Objective
The Langlands program, often called a ``grand unified theory of mathematics'', predicts reciprocity laws that relate very different kinds of mathematical objects: automorphic forms and Galois representations. Shimura varieties play a fundamental role in constructing and propagating instances of Langlands reciprocity.
In the past decade, advances in p-adic geometry have revolutionised the study of Shimura varieties. This recently led to the introduction of certain p-adic analytic varieties called Igusa stacks, which connect Shimura varieties to ideas from geometric Langlands via the work of Fargues--Scholze. This gives access to powerful new tools and makes Igusa stacks seem as fundamental as the Shimura varieties themselves.
This proposal will realise the potential of p-adic Igusa stacks and of the closely related Igusa varieties to give a systematic understanding of the cohomology of Shimura varieties with l-torsion coefficients and of congruences modulo l between automorphic forms on Shimura varieties. There are two components to this proposal, depending on whether or not l = p.
When l is not p, I will construct the relative intersection cohomology of the Igusa stack and describe it in terms of purely local (generalised) eigensheaves, expanding on and proving a conjecture of Fargues. This will lead to a description of torsion in the intersection cohomology of Shimura varieties, reminiscent of Arthur's conjectures in characteristic 0, and to an axiomatic approach to congruences, such as level-raising, when the tame level varies.
When l is p, I will prove an integral Eichler--Shimura comparison theorem for ordinary p-adic automorphic forms. Initially, my approach will exploit the ordinary part of the Igusa stack, then move into deeper Newton strata for generalisations. This theorem will unify geometric and representation-theoretic perspectives on ordinary p-adic automorphic forms and will have major applications to Euler systems and Iwasawa theory.
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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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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.
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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-2025-COG
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SW7 2AZ London
United Kingdom
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