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The Relative Trace Formula in Analytic Number Theory

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Relative Trace Formulas

Part of the book series: Simons Symposia ((SISY))

Abstract

We discuss a variety of applications of the relative trace formula of Kuznetsov type in analytic number theory and the theory of automorphic forms.

Author partially supported by DFG grant BL 915/2-2.

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Notes

  1. 1.

    Which follows from Sarnak’s argument [Sa1] with small modifications.

  2. 2.

    It is also possible to detect the condition additive shift by some variant of the circle method. After some elementary Fourier analysis this also leads to Kloosterman sums. Although relatively similar, in many cases the above approach has slight advantages.

  3. 3.

    At the time of writing, the current record is slightly over 41%.

  4. 4.

    For any prime p ≡ 1 (mod 4) choose 1 ≤ n < p with n 2 ≡−1 (mod p).

  5. 5.

    This can be phrased as an analogue of Fermi’s golden rule.

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Acknowledgement

The author was partially supported by DFG grant BL 915/2-2.

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  1. Mathematisches Institut, Bonn, Germany

    Valentin Blomer

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  1. Mathematical Institute, University of Bonn, Bonn, Nordrhein-Westfalen, Germany

    Werner Müller

  2. Department of Mathematics, University of California, Berkeley, CA, USA

    Sug Woo Shin

  3. Department of Mathematics, Malott Hall, Cornell University, Ithaca, NY, USA

    Nicolas Templier

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Blomer, V. (2021). The Relative Trace Formula in Analytic Number Theory. In: Müller, W., Shin, S.W., Templier, N. (eds) Relative Trace Formulas. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-030-68506-5_2

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