Project description
Exponential motives along with the arithmetic Gevrey series
Exponential motives are a key tool in researching and studying a vast variety of host objects that are attached to algebraic varieties along with their regular function. They often play essential roles in both analytic number theory and Landau-Ginzburg models in mirror symmetry. The ERC-funded EMOTIVE project will enhance the understanding of their role and application in concrete problems related to the arithmetic Gevrey series, a series of algebraic coefficients satisfying their differential equation. These efforts will focus on studying various functions and key questions around them, as well as the Gevrey series. The insights gained in this project could help revolutionise arithmetic and provide countless novel insights.
Objective
Exponential motives are a powerful tool for the study of a host of objects attached to an algebraic variety along with a regular function, ranging from exponential sums over finite fields in analytic number theory to Landau-Ginzburg models in mirror symmetry. This project revolves around applications of the abstract theory of exponential motives to concrete problems pertaining to arithmetic Gevrey series, after my recent breakthrough in solving a 1929 question by Siegel.
Arithmetic Gevrey series are power series with algebraic coefficients that satisfy a differential equation and certain growth conditions of arithmetic nature. Depending on the specific shape of these conditions, they come into three main flavours: G-functions, E-functions, and Э-functions. We plan to make significant progress on three interrelated questions about arithmetic Gevrey series: What are the transcendence properties of their special values? What is the nature of the differential equations they satisfy? Do they admit integral representations coming from geometry?
The most important examples of G-functions arise from period functions of one-parameter families of algebraic varieties. A geometric interpretation of E-functions and their differential equations was lacking until exponential motives entered the scene. We will systematically exploit the new possibilities they offer to make the differential Galois group act on special values of E-functions, elucidate the local-to-global nature of index theorems for E-operators, prove general structure results for Hodge loci of exponential motives, and reconcile the Siegel-Shidlovsky theorem with Wüstholz’s analytic subgroup theorem, with a view to separating special values of E-functions and G-functions. We will also advance our understanding of the relation between G-functions and hypergeometric series, as well as the arithmetic of regularised special values of Э-functions.
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 arithmetics
- natural sciences mathematics pure mathematics geometry
- natural sciences mathematics pure mathematics algebra algebraic geometry
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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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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.1 - European Research Council (ERC)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
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Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
HORIZON-ERC - HORIZON ERC Grants
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Call for proposal
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(opens in new window) ERC-2024-COG
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75006 PARIS
France
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