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Najmuddin Fakhruddin

Publications and source records attributed to Najmuddin Fakhruddin.

At least 19 recordsLinked to original sources
arXiv2023

Motivic factorisation of KZ local systems and deformations of representation and fusion rings

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$. The KZ connection is a connection on the constant bundle associated to a set of $n$ finite dimensional irreducible representations of $\mathfrak{g}$ and a nonzero $κ\in \mathbb{C}$, over the configuration space of $n$-distinct points on the affine line. Via the work of Schechtman--Varchenko and Looijenga, when $κ$ is a rational number the associated local systems can be seen to be realisations of naturally defined motivic local systems. We prove a basic factorisation for the nearby cycles of these motivic local systems as some of the $n$ points coalesce. This leads to the construction of a family (parametrised by $κ$) of deformations over $\mathbb{Z}[t]$ of the representation ring of $\mathfrak{g}$--we call these enriched representation rings--which allows one to compute the ranks of the Hodge filtration of the associated variations of mixed Hodge structure; in turn, this has applications to both the local and global monodromy of the KZ connection. In the case of $\mathfrak{sl}_n$ we give an explicit algorithm for computing all products in the enriched representation rings, which we use to prove that if $1/κ$ is an integer then the global monodromy is finite and scalar. We also prove a similar factorisation result for motivic local systems associated to conformal blocks in genus $0$; this leads to the construction of a family of deformations of the fusion rings. Computations in these rings have potential applications to finiteness of global monodromy. Several open problems and conjectures are formulated. These include questions about motivic BGG-type resolutions and the relationship between the Hodge filtration and the filtration by conformal blocks at varying levels.

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arXiv2023

Conformal blocks in genus zero and the KZ connection

We survey some recent work on conformal blocks in genus zero, focussing on (1) Chern classes, global generation and morphisms, and (2) the Knizhnik--Zamolodchikov connection on conformal blocks (and invariants), their motivic realizations, and unitarity.

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arXiv2022

Singularities of local models

We construct local models of Shimura varieties and investigate their singularities, with special emphasis on wildly ramified cases. More precisely, with the exception of odd unitary groups in residue characteristic $2$ we construct local models, show reducedness of their special fiber, Cohen$-$Macaulayness and in equi-characteristic also (pseudo-)rationality. In mixed characteristic we conjecture their (pseudo-)rationality. This is based on the construction of parahoric group schemes over two-dimensional bases for wildly ramified groups and an analysis of singularities of the attached Schubert varieties in positive characteristic using perfect geometry.

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arXiv2022

Trianguline lifts of global mod $p$ Galois representations

We show that under a suitable oddness condition, irreducible mod $p$ representations of the absolute Galois group of an arbitrary number field have characteristic zero lifts which are unramified outside a finite set of primes and trianguline at all primes of $F$ dividing $p$. We also prove variants of this result for representations valued in connected reductive groups.

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arXiv2021

Relative deformation theory, relative Selmer groups, and lifting irreducible Galois representations

We study irreducible odd mod $p$ Galois representations $\barρ \colon \mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_p)$, for $F$ a totally real number field and $G$ a general reductive group. For $p \gg_{G, F} 0$, we show that any $\barρ$ that lifts locally, and at places above $p$ to de Rham and Hodge-Tate regular representations, has a geometric $p$-adic lift. We also prove non-geometric lifting results without any oddness assumption.

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arXiv2021

Lifting and automorphy of reducible mod p Galois representations over global fields

We extend the lifting methods of our previous paper to lift reducible odd representations $\barρ:\mathrm{Gal}(\overline{F}/F) \to G(k)$ of Galois groups of global fields $F$ valued in Chevalley groups $G(k)$. Lifting results, when combined with automorphy lifting results pioneered by Wiles in the number field case and the results on the global Langlands correspondence proved by Drinfeld and L. Lafforgue in the function field case, give the only known method to access modularity of mod $p$ Galois representations in both reducible and irreducible cases. In the reducible case this allows one to show that the actual representation, rather than just its semisimplification, arises from reduction of the geometric representation attached to a cuspidal automorphic representation on the dual group of $G$. As a particularly concrete application, we get a version of Serre's modularity conjecture for reducible, odd representations $\barρ: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_2(k)$. This extends earlier results of Hamblen and Ramakrishna in this classical case and proves modularity of infinitely many extensions of fixed characters that are not covered by loc. cit.

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arXiv2021

Finite groups scheme actions and incompressibility of Galois covers: beyond the ordinary case

Inspired by recent work of Farb, Kisin and Wolfson, we develop a method for using actions of finite group schemes over a mixed characteristic dvr R to get lower bounds for the essential dimension of a cover of a variety over K = Frac(R). We then apply this to prove p-incompressibility for congruence covers of a class of unitary Shimura varieties for primes p at which the reduction of the Shimura variety (at any prime of the reflex field over p) does not have any ordinary points. We also make some progress towards a conjecture of Brosnan on the p-incompressibility of the multiplication by p map of an abelian variety.

