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Langlands Functoriality
Langlands functoriality has been a driving force in number theory, representation theory, and beyond since it was first posed in a famous letter of...
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On non-abelian higher special elements of p-adic representations
We develop a theory of ‘non-abelian higher special elements’ in the non-commutative exterior powers of the Galois cohomology of p -adic...
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A Survey of the Additive Dilogarithm
Borel’s construction of the regulator gives an injective map from the algebraic K–groups of a number field to its Deligne–Beilinson cohomology...
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Eventually homological isomorphisms and Gorenstein projective modules
We prove that a certain eventually homological isomorphism between module categories induces triangle equivalences between their singularity...
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The Moduli Space of Stable Coherent Sheaves via Non-archimedean Geometry
We provide a construction of the moduli space of stable coherent sheaves in the world of non-archimedean geometry, where we use the notion of...
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On the Relative Twist Formula of ℓ-adic Sheaves
We propose a conjecture on the relative twist formula of ℓ-adic sheaves, which can be viewed as a generalization of Kato–Saito’s conjecture. We...
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Smith–Treumann theory and the linkage principle
We apply Treumann’s “Smith theory for sheaves” in the context of the Iwahori–Whittaker model of the Satake category. We deduce two results in the...
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Independence of algebraic monodromy groups in compatible systems
In this paper we develop a general method to prove independence of algebraic monodromy groups in compatible systems of representations, and we apply...
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Supersingular Irreducible Symplectic Varieties
In complex geometry, irreducible holomorphic symplectic varieties, also known as compact hyper-Kähler varieties, are natural higher-dimensional...
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Basic Properties of Drinfeld Modules
A Drinfeld module over a field K, our principal object of study in this book, is a field K equipped with an action of the ring of polynomials...
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Computational Number Theory, Past, Present, and Future
A survey of computational number theory viewed through the prism of my own experience and the Pari/GP software.
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Hecke symmetries: an overview of Frobenius properties
This paper improves several previously known results. First, the results describing the R -skewsymmetric algebra and the quadratic dual of the R -symmet...
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Grothendieck: A Short Guide to His Mathematical and Philosophical Work (1949–1991)
We present a short survey of Grothendieck’s “known” work (1949–1991), before his seclusion in the Pyrenees (1991–2014). Three main periods are...
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LATTICE VERTEX ALGEBRAS AND LOOP GRASSMANNIANS
For an integral symmetric matrix κ we construct a new “nonabelian homology localization” of the lattice vertex algebra
L κ on the corresponding loop... -
Divisibility of Frobenius eigenvalues on \(\varvec{\ell }\)-adic cohomology
For a projective variety defined over a finite field with q elements, it is shown that as algebraic integers, the eigenvalues of the geometric...
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Axiomatics as a Functional Strategy for Complex Proofs: The Case of Riemann Hypothesis
My purpose is to comment some claims of André Weil (1906–1998) in his letter of March 26, 1940 to his sister Simone, in particular, the following...