Cone theta functions and what they tell us about the irrationality of spherical polytope volumes Universite de Bordeaux Algorithmic Number Theory Seminar June 2013 Sinai Robins CNRS/LAAS, Toulouse France and NTU, Singapore Joint work with Amanda Folsom and Winfried Kohnen Tuesday, June 18, 2013
Cone K θ vertex Tuesday, June 18, 2013
An angle can be thought of as the measure of the intersection of a cone with a sphere, centered at the vertex of the cone. Cone K θ vertex Tuesday, June 18, 2013
An angle can be thought of as the measure of the intersection of a cone with a sphere, centered at the vertex of the cone. Cone K What is a higher-dim’l angle? θ vertex Tuesday, June 18, 2013
A cone is the non-negative real K ⊂ R d span of a finite number number of vectors in Euclidean space. That is, a cone is defined by: K = { λ 1 W 1 + . . . + λ d W d | all λ j ≥ 0 } where we assume that the vectors W 1 , . . . , W d are linearly independent in . R d Tuesday, June 18, 2013
Example: a 3-dimensional cone. v Cone K Tuesday, June 18, 2013
How do we describe an angle analytically in higher dimensions? Tuesday, June 18, 2013
A nice analytic description of an angle is given by: � e − π ( x 2 + y 2 ) dxdy angle = K The solid angle � e − π || x || 2 dx at the vertex = ω K ( v ) = v of a cone in R d K K Tuesday, June 18, 2013
A solid angle in dimension d is equivalently: The proportion of a sphere, centered at the 1 . vertex of a cone, which intersects the cone. The probability that a randomly chosen point in 2 . Euclidean space, chosen from a fixed sphere centered at the vertex of K, will lie inside K. � e − π ( x 2 + y 2 ) dxdy Solid angle = 3 . K The volume of a spherical polytope. 4 . Tuesday, June 18, 2013
Example: defining the solid angle at a vertex of a 3-dimensional cone. v Cone K v Tuesday, June 18, 2013
Example: defining the solid angle at a vertex of a 3-dimensional cone. sphere centered at vertex v v Cone K v Tuesday, June 18, 2013
Example: defining the solid angle at a vertex of a 3-dimensional cone. sphere centered at vertex v this is a geodesic triangle on the sphere, v representing the solid angle at vertex v. Cone K v Tuesday, June 18, 2013
The moral: a solid angle in higher dimensions is really the volume of a spherical polytope. Tuesday, June 18, 2013
To help us analyze solid angles, we introduced the following cone theta function for a cone K, and a full rank lattice L: Definition. e π i τ || m || 2 , � Φ K, L ( τ ) := m ∈ L ∩ K where τ is in the upper complex half plane. Tuesday, June 18, 2013
Example. For the cone theta function of the positive orthant K 0 := R d ≥ 0 , and with L the integer lattice, we claim that 2 d ( θ ( τ ) + 1) d , 1 Φ K 0 ( τ ) = n ∈ Z e π i τ n 2 , the classical weight 1 / 2 where θ ( τ ) := � modular form. In particular, � d � d 1 � θ k ( τ ) , Φ K 0 ( τ ) = 2 d k =0 k Tuesday, June 18, 2013
There is an analytic link between solid angles and these conic theta functions, given by: Lemma. d 2 Φ K, L ( it ) ∼ ω K t | detK | , as t → 0 + . Tuesday, June 18, 2013
What are tangent cones? Tuesday, June 18, 2013
Example: If the face F is a vertex, what does the tangent cone at the vertex look like? Face = v, a vertex Tuesday, June 18, 2013
Example: If the face F is a vertex, what does the tangent cone at the vertex look like? y 1 Face = v, a vertex Tuesday, June 18, 2013
y 2 y 1 Face = v, a vertex Tuesday, June 18, 2013
y 2 y 1 y 5 Face = v, a vertex Tuesday, June 18, 2013
K F y 3 y 4 y 2 y 1 y 5 Face = v, a vertex The tangent cone is the union K F of all of these rays from the face F = vertex v Tuesday, June 18, 2013
K F Face = v, a vertex The tangent cone is the union K F of all of these rays from the face F = vertex v Tuesday, June 18, 2013
Definition. The tangent cone K F of a face is defined by F ⊂ P K F = { x + λ ( y − x ) | x ∈ F, y ∈ P, and λ ≥ 0 } . Intuitively, the tangent cone of F is the union of all rays that have a base point in F and point ‘towards P’. We note that the tangent cone of F contains the affine span of F . Tuesday, June 18, 2013
