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Chandrashekhar B. Khare

Publications and source records attributed to Chandrashekhar B. Khare.

14 recordsLinked to original sources
arXiv2026

The commutative algebra of congruence ideals and applications to number theory

In his proof of Fermat's Last Theorem, Wiles deployed a commutative algebra technique, namely a numerical criterion for detecting isomorphisms of rings. In our recent work we pick up on Wiles' work and generalize the numerical criterion to ``higher codimension''. A critical ingredient is a notion of congruence module in higher codimension: this has turned out to be a key definition whose utility extends beyond the role it plays in the numerical criterion. In this paper we trace the origin of some of the ideas that led to our work, both in number theory and commutative algebra, and new directions that emerge from it. We introduce a related notion of a congruence ideal. When applied to deformation theory of Galois representations and Hecke algebras, which is the setting of Wiles's work on Fermat's Last Theorem, our work leads to the notion of congruence ideals for local deformation rings. This sheds light on the classically studied congruence ideals for global deformation rings and Hecke algebras. We outline applications of the commutative algebra we have developed to: (i) integral modularity lifting theorems in the context of weight one forms, and (ii) factorization formulas for congruence ideals of global deformation rings at augmentations induced by newforms in which local congruence ideals enter as the local terms. The latter leads to surprising relations between these local congruence ideals and local Tamagawa ideals of Bloch-Kato associated to the rank 3 adjoint motive of $f$.

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arXiv2025

A freeness criterion for complexes with derived actions

Inspired by the patching method of Calegari and Geraghty, and a conjecture of de Smit that has been proved by the first author, we present a conjectural freeness criterion without patching for complexes over commutative noetherian local rings with derived actions, and verify it in several cases.

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arXiv2024

Congruence modules and the Wiles-Lenstra-Diamond numerical criterion in higher codimensions

We define a congruence module $Ψ_A(M)$ associated to a surjective $\mathcal O$-algebra morphism $λ\colon A \to \mathcal{O}$, with $\mathcal{O}$ a discrete valuation ring, $A$ a complete noetherian local $\mathcal{O}$-algebra regular at $\mathfrak{p}$, the kernel of $λ$, and $M$ a finitely generated $A$-module. We establish a numerical criterion for