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Documents  Bringmann, Kathrin | enregistrements trouvés : 7

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Number Theory

We study the smallest parts function introduced by Andrews. The associated generating function forms a component of a natural mock modular form of weight 3/2 whose shadow is the Dedekind eta function. We obtain an exact formula and an algebraic formula for each value of the smallest parts function; these are analogues of the formulas of Rademacher and Bruinier-Ono for the ordinary partition function. The convergence of our expression is non-trivial; the proof relies on power savings estimates for weighted sums of generalized Kloosterman sums which follow from spectral methods. We study the smallest parts function introduced by Andrews. The associated generating function forms a component of a natural mock modular form of weight 3/2 whose shadow is the Dedekind eta function. We obtain an exact formula and an algebraic formula for each value of the smallest parts function; these are analogues of the formulas of Rademacher and Bruinier-Ono for the ordinary partition function. The convergence of our expression is ...

11F37 ; 11P82

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Number Theory

In parallel to the Gross-Kohnen-Zagier theorem, Zagier proved that the traces of the values of the j-function at CM points are the coefficients of a weakly holomorphic modular form of weight 3/2. Later this result was generalized in different directions and also put in the context of the theta correspondence. We recall these results and report on some newer aspects.

11Fxx ; 11F37

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Number Theory

I will discuss recent progress in the analytic study of classical partition identities, including the famous " sum-product " formulas of Rogers-Ramanujan, Schur, and Capparelli. Such identities are rich in automorphic objects such as Jacobi theta functions, mock theta functions, and false theta functions. Furthermore, there are interesting connections to the combinatorics of multi-colored partitions, and the calculation of standard modules for Lie algebras and vertex operator theory. I will discuss recent progress in the analytic study of classical partition identities, including the famous " sum-product " formulas of Rogers-Ramanujan, Schur, and Capparelli. Such identities are rich in automorphic objects such as Jacobi theta functions, mock theta functions, and false theta functions. Furthermore, there are interesting connections to the combinatorics of multi-colored partitions, and the calculation of standard modules for ...

11Pxx ; 11P81 ; 11P82 ; 11P84

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The genus-two Kawazumi-Zhang (KZ) invariant is a real-analytic modular function on the Siegel upper half-plane of degree two, which plays an important role in arithmetic geometry. In String theory, it appears as part of the integrand in two-loop four-graviton scattering amplitudes. With hindsight from String theory, I will show that the KZ invariant can be obtained as a generalized Borcherds lift from a weak Jacobi form of index 1 and weight 2. This implies that the KZ invariant is an eigenmode of the quadratic and quartic Casimir operators, and gives access to the full asymptotic expansion in all possible degeneration limits. It also reveals a mock-type holomorphic Siegel modular form underlying the KZ invariant. String theory amplitudes involves modular integrals of the KZ invariant (times lattice partition functions) on the Siegel upper half-plane, which provide new examples of automorphic objects on orthogonal Grassmannians, beyond the usual Langlands-Eisenstein series.

Note: this talk is based on the preprint "A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces" available on arXiv:1504.04182. Following up on a question asked during the talk (which was answered very poorly), the author obtained shortly after a proof of the conjecture stated in this preprint and during the talk. The proof is available in the revised version arXiv:1504.04182v2.
The genus-two Kawazumi-Zhang (KZ) invariant is a real-analytic modular function on the Siegel upper half-plane of degree two, which plays an important role in arithmetic geometry. In String theory, it appears as part of the integrand in two-loop four-graviton scattering amplitudes. With hindsight from String theory, I will show that the KZ invariant can be obtained as a generalized Borcherds lift from a weak Jacobi form of index 1 and weight 2. ...

81T30 ; 83E30

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Number Theory

Fourier coefficients of meromorphic Jacobi forms show up in, for example, the study of mock theta functions, quantum black holes and Kac-Wakimoto characters. In the case of positive index, it was previously shown that they are the holomorphic parts of vector-valued almost harmonic Maass forms. In this talk, we give an alternative characterization of these objects by applying the Maass lowering operator to the completions of the Fourier coefficients. Further, we'll also describe the relation of Fourier coefficients of negative index Jacobi forms to partial theta functions. Fourier coefficients of meromorphic Jacobi forms show up in, for example, the study of mock theta functions, quantum black holes and Kac-Wakimoto characters. In the case of positive index, it was previously shown that they are the holomorphic parts of vector-valued almost harmonic Maass forms. In this talk, we give an alternative characterization of these objects by applying the Maass lowering operator to the completions of the Fourier ...

11F27 ; 11F30

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- viii; 190 p.
ISBN 978-3-03719-142-2

EMS series of lectures in mathematics

Localisation : Ouvrage RdC (FOUR)

théorie des nombres # transcendance # nombre algébrique # théorème subspatial de Schmidt # fraction continue # expansion digitale # approximation Diophantienne

11-02 ; 11J81 ; 11A63 ; 11J04 ; 11J13 ; 11J68 ; 11J70 ; 11J87 ; 68R15

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- xv; 391 p.
ISBN 978-1-4704-1944-8

American mathematical society colloquium publications , 0064

Localisation : Collection 1er étage

théorie des nombres # groupe discontinu # forme automorphe # forme modulaire # forme de Jacobi # représentation de Weil # correspondance Thêta # coefficient de Fourier

11F03 ; 11F11 ; 11F27 ; 11F30 ; 11F37 ; 11F50 ; 11-02 ; 11Fxx ; 14K25 ; 11M41

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