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arXiv2020

Lifting $G$-irreducible but $\mathrm{GL}_n$-reducible Galois representations

In recent work, the authors proved a general result on lifting $G$-irreducible odd Galois representations $\mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_{\ell})$, with $F$ a totally real number field and $G$ a reductive group, to geometric $\ell$-adic representations. In this note we take $G$ to be a classical group and construct many examples of $G$-irreducible representations to which these new lifting methods apply, but to which the lifting methods provided by potential automorphy theorems do not.

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arXiv2020

Fixed points, local monodromy, and incompressibility of congruence covers

We prove a fixed point theorem for the action of certain local monodromy groups on étale covers and use it to deduce lower bounds in essential dimension. In particular, we give more geometric proofs of many (but not all) of the results of the preprint of Farb, Kisin and Wolfson, which uses arithmetic methods to prove incompressibility results for Shimura varieties and moduli spaces of curves. Our method allows us to prove results for exceptional groups, and also for the reduction modulo good primes of Shimura varieties and moduli spaces of curves.

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arXiv2020

Quantitative level lowering for Galois representations

We use Galois cohomology methods to produce optimal mod $p^d$ level lowering congruences to a $p$-adic Galois representation that we construct as a well chosen lift of a given residual mod $p$ representation. Using our explicit Galois cohomology methods, we construct for a reductive group $G$ and a given residual representation $\barρ: Γ_F \to G(k)$, ramified at a finite set of primes $S$, in favorable conditions that we identify, a finite set of lifts $ρ$, $\{ρ^q\}$ of $\barρ$ to $G(W(k))$ with the following properties: $ρ: Γ_F \to G(W(k))$ is ramified precisely at $S \cup Q$, with $Q$ a finite set of primes disjoint from $S$. For $q \in Q$, $ρ^q:G_F \to G(W(k))$ is unramified outside $S \cup Q \backslash \{q\}$ and $ρ$ and $ρ^q$ are congruent mod $p^d$ if $ρ$ mod $p^d$ is unramified at $q$. Furthermore, the Galois representations $\{ρ^q\}$ are "independent".

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arXiv2019

Hecke operators and the coherent cohomology of Shimura varieties

We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain $p$-adic estimates of Satake parameters of certain non-regular self dual automorphic representations of $\mathrm{GL}_n$.

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arXiv2019

Finite group schemes of essential dimension one

We prove that if a finite group scheme $G$ over a field $k$ has essential dimension one, then it embeds in $PGL_{2/k}$. We use this to give an explicit classification of all infinitesimal group schemes of essential dimension one over any field and a characterisation of all finite group schemes of essential dimension one over algebraically closed fields.

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arXiv2018

Lifting irreducible Galois representations

We study irreducible mod p representations, valued in general reductive groups, of the Galois group of a number field. When the number field is totally real, we show that odd representations satisfying local ramification hypotheses and a certain multiplicity-free condition on the adjoint representation admit geometric lifts. For general number fields, we show without any oddness or multiplicity condition that the representation admits a p-adic lift if it does everywhere locally.

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arXiv2018

Triviality properties of principal bundles on singular curves-II

For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\to S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D \subset X$. We show, by constructing explicit examples, that the obstruction is nontrivial if $G$ is not simply connected but it can be made to vanish, if $S$ is the spectrum of a dvr (and some other hypotheses), by a faithfully flat base change. The vanishing of this obstruction is shown to be a sufficient condition for etale local triviality if $S$ is a smooth curve, and the singular locus of $X-D$ is finite over $S$.

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arXiv2017

Triviality properties of principal bundles on singular curves

We show that principal bundles for a semisimple group on an arbitrary affine curve over an algebraically closed field are trivial, provided the order of $π_1$ of the group is invertible in the ground field, or if the curve has semi-normal singularities. Several consequences and extensions of this result (and method) are given. As an application, we realize conformal blocks bundles on moduli stacks of stable curves as push forwards of line bundles on (relative) moduli stacks of principal bundles on the universal curve.

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arXiv2014

Exceptional collections on 2-adically uniformised fake projective planes

We show that there exist exceptional collections of length 3 consisting of line bundles on the three fake projective planes that have a 2-adic uniformisation with torsion free covering group. We also compute the Hochschild cohomology of the right orthogonal of the subcategory of the bounded derived category of coherent sheaves generated by these exceptional collections.

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arXiv2013

The algebraic dynamics of generic endomorphisms of P^n

We investigate some general questions in algebraic dynamics in the case of generic endomorphisms of projective spaces over a field of characteristic zero. The main results that we prove are that a generic endomorphism has no non-trivial preperiodic subvarieties, any infinite set of preperiodic points is Zariski dense and any infinite subset of a single orbit is also Zariski dense, thereby verifying the dynamical "Manin--Mumford" conjecture of Zhang and the dynamical "Mordell--Lang" conjecture of Denis and Ghioca--Tucker in this case.

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