The algebraic functional equation of Riemann’s theta function
Annales de l'Institut Fourier, Volume 70 (2020) no. 2, pp. 809-830.

We give an algebraic analog of the functional equation of Riemann’s theta function. More precisely, we define a “theta multiplier” line bundle over the moduli stack of principally polarized abelian schemes with theta characteristic and prove that its dual is isomorphic to the determinant bundle over the moduli stack. We do so by explicit computations involving the Picard group of the moduli stack. This is all done over the ring R=[1/2,i]: passing to the complex numbers, we recover the classical functional equation.

Nous donnons un analogue algébrique de l’équation fonctionnelle de la fonction thêta de Riemann. Plus précisément, nous définissons un fibré en droites « multiplicateur thêta » sur le champ de modules de schémas abéliens principalement polarisés avec une caractéristique thêta et prouvons que son dual est isomorphe au fibré déterminant sur le champ de modules. Nous le faisons par des calculs explicites impliquant le groupe de Picard du champ de modules. Tout cela se fait sur l’anneau R=[1/2,i] : en passant aux nombres complexes, on retrouve l’équation fonctionnelle classique.

Received: 2017-06-13
Revised: 2018-03-19
Accepted: 2018-12-27
Published online: 2020-05-28
DOI: https://doi.org/10.5802/aif.3324
Classification: 11F03,  11F27,  11G10
Keywords: theta functions, modular forms, abelian varieties
@article{AIF_2020__70_2_809_0,
     author = {Candelori, Luca},
     title = {The algebraic functional equation of Riemann's theta function},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {2},
     year = {2020},
     pages = {809-830},
     doi = {10.5802/aif.3324},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2020__70_2_809_0/}
}
Candelori, Luca. The algebraic functional equation of Riemann’s theta function. Annales de l'Institut Fourier, Volume 70 (2020) no. 2, pp. 809-830. doi : 10.5802/aif.3324. https://aif.centre-mersenne.org/item/AIF_2020__70_2_809_0/

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