Project description
Advancing key areas of research related to the arithmetic of curves
Algebraic curves, formally defined as an algebraic variety of dimension one, include circles, ellipses, parabolas and hyperbolas. Some algebraic curves have many rational points (points for which all coordinates are rational numbers), but some curves have only a few. Algebraic curves and the nature of their rational points, an interplay of arithmetic and algebra, have fascinated mathematicians for centuries. The ERC-funded CurveArithmetic project will progress key areas of research related to the arithmetic of curves: a Mazur-type theorem for a family of unitary Shimura curves, the Poonen-Rains heuristics for elliptic curves and certain instances of the Beilinson-Bloch conjecture.
Objective
The study of the arithmetic of curves is as old as mathematics itself and takes on many forms. In some cases, such as Fermat's Last Theorem or Mazur's torsion theorem, one tries to prove that a sequence of curves with growing genus has no interesting rational points. In other cases, such as the study of rational points in families of elliptic curves, there is no way to classify all solutions, but one tries to understand what is happening on average. A third approach aims to link the existence of rational points on a given curve to the preponderance of points on the curve modulo larger and larger prime numbers. This is the idea behind the Birch and Swinnerton-Dyer conjecture, and its generalization, the Beilinson-Bloch conjecture.
The proposed research makes progress in each of the three paradigms above. In corresponding order, we propose a Mazur-type theorem for a family of unitary Shimura curves, by exploiting the Jacquet-Langlands correspondence and a connection with Prym varieties. A special case of this result would give a classification of torsion points in a family of genus three bielliptic Jacobians. Second, we propose an approach to the Poonen-Rains heuristics for elliptic curves by combining twisting methods with Bhargava's geometry-of-numbers methods for universal families. Using similar methods, we aim to show that Hilbert's tenth problem has a negative answer over every number field. Third, we study certain instances of the Beilinson-Bloch conjecture for the degree 3 motive of the Jacobian of a curve with complex multiplication. The strategy involves the construction of an Euler system composed of CM Ceresa cycles. Related work will explore torsion and infinite generation phenomena for Ceresa cycles, as well.
Fields of science (EuroSciVoc)
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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.
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Keywords
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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)
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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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Topic(s)
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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.
Funding Scheme
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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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Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2022-STG
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91904 JERUSALEM
Israel
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