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Motivic Mellin transforms and exponential sums through non-archimedean geometry

Project information

Grant agreement ID: 615722

Status

Closed project

  • Start date

    1 March 2014

  • End date

    28 February 2019

Funded under:

FP7-IDEAS-ERC

  • Overall budget:

    € 912 000

  • EU contribution

    € 912 000

Hosted by:

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS

France

Objective

"We aim to create a new and powerful theory of motivic integration which incorporates Mellin transforms. The absence of motivic Mellin transforms is a major drawback of the existing theories. Classical Mellin transforms are in essence Fourier transforms on the multiplicative group of local fields. We aim to apply this theory to study new motivic Poisson summation formulas, new transfer principles, and applications of these. All of this has so far only been studied in the presence of additive characters, and remains completely open for multiplicative characters. Understanding all this at a motivic level yields a uniform understanding when the local field varies and will require an approach using non-archimedean geometry. We will open up possibilities for applications via new transfer principles and will give access to motivic Poisson formulas of other groups than the additive group. For these applications it is important that Fubini Theorems are present at the level of the motivic integrals, which we aim to develop. We will overcome the major obstacle of the totally different nature of the dual group of the multiplicative group by a proposed sequence of germs of ideas by the author. The importance of our work on motivic Fourier transforms on the additive group is already widely recognized, and this proposal will complement it by exploring the new territory of motivic multiplicative characters. A final topic is the study of the highly non-understood exponential sums modulo powers of primes, in relation with Igusa's foundational work. We will try to discover a deeper understanding of the uniform behavior of these sums when the prime number varies. These sums are linked to geometrical concepts like the log-canonical threshold, and also to Poisson summation, after the work by Igusa. We will aim to prove a highly generalized form of Igusa's conjecture on exponential sums."

Principal Investigator

Raf Cluckers (Prof.)

Host institution

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS

Address

Rue Michel Ange 3
75794 Paris

France

Activity type

Research Organisations

EU Contribution

€ 912 000

Principal Investigator

Raf Cluckers (Prof.)

Administrative Contact

Bénédicte Samyn (Mrs.)

Beneficiaries (1)

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS

France

EU Contribution

€ 912 000

Project information

Grant agreement ID: 615722

Status

Closed project

  • Start date

    1 March 2014

  • End date

    28 February 2019

Funded under:

FP7-IDEAS-ERC

  • Overall budget:

    € 912 000

  • EU contribution

    € 912 000

Hosted by:

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS

France