F = an edge K F Example. when the face is a 1-dimensional edge F of a polygon, the tangent cone of is a half-plane. F Tuesday, June 18, 2013
F = an edge K F Example. when the face is a 1-dimensional edge F of a polygon, the tangent cone of is a half-plane. F Tuesday, June 18, 2013
Question 1. Which lattice polyhedral cones K give rise to spherical polytopes with a rational volume? Tuesday, June 18, 2013
Question 1. Which lattice polyhedral cones K give rise to spherical polytopes with a rational volume? Question 2. Analyzing the cone theta function Φ K attached to a polyhedral cone K , how ‘close’ is Φ K to being modular? Tuesday, June 18, 2013
For each even integral lattice L , we define its usual theta function by: n ∈ L e π i τ || n || 2 , Θ L ( τ ) := � where τ lies in the upper half plane H . Tuesday, June 18, 2013
For each even integral lattice L , we define its usual theta function by: n ∈ L e π i τ || n || 2 , Θ L ( τ ) := � where τ lies in the upper half plane H . It is a standard fact that the theta function Θ L ( τ ) turns out to be a modular form, of weight d 2 and level N , where N divides | det( A ) | . Tuesday, June 18, 2013
We define R to be the ring of all finite, rational linear combinations of theta functions Θ L , for any d -dimensional even integral lattice L , where we vary over all dimensions d . Tuesday, June 18, 2013
Theorem (Folsom, Kohnen, R.) If the polyhedral cone K is the Weyl chamber of a finite reflection group W , then the cone theta function Φ K, 2 L root ( τ ) is in the graded ring R . Tuesday, June 18, 2013
The spirit of this result is that enough symmetry of the integer cone K will be reflected in some functional relations between the associated cone theta functions Φ K, L j , for various j -dimensional lattices L j which lie on the boundaries of K ∩ L . Tuesday, June 18, 2013
On the other hand, we have the following result, showing that conic theta functions are ‘usually’ very far from being in R. Tuesday, June 18, 2013
Theorem (Folsom, Kohnen, R.) Suppose that the d -dim’l polyhedral cone K has the solid angle ω K at its vertex, located at the origin, and that L := A ( Z d ) is an even integral lattice of full rank. If | detA | is irrational, then Φ K, L ( τ ) is not ω K a modular form of weight k on any congruence subgroup, and for any k ∈ 1 2 Z , k ≥ 1 2 . Tuesday, June 18, 2013
In the 2-dimensional case, we can classify the integer cones that have an irrational angle. As a consequence: Tuesday, June 18, 2013
In the 2-dimensional case, we can classify the integer cones that have an irrational angle. As a consequence: Theorem (Folsom, Kohnen, R.) Suppose we are given an integer cone K ⊂ R 2 , with integer edge vectors w 1 , w 2 ∈ Z 2 . Then Φ K, Z 2 ( τ ) is not a modular form of weight 1 for any congruence subgroup. Tuesday, June 18, 2013
Open Problems Problem 1. What are the necessary and su ffi cient conditions on the geometry of the cones K whose cone theta function belongs to the graded ring R ? Tuesday, June 18, 2013
Open Problems Problem 1. What are the necessary and su ffi cient conditions on the geometry of the cones K whose cone theta function belongs to the graded ring R ? Problem 2. For the case that | detA | ∈ Q , we don’t yet ω K have any proofs of non-modularity for Φ K, L , except in some special two-dim’l cases. Tuesday, June 18, 2013
Open Problems Problem 1. What are the necessary and su ffi cient conditions on the geometry of the cones K whose cone theta function belongs to the graded ring R ? Problem 2. For the case that | detA | ∈ Q , we don’t yet ω K have any proofs of non-modularity for Φ K, L , except in some special two-dim’l cases. Problem 3. Which integer 3-dimensional cones have a rational spherical volume? (This is closely related to the Cheeger-Simons rational simplex conjecture, so it is most likely quite challenging.) Tuesday, June 18, 2013
Thank you Tuesday, June 18, 2013
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