Symplectic Kloosterman sums and Poincaré series

We prove power-saving bounds for general Kloosterman sums on $${\text {Sp}}(4)$$ Sp ( 4 ) associated to all Weyl elements via a stratification argument coupled with p-adic stationary phase methods. We relate these Kloosterman sums to the Fourier coefficients of $${\text {Sp}}(4)$$ Sp ( 4 ) Poincare series.

AdS3 gravity and RCFT ensembles with multiple invariants

We use the Poincaré series method to compute gravity partition functions associated to SU(N)1 WZW models with arbitrarily large numbers of modular invariants. The result is an average over these invariants, with the weights being given by inverting a matrix whose size is of order the number of invariants. For the chosen models, this matrix takes a special form that allows us to invert it for arbitrary size and thereby explicitly calculate the weights of this average. For the identity seed we find that the weights are positive for all N, consistent with each model being dual to an ensemble average over CFT’s.

Extremal Betti Numbers of Some Cohen–Macaulay Binomial Edge Ideals

We provide the regularity and the Cohen–Macaulay type of binomial edge ideals of Cohen–Macaulay cones, and we show the extremal Betti numbers of some classes of Cohen–Macaulay binomial edge ideals: Cohen–Macaulay bipartite and fan graphs. In addition, we compute the Hilbert–Poincaré series of the binomial edge ideals of some Cohen–Macaulay bipartite graphs.

Poincaré series, 3d gravity and averages of rational CFT

We investigate the Poincaré series approach to computing 3d gravity partition functions dual to Rational CFT. For a single genus-1 boundary, we show that for certain infinite sets of levels, the SU(2)k WZW models provide unitary examples for which the Poincaré series is a positive linear combination of two modular-invariant partition functions. This supports the interpretation that the bulk gravity theory (a topological Chern-Simons theory in this case) is dual to an average of distinct CFT’s sharing the same Kac-Moody algebra. We compute the weights of this average for all seed primaries and all relevant values of k. We then study other WZW models, notably SU(N)1 and SU(3)k, and find that each class presents rather different features. Finally we consider multiple genus-1 boundaries, where we find a class of seed functions for the Poincaré sum that reproduces both disconnected and connected contributions — the latter corresponding to analogues of 3-manifold “wormholes” — such that the expected average is correctly reproduced.

On coefficients of Poincaré series and single-valued periods of modular forms

We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level $$\varGamma _0(N)$$ Γ 0 ( N ) and integral weight $$k\ge 2$$ k ≥ 2 coincides with the field generated by the single-valued periods of a certain motive attached to $$\varGamma _0(N)$$ Γ 0 ( N ) . This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres–Broadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann–Ono’s construction of harmonic lifts of Poincaré series.