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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 10th 2014
   • (edited Dec 10th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAdded pointers to _[[Selberg zeta function]]_ for the fact that, under suitable conditions over a 3-manifold, the exponentiated eta function $\exp(i \pi \, \eta_D(0))$ equals the Selberg zeta function of odd type. Together with the fact at _[eta invariant -- For manifolds with boundaries](http://ncatlab.org/nlab/show/eta+invariant#OnManifoldsWithBoundary)_ this says that the Selberg zeta function of odd type constitutes something like an Atiyah-style TQFT which assigns determinant lines to surfaces and Selberg zeta functions to 3-manifolds. This brings me back to that notorious issue of whether to think of arithmetic curves as "really" being 2-dimensional or "really" being 3-dimensional: what is actually more like a Dedekind zeta function: the Selberg zeta functions of even type or those of odd type?

   Added pointers to Selberg zeta function for the fact that, under suitable conditions over a 3-manifold, the exponentiated eta function $\exp(i \pi \, \eta_D(0))$ equals the Selberg zeta function of odd type.

   Together with the fact at eta invariant – For manifolds with boundaries this says that the Selberg zeta function of odd type constitutes something like an Atiyah-style TQFT which assigns determinant lines to surfaces and Selberg zeta functions to 3-manifolds.

   This brings me back to that notorious issue of whether to think of arithmetic curves as “really” being 2-dimensional or “really” being 3-dimensional: what is actually more like a Dedekind zeta function: the Selberg zeta functions of even type or those of odd type?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 11th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexJohn Baez kindly points out that the analogy between the Selberg zeta and the Artin L given in the $n$Lab [here](http://ncatlab.org/nlab/show/Artin%20L-function#AnalogyWithSelbergZeta) had been highlighed much in * Darin Brown, _Lifting properties of prime geodesics_, Rocky Mountain J. Math. Volume 39, Number 2 (2009), 437-454 ([euclid](http://projecteuclid.org/euclid.rmjm/1239113439)) Page 9 there has a table with all the key ingredients. Except maybe for one detail: there the analogy is made between number fields and hyperbolic _surfaces_. Whereas I think now it works a bit better still for hyperbolic 3-manifolds.

   John Baez kindly points out that the analogy between the Selberg zeta and the Artin L given in the $n$Lab here had been highlighed much in

   • Darin Brown, Lifting properties of prime geodesics, Rocky Mountain J. Math. Volume 39, Number 2 (2009), 437-454 (euclid)

   Page 9 there has a table with all the key ingredients.

   Except maybe for one detail: there the analogy is made between number fields and hyperbolic surfaces. Whereas I think now it works a bit better still for hyperbolic 3-manifolds.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeDec 11th 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexPerhaps you could explain the notation $n_\Gamma(g)$?

   Perhaps you could explain the notation $n_\Gamma(g)$?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 11th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexRight, sorry, I still need to add definition of this and a few other terms.

   Right, sorry, I still need to add definition of this and a few other terms.