ) if (cid:178) (cid:48) is small enough. Indeed, V ( (cid:82) ) is finite dimensional so that all norms are equivalent toeach other. We remark that E ( (cid:178) ) can also be understood in terms of cross sections as at the beginningof the proof of Lemma 2.2. Lemma 2.4.
Given a non-degenerate and non-negative polyhedral cone C ( W ) , for every (cid:178) > ,C ( W ) \ (cid:91) E ∈ E ( W ) E ( (cid:178) ) ⊆ (cid:80) ( V ( (cid:82) )) is closed and positive.Proof. Consider the set in the statement. It is closed since E ( (cid:178) ) is open in C ( W ). It contains no isotropicpoints by Lemma 2.3, so it is positive. – – ndefinite Theta Series on Cones M. Westerholt-Raum
Consider a non-degenerate and non-negativecone C ( W ). For any E ∈ E ( W ) , write E ( (cid:82) ) : = span (cid:82) E and E ( (cid:81) ) = E ( (cid:82) ) ∩ V . Write E ( (cid:81) ) for the maximaltotally isotropic subspace of V ( (cid:81) ) ∩ E ( (cid:82) ), which is unique as E ( (cid:82) ) contains no negative vectors. Recallthat the Witt index of a quadratic space is the dimension of a maximal isotropic subspace. We saythat E ∈ E ( W ) is a rational edge if the Witt index of E ( (cid:82) ) and the dimension of E ( (cid:81) ) agree. Thenext lemma says that we can find neighborhoods of the rational isotropic edges of C ( W ) in term ofpolyhedral cones. Lemma 2.5.
Fix a polyhedral, non-degenerate, and non-negative cone C ( W ) and an edge E ∈ E ( W ) that is rational. Let E ( (cid:81) ) − ⊂ E ( (cid:81) ) be a negative definite subspace. There is a set E (cid:48) ⊂ ( E ( (cid:81) ) − ) ⊥ ∩ V ( (cid:81) ) of dim E ( (cid:81) ) linearly independent vectors such that span E (cid:48) is negative definite and C ( E ∩ E (cid:48) ) ⊇ E ( (cid:178) ) forsome (cid:178) > .Proof. Without loss of generality, we can assume that E ( (cid:81) ) − is trivial by projecting to its orthogonalcomplement. We can further assume that E ( (cid:81) ) = E ( (cid:81) ) by projecting to the orthogonal complementof a maximal negative definite and rational subspace of E ( (cid:82) ). By that orthogonal projection E ( (cid:81) ) ismapped to a non-negative subspace of V ( (cid:81) ). This suffices as an assumption on E ( (cid:81) ) in the followingargument.Set n : = dim E ( (cid:81) ) , and observe that we have d + , d − ≥ n . Therefore there is a negative definite, n -dimensional subspace W ⊂ V ( (cid:81) ) such that the pairing 〈 · , · 〉 between E ( (cid:81) ) and W is non-degenerate.Fix a rational basis v , . . . , v n of E ( (cid:81) ) and a dual basis w , . . . , w n of W . Set e (cid:48) i = e (cid:48) i ( t ) = t w i − v i for1 ≤ i ≤ n and t >
0. There is a t > ≤ i ≤ d , we have q ( e (cid:48) i ) = t q ( w i ) − t <
0, and thematrix (cid:161) 〈 e (cid:48) i , e (cid:48) j 〉 (cid:162) i , j is negative definite. Fix such a t , and observe that 〈 v j , e (cid:48) i 〉 ≥ ≤ i , j ≤ n . Thatis, E (cid:48) : = (cid:169) t w i − v i : 1 ≤ i ≤ n (cid:170) is a valid choice of E (cid:48) .To make full use of Lemma 2.5, we construct a well-behaved covering of E ( (cid:178) ) by suitable cones C ( E (cid:48) i ). Lemma 2.6.
