Project description
Research tackles function field challenges
A key challenge in multiplicative number theory is to understand how numbers can be broken down into their prime factors within certain sets. Researchers also look at complex problems involving factorisation in function fields, which are similar to number fields but involve functions instead of numbers. The ERC-funded Function Fields project introduces a groundbreaking approach to these problems. By limiting the size of cohomology groups related to sheaves through Massey’s bound and the characteristic cycle, the project helps simplify complex structures. Coupled with the circle method, the proposed approach should offer efficient solutions to function field problems. While it does not directly resolve number field issues, the project opens new avenues in geometry, topology, and homological stability.
Objective
An archetypal problem in multiplicative number theory is to determine the factorization statistics in a given set, such as the set of values of an integral polynomial, or of the function raising an integer to a non-integral power and taking the integral part, the set of integers n for which there exists an integer 0 < a < n/10 such that n divides a^3-2, or the set of discriminants of cubic extensions of a number field. One is particularly interested in estimating the number of primes in such sets. We study analogs of such problems over function fields (in one variable over a finite field). Almost every problem over number fields admits a sensible (although not necessarily obvious) analog over function fields, and solutions to such problems carry over as well. On the other hand, the Riemann Hypothesis has been resolved in a most definitive form by Deligne in the function field setting. While solutions of problems over function fields do not translate to solutions of analogous problems over number fields, new insights are gained, and connections to geometry, topology, and homological stability emerge. Function field analytic number theory problems often reduce to obtaining cancellation in sums of trace functions of l-adic etale sheaves over the points of a variety over a finite field. The Grothendieck--Lefschetz trace formula, in conjunction with Delignes theorem, gives us cancellation once strong upper bounds on the dimensions of the cohomology groups of our sheaves are available. The main proposed innovation is a bound on the dimensions of cohomology groups of sheaves built using the six operations from more basic sheaves, approached using Massey's bound involving the characteristic cycle. Our methods involve also judicious choices of l for which the reduction of our sheaves mod l simplifies them. These will be combined with the circle method to serve as an off-the-shelf approach to function field problems.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
- natural sciences mathematics pure mathematics topology
- natural sciences mathematics pure mathematics geometry
- natural sciences mathematics pure mathematics arithmetics prime numbers
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Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
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HORIZON.1.1 - European Research Council (ERC)
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Call for proposal
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(opens in new window) ERC-2024-STG
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7610001 Rehovot
Israel
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