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Quadratic Forms and Automorphic Forms

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Quadratic and Higher Degree Forms

Part of the book series: Developments in Mathematics ((DEVM,volume 31))

Abstract

These notes give a friendly four-part introduction to various aspects of the arithmetic and analytic theories of quadratic forms, aimed at a graduate-level audience. The main themes discussed are: geometry and local-global methods, theta functions and Siegel’s theorem, Clifford algebras and spin groups, and adelic theta liftings via the Weil representation.

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Notes

  1. 1.

    We could also have defined an inner product using the Hessian bilinear form, but this choice is less standard in the literature and the difference will not matter for our purposes in this section.

  2. 2.

    For any subset S ⊆ V the set H(S, L) is an R-module, and so it is natural to consider maximal subsets of V where H(S, L) is a fixed R-module. From the bilinearity of H, we see that these maximal sets S are also R-modules.

  3. 3.

    To justify this, notice that both the trivial and theta multiplier systems have value 1 on this element.

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    Jonathan Hanke

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  1. Department of Mathematics, University of Florida Department of Mathematics, Gainesville, Florida, USA

    Krishnaswami Alladi

  2. Department of Mathematics, Princeton University, Princeton, New Jersey, USA

    Manjul Bhargava

  3. Department of Mathematics, University of Arizona, Tucson, Arizona, USA

    David Savitt

  4. Department of Mathematics, University of Arizona, Tuscon, Arizona, USA

    Pham Huu Tiep

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Hanke, J. (2013). Quadratic Forms and Automorphic Forms. In: Alladi, K., Bhargava, M., Savitt, D., Tiep, P. (eds) Quadratic and Higher Degree Forms. Developments in Mathematics, vol 31. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7488-3_5

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