Let C ( W ) be a polyhedral non-degenerate and non-negative cone. Fix E ∈ E ( W ) that isrational. Then there is a finite collection of finite sets E (cid:48) i ⊂ V ( (cid:82) ) such that for all (cid:178) i > we haveE ( (cid:178) ) ⊂ (cid:91) E (cid:48) i ( (cid:178) i ) for some (cid:178) > and the E (cid:48) i satisfy:(i) We have E (cid:48) i = dim E (cid:48) i ( (cid:81) ) = d − .(ii) We have E (cid:48) i ( (cid:81) ) = E ( (cid:81) ) .(iii) The E (cid:48) i ( (cid:178) i ) are non-negative for sufficiently small (cid:178) i > .(iv) There is E (cid:48)− i ⊆ E (cid:48) i ∩ V ( (cid:81) ) such that span E (cid:48)− i is negative definite and E (cid:48) i ( (cid:81) ) = E ( (cid:81) ) + span E (cid:48)− i .Proof. We may replace V ( (cid:81) ) by ( E ( (cid:81) ) ) ⊥ / E ( (cid:81) ) , and thus assume that E ( (cid:81) ) = {0}.Choose (cid:178) > E ( (cid:178) ) is positive. We consider E as a point in the space of maximal,framed, negative definite subspaces in V ( (cid:82) ). Fix a closed neighborhood U of E ( (cid:178) ) in [ V ( (cid:82) ) + ]. – – ndefinite Theta Series on Cones M. Westerholt-Raum
For any [ v ] ∈ [ V ( (cid:82) ) + ], there is one maximal, framed, negative definite subspace E (cid:48) and (cid:178) (cid:48) > E (cid:48) ( (cid:178) (cid:48) ) ⊆ V ( (cid:82) ) + such that v ∈ E (cid:48) ( (cid:178) (cid:48) ). Rational points are dense in the space of maximal, framed, negativedefinite subspaces of V ( (cid:82) ). In particular, the compact set U can be covered by the countable set ofinteriors of C ( E (cid:48) ) for rational E (cid:48) . Finitely many suffice to cover C ( E ) ∩ U , and the set of these E (cid:48) yieldsthe E (cid:48) i in the statement. Theorem 2.7.
Fix a polyhedral cone C ( W ) that is non-degenerate and non-negative. If every E ∈ E ( W ) is rational, then the theta series θ L ( C ( W )) converges locally absolutely and uniformly on (cid:169) ( τ , z ) ∈ (cid:72) J ( V ) : ∀ E ∈ E ( W ) : (cid:59) = (cid:90) ∩ 〈 L + Im( z )/Im( τ ), L ∩ E ( (cid:81) ) 〉 (cid:170) .Proof. Using notation from (2.1), let C ( (cid:178) ) + = C ( W ) \ (cid:83) E ∈ E ( W ) E ( (cid:178) ) for (cid:178) >
0. To simplify notation, set E ( (cid:178) ) = C ( E ( (cid:178) )) and C ( (cid:178) ) + = C ( C ( (cid:178) ) + ). If (cid:178) is sufficiently small, we can decompose θ L ( C ( W )) as θ L ( C ( W ); τ , z ) = θ L ( C ( (cid:178) ) + ; τ , z ) + (cid:88) E ∈ E ( W ) θ L ( E ( (cid:178) ); τ , z ).Since C ( (cid:178) ) + is positive and closed in V ( (cid:82) ) the first summand converges absolutely and uniformlyon (cid:72) J ( V ) by Theorem 2.1. It suffices to demonstrate convergence of θ L ( E ( (cid:178) )) for each E .We next employ Lemma 2.6 to reduce our consideration to E such that there is a set of rational vec-tors E − ⊂ E with negative definite span E − and E ( (cid:81) ) = span E − + E ( (cid:81) ) . For such E we obtain fromLemma 2.5 an auxiliary set E (cid:48) with span E (cid:48) negative definite, E (cid:48) ⊥ E − , and C ( E ∪ E (cid:48) ) ⊇ E ( (cid:178) ) for suffi-ciently small (cid:178) . The pairing of E ( (cid:81) ) and E (cid:48) ( (cid:81) ) is non-degenerate. We may