Formalization of Langlands’s Second Main Lemma for Local Epsilon Factors
A Lean 4 formalization of Langlands’s Second Main Lemma following the companion local proof.
PDF · arXiv / full text · HAL · Lean code
Fukuhiro Ueda
Research Institute for Mathematical Sciences (RIMS), Kyoto University
Research
My research is in arithmetic geometry and number theory. I work on p-adic Hodge theory, Galois representations, and Langlands Correspondence.
Preprints
A Local Proof of Langlands’s Second Main Lemma for Local Epsilon Factors over Nonarchimedean Local Fields
A local proof of Langlands’s Second Main Lemma in mixed and equal characteristic. The new point is the wild dyadic case, where the remaining sign is determined by comparing the finite sums in Lamprecht’s formula.
PDF · arXiv / full text · HAL · Langlands’s comments on the Second Main Lemma
Formalization of Langlands’s First Main Lemma for Local Epsilon Factors
A Lean 4 formalization of the First Main Lemma following the companion local proof.
A Local Proof of Langlands’s First Main Lemma for Local Epsilon Factors over Nonarchimedean Local Fields
A complete purely local proof of Langlands’s First Main Lemma in both mixed and equal characteristic, reconstructing the Dwork–Langlands method.
PDF · arXiv / full text · HAL
Published papers
Crystalline comparison isomorphisms in p-adic Hodge theory: the absolutely unramified case
Constructs crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base, allowing nontrivial coefficients and a relative version, using the pro-étale site.
The Breuil–Mézard conjecture for non-scalar split residual representations
Proves the Breuil–Mézard conjecture for split non-scalar residual representations of the local Galois group by local methods; together with earlier cases this completes the conjecture for p ≥ 5. As a consequence, it removes the remaining local restriction in Kisin’s proof of the Fontaine–Mazur conjecture, yielding the corresponding two-dimensional modularity theorem.
Overconvergent Families of Siegel–Hilbert Modular Forms
Constructs one-parameter families of overconvergent Siegel–Hilbert modular forms, with applications to Galois representations attached to automorphic forms of non-cohomological weights.
Modularity of Calabi–Yau varieties and conformal field theory
Studies modularity of Calabi–Yau varieties from the viewpoint of conformal field theory, expressing the modular forms of one-dimensional Calabi–Yau orbifolds as products of Dedekind eta functions.
Thesis
Families of p-adic Galois Representations
Generalizes Kisin’s theory of finite-slope subspaces to arbitrary p-adic fields and studies finite-slope deformation rings, local Galois eigenvarieties, their geometry near de Rham points, and an infinite fern in local Galois deformation space.