therefore replace L by L ∩ E (cid:48) ( (cid:81) ) ⊥ ⊕ L ∩ E ( (cid:81) ) ,which has finite index in L . In this situation, it suffices to show convergence ofexp(2 π q (Im( v ))) (cid:175)(cid:175)(cid:175) θ L ( C ( E ∪ E (cid:48) ); τ , z ) (cid:175)(cid:175)(cid:175) ≤ (cid:88) l ∈ L ∩ E (cid:48) ( (cid:81) ) ⊥ l + Im( z )/Im( τ ) ∈ C ( E − ) (cid:88) l ∈ L ∩ E ( (cid:81) ) l + Im( z )/Im( τ ) ∈ C ( E (cid:48) ) exp (cid:161) − π y ( q ( l + Im( z )) + 〈 l , l + Im( z ) 〉 ) (cid:162) .As in Zwegers’s thesis [Zwe02], for fixed l the sum over l is bounded, since 〈 l , l + v 〉 is bounded awayfrom 0. Then the sum over l converges, since C ( E ) is a positive cone in the orthogonal complementof E (cid:48) . We let sgn be the sign function, with sgn(0) : =
0. Recall from the introduction thatsgn E ( v ) : = (cid:89) w ∈ E sgn 〈 v , w 〉 for any finite subset E of V ( (cid:82) ). Notice that sgn E is symmetric in the elements of E . – – ndefinite Theta Series on Cones M. Westerholt-Raum
Assuming that V is negative definite, the normalized convolution (cid:100) sgn E ( v ) : = (cid:90) V ( (cid:82) ) sgn E ( v (cid:48) ) exp(4 π q ( v (cid:48) − v )) dv (cid:48) (cid:46) (cid:90) V ( (cid:82) ) exp(4 π q ( v (cid:48) )) dv (cid:48) (3.1)is our approximation to sgn E (cf. Proposition 3.4). Notice that (cid:100) sgn E ( v ) it smooth and depends only theprojection of v to span E ⊆ V ( (cid:82) ). Remark 3.1.
Up to a change of coordinates and the associated renormalization this smoothened signis the same as the “higher error function” that appears in Equation (3.44) of [Ale+16] and Equation (4)of [Naz16].If V is negative semi-definite, then V ( (cid:82) ) ⊆ V ( (cid:82) ) is a subspace and V ( (cid:82) )/ V ( (cid:82) ) is a negative definitequadratic space. In this situation, we define (cid:100) sgn E ( v ) : = (cid:100) sgn E − ( π ( v )) sgn E ( v ), where E = { w ∈ E : q ( w ) = E − = { w ∈ E : q ( w ) (cid:54)= π : V ( (cid:82) ) → V ( (cid:82) )/ V ( (cid:82) ) is the canonical projection. For general V and negative semi-definitespan E , we define (cid:100) sgn E ( v ) : = (cid:100) sgn E ( v E ), (3.2)where the right hand side is computed as with respect to the quadratic space span E . To give a decomposition of (cid:100) sgn E that exhibits its asymptotic be-havior in a precise way, we employ a smoothened version of a volume function. Assuming that V isnegative definite, for any finite, non-empty set E ⊂ V ( (cid:82) ), we let S E ( v ) : = (cid:169) v (cid:48) ∈ V ( (cid:82) ) : ∀ w ∈ E : sgn 〈 v (cid:48) , w 〉 (cid:54)= sgn 〈 v , w 〉 (cid:170) (3.3)be the sector in V ( (cid:82) ) that is opposite to v with respect to the walls corresponding to E . Denote itsvolume by vol E ( v ) : = vol (cid:161) S E ( v ) (cid:162) , (3.4)where the volume is taken with respect to the quadratic form − q on V ( (cid:82) ). Our smoothening (cid:99) vol E ( v ) : = (cid:90) S E ( v ) exp(4 π q ( v (cid:48) − v )) dv (cid:48) (cid:46) (cid:90) V ( (cid:82) ) exp(4 π q ( v (cid:48) )) dv (cid:48) depends only on the geometry with respect to span E . If E = (cid:59) , we let vol E ( v ) = (cid:99) vol E ( v ) = Proposition 3.2.
For every v ∈ V ( (cid:82) ) , we have vol E ( v ) = vol E ( v E ) .Proof. From its definition, we see that S E ( v ) = S E ( v E ). When applying an element of g ∈ GL( E ⊥ ) to v (cid:48) , the normalizing factor det( g ) emerges in both the numerator and the denominator of the definingexpression for (cid:99) vol E ( v ), and thus cancels. – – ndefinite Theta Series on Cones M. Westerholt-Raum
The asymptotic properties of (cid:99) vol E are described in the next statement. Proposition 3.3.
For E (cid:54)= (cid:59) , we have (cid:99) vol E ( v ) (cid:191) D d − exp (cid:161) − π D (cid:162) , D : = dist( v , E ⊥ ) .Proof. We observe that the ball B D ( v ) does not intersect S E ( v ). Therefore, it suffices to estimate (cid:90) V ( (cid:82) )\ B D ( v ) exp(4 π q ( v (cid:48) − v )) dv (cid:48) (cid:46) (cid:90) V ( (cid:82) ) exp(4 π q ( v (cid:48) )) dv (cid:48) = (cid:90) V ( (cid:82) )\ B D (0) exp(4 π q ( v (cid:48) )) dv (cid:48) (cid:46) (cid:90) V ( (cid:82) ) exp(4 π q ( v (cid:48) )) dv (cid:48) .This is achieved in Lemma 1.2. (cid:100) sgn E We write the smoothened sign function as a linear composition of vol-ume functions.
Proposition 3.4.
If V is negative definite, we have (cid:100) sgn E ( v ) = (cid:88) E (cid:48) ⊆ E E (cid:48) sgn E \ E (cid:48) ( v ) (cid:99) vol E (cid:48) ( v ) .Proof. We refine the sectors S E (cid:48) ( v ) to S EE (cid:48) ( v ) : = (cid:169) v (cid:48) ∈ V ( (cid:82) ) : ∀ w ∈ E (cid:48) : sgn 〈 v (cid:48) , w 〉 (cid:54)= sgn 〈 v , w 〉 , ∀ w ∈ E \ E (cid:48) : sgn 〈 v (cid:48) , w 〉 = sgn 〈 v , w 〉 (cid:170) , (3.5)the difference being that we now enforce the sign of 〈 v (cid:48) , w 〉 also for w (cid:54)∈ E (cid:48) . Set (cid:99) vol EE (cid:48) ( v ) = (cid:90) S EE (cid:48)(cid:48) ( v ) exp(4 π q ( v (cid:48) − v )) dv (cid:48) (cid:46) (cid:90) V ( (cid:82) ) exp(4 π q ( v (cid:48) )) dv (cid:48) ,and notice that (cid:99) vol E (cid:48) ( v ) = (cid:88) E (cid:48) ⊆ E (cid:48)(cid:48) ⊆ E (cid:99) vol EE (cid:48)(cid:48) ( v ). (3.6)Define a n ( m ) : = a n − − (cid:195) mn − (cid:33) a n − ( n − m ≥ n ≥ a ( m ) = ( − m . (3.7)Notice that a m ( m ) = ( − m , which can be proved by induction. Indeed, we have a m ( m ) = ( − m − m − (cid:88) n = (cid:195) mn (cid:33) a n ( n ) = ( − m − m − (cid:88) n = ( − n = ( − m − ( − m + ( − m .Fix v and assume that sgn E ( v ) >
0. We will show by induction that for every 0 ≤ n ≤ + d − , we have (cid:100) sgn E ( v ) = (cid:88) E (cid:48) ⊆ E E (cid:48) < n ( − E (cid:48) (cid:99) vol E (cid:48) ( v ) + (cid:88) E (cid:48) ⊆ E E (cid:48) ≥ n a n ( E (cid:48) ) (cid:99) vol EE (cid:48) ( v ). (3.8) – – ndefinite Theta Series on Cones M. Westerholt-Raum If n =
0, then (3.8) is true by the definitions of S EE (cid:48) and a ( m ). We assume that it is true for n and showthat it holds for n (cid:32) n +
1. Rewrite the contribution of E (cid:48) with E (cid:48) = n to the second summand of (3.8): a n ( n ) (cid:99) vol EE (cid:48) ( v ) = a n ( n ) (cid:99) vol E (cid:48) ( v ) − a n ( n ) (cid:88) E (cid:48) (cid:40) E (cid:48)(cid:48) ⊆ E (cid:99) vol EE (cid:48)(cid:48) ( v ).After inserting this into (3.8), it is straightforward to compare the arising contributions to (3.7).The case of sgn E ( v ) < E ( v ) =
0, observe that contribu-tions of (cid:99) vol E (cid:48) vanish due to symmetry for E (cid:48) that satisfy 〈 v , w 〉 = w ∈ E \ E (cid:48) . (cid:100) sgn E in families We consider the behavior of (cid:100) sgn E ( t ) for families E ( t ), t ∈ [0, ∞ ).We call such families negative if E ( t ) = dim span E ( t ) is constant on (0, ∞ ) and span E ( t ) is negativedefinite for all t ∈ (0, ∞ ). Proposition 3.5.
Assume that E ( t ) = E ∪ { w ( t )} is a negative family. If q ( w ( t )) → as t → and v (cid:54)∈ w (0) ⊥ , then locally around v, we have (cid:99) vol E ( t ) ( v ) (cid:191) exp (cid:161) π 〈 w ( t ), v 〉 q ( w ( t )) (cid:162) → as t → .Proof. This follows when estimating dist( v , w ⊥ ) , w ∈ E ( t ) by q (cid:179) 〈 w ( t ), v 〉 q ( w ( t )) w ( t ) (cid:180) = 〈 w ( t ), v 〉 q ( w ( t )) ,and then applying Proposition 3.3. Corollary 3.6.
With assumptions as in Proposition 3.5, we have locally around v | (cid:100) sgn E ( t ) ( v ) − (cid:100) sgn E (0) ( v ) | (cid:191) exp (cid:161) π 〈 w ( t ), v 〉 q ( w ( t )) (cid:162) as t → . Proposition 3.7.
Assume that E ( t ) = E − ∪ E ( t ) is a negative family, where the Witt index of E (0)( (cid:82) ) equals E ( t ) . Then (cid:100) sgn E ( t ) is a (one-side) differentiable family of smooth functions.Proof. This can be directly seen from the decomposition in Proposition 3.4.
We verify Vignéras’s differential equation.
Lemma 3.8.
Assuming that V ( (cid:82) ) is negative definite, we have (cid:161) π E − ∆ (cid:162) (cid:100) sgn E = .Proof. We consider the defining integral of (cid:100) sgn E . The integrand decays exponentially, so we can dif-ferentiate it directly. This shows that the differential (cid:161) π E − ∆ (cid:162) (cid:100) sgn E is the integral over a Schwartzfunction. Now, write V ( (cid:82) ) as the union of U (cid:178) and V ( (cid:82) ) \ U (cid:178) for an (cid:178) -neighborhood (with respect to thedistance coming from q ) U (cid:178) of { v ∈ V : ∃ w ∈ E : 〈 v , w 〉 = V \ U (cid:178) and sgn E is locally constant on there. We therefore see that (2 π E − ∆ ) (cid:100) sgn E ( v ) equals (cid:90) U (cid:178) sgn( v (cid:48) ) (cid:161) π E − ∆ (cid:162) exp(4 π q ( v (cid:48) − v )) dv (cid:48) + (cid:90) V ( (cid:82) )\( U (cid:178) − v ) (cid:161) π E − ∆ (cid:162) sgn( v (cid:48) + v ) exp(4 π q ( v (cid:48) )) dv (cid:48) .As (cid:178) →
0, this tends to 0, proving the statement. – – ndefinite Theta Series on Cones M. Westerholt-Raum
Recall from the introduction that we say that a poly-hedral cone C = C ( W ) is determined by its face indicators if there is a linear combination sgn + C ofsgn E for E ⊂ V ( (cid:82) ) with E ≤ d − such thatinterior( C ) ⊆ support(sgn + C ) ⊆ closure( C ) and sgn + C | interior( C ) = ± E as above.If C is a cone that is determined by face indicators, we set (cid:100) sgn + C ( v ) = (cid:88) E a E (cid:100) sgn E ( v ), where sgn + C ( v ) = (cid:88) E a E sgn E ( v ), a E ∈ (cid:67) . (3.9) Proposition 3.9.
Cubical cones are determined by their face indicators.Proof.
Let C = C ( W ) be a cubical cone. By definition there is a partition S of W into d − pairs suchthat sgn + C ( v ) = − W /2 (cid:89) ( w , w (cid:48) ) ∈ S (cid:161) sgn 〈 v , w 〉 + sgn 〈 v , w (cid:48) 〉 (cid:162) satisfies the desired properties. Proposition 3.10.
Tetrahedral cones are determined by their face indicators.Proof.
Given a tetrahedral cone C = C ( W ), define (cid:103) sgn + C ( v ) : = (cid:89) w , w (cid:48) ∈ Ww (cid:54)= w (cid:48) (cid:161) sgn 〈 v , w 〉 + sgn 〈 v , w (cid:48) 〉 (cid:162) · (cid:161) ( W − W (cid:162) − , if W is odd; W − (cid:81) sgn 〈 v , w 〉 , if W is even. (3.10)Normalization is chosen in such a way that (cid:103) sgn + C ( v ) = ± C ( W ).Let R = (cid:67) [ s w : w ∈ W ] be the polynomial ring in formal variables indexed by W and I = R ( s w : w ∈ W )be an ideal. There is a unique polynomial (cid:101) p ∈ R such that (cid:103) sgn + C ( v ) = (cid:101) p (cid:161) s w → sgn 〈 v , w 〉 (cid:162) .Further, there is a unique lift p ∈ R of p (mod I ) to R that is linear in each of of the variables. We setsgn + C ( v ) = p (cid:161) s w → sgn 〈 v , w 〉 (cid:162) .Checking the total degree of p , we conclude that sgn + C is a linear combination of functions sgn E for E ⊂ V ( (cid:82) ) of size at most d − .We have to verify the support of sgn + C to finish the proof. Observe that sgn + C coincides with (cid:103) sgn + C with possible exceptions on ∪ w w ⊥ . Given v , let E = { w ∈ W : 〈 v , w 〉 = – – ndefinite Theta Series on Cones M. Westerholt-Raum contributions to sgn + C which do not involve sgn 〈 v , w 〉 for w ∈ E . Collecting these terms in the definitionof (cid:103) sgn + C , we see that (cid:103) sgn + C ( v ) : = (cid:89) w , w (cid:48) ∈ Ww ∈ E ∨ w (cid:48) ∈ Ew (cid:54)= w (cid:48) (cid:161) sgn 〈 v , w 〉 + sgn 〈 v , w (cid:48) 〉 (cid:162)(cid:89) w , w (cid:48) ∈ W \ Ew (cid:54)= w (cid:48) (cid:161) sgn 〈 v , w 〉 + sgn 〈 v , w (cid:48) 〉 (cid:162) · (cid:161) ( W − W (cid:162) − , if W is odd; W − (cid:81) sgn 〈 v , w 〉 , if W is even.If the second factor does not vanish, then v ∈ C ( W ) ∩ E ⊥ . The value of the corresponding product in R / I remains the same, since sgn 〈 v , w 〉 = w (cid:54)∈ E . This shows that v ∈ E ⊥ ∩ C ( W \ E ) ⊆ C ( W ). In this section, we drop notation v for elements in V ( (cid:82) ), and write z = u + i v instead. For polyhedralcones C that are determined by face indicators, we set θ L (sgn + C ; τ , z ) = (cid:88) l ∈ L ∨ e l sgn + C (cid:161) (cid:112) y ( l + vy ) (cid:162) exp (cid:161) π i ( q ( l ) τ + 〈 z , l 〉 ) (cid:162) . (4.1)As a direct consequence of Theorem 2.7, we find: Corollary 4.1.
Let C = C ( W ) be a polyhedral cone that is non-degenerate, non-negative, and deter-mined by face indicators. If every E ∈ E ( W ) is rational, then the theta series θ L (sgn + W ) converges locallyabsolutely and uniformly on (cid:169) ( τ , z ) ∈ (cid:72) J ( V ) : ∀ E ∈ E ( W ) : (cid:59) = (cid:90) ∩ 〈 L ∨ + vy , L ∨ ∩ E ( (cid:81) ) 〉 (cid:170) . For cones determined by their face indicators a completed version of θ L (sgn + C ) arises from (cid:100) sgn + C : θ L ( (cid:100) sgn + C ; τ , z ) : = (cid:88) l ∈ L ∨ e l (cid:100) sgn + C (cid:161) (cid:112) y ( l + vy ) (cid:162) exp (cid:161) π i ( q ( l ) τ + 〈 z , l 〉 ) (cid:162) . (4.2) Theorem 4.2.
Let C = C ( W ) be a non-degenerate, non-negative tetrahedral or cubical cone such thatevery E ∈ E ( W ) is rational. Then θ L ( (cid:100) sgn + C τ , z ) with the choice of sgn + C as in Propositions 3.9 and 3.10converges locally absolutely on (cid:169) ( τ , z ) ∈ (cid:72) J ( V ) : ∀ E ∈ E ( W ) : (cid:59) = (cid:90) ∩ 〈 L ∨ + vy , L ∨ ∩ E ( (cid:81) ) 〉 (cid:170) .It is a real-analytic Jacobi form with meromorphic singularities of weight ( d + + d − )/2 , index L, andtype ρ L .Proof. We first establish convergence. To see that (4.2) converges locally absolutely, use the decom-position of (cid:100) sgn E in Proposition 3.4 and the asymptotic behavior of (cid:99) vol E (cid:48) in Proposition 3.3. In the – – ndefinite Theta Series on Cones M. Westerholt-Raum tetrahedral case, fix E (cid:48) ⊂ W with E (cid:48) < W . Such E (cid:48) index all possible (cid:99) vol E (cid:48) terms in (cid:100) sgn + W , w . Thecontribution to (4.2) is (cid:88) l ∈ L ∨ e l (cid:88) E ∈ E ( W ) p E (cid:48) ( l + vy ) sgn + E \ E (cid:48) ( l + vy ) (cid:99) vol E (cid:48) (cid:161) (cid:112) y ( l + vy ) (cid:162) exp (cid:161) π i ( q ( l ) τ + 〈 z , l 〉 ) (cid:162) ,where p E (cid:48) ( l ) is a polynomial in sgn 〈 v , w 〉 , w ∈ E (cid:48) and sgn + E \ E (cid:48) is the indicator for the tetrahedral conespanned by E \ E (cid:48) . After using the asymptotic behavior of (cid:99) vol E (cid:48) to obtain a partial majorant of q , wecan invoke Corollary 4.1. Convergence in the cubical case follows along the same lines.As for modularity, in the case of positive C ( W ), Vignéras’s modularity theorem suffices, when usingLemma 3.8, because (cid:100) sgn + C ( v ) exp( − π q ( v )) is a Schwartz function on V ( (cid:82) ). In the general case, weexpress θ L ( (cid:100) sgn + C ) as a uniform limit of modular theta series: We first treat the case of isotropic vectorsin E . This step is completely analogous to [Zwe02]. We let E ( t ) be a linear approximation of E (0). Thenthe remainder terms in Corollary 3.6 are sufficient to obtain uniform convergence, when decomposing V ( (cid:82) ) as in the proof of Proposition 2.7 in [Zwe02]. To also cover the case of E ( (cid:81) ) that is not spannedby isotropic vectors in E , we can use Proposition 3.7 directly. [Ale+16] S. Alexandrov, S. Banerjee, J. Manschot,and B. Pioline. Indefinite Theta Series andGeneralized Error Functions .arXiv:1606.05495. 2016.[AMP13] S. Alexandrov, J. Manschot, andB. Pioline. “D3-instantons, mock thetaseries and twistors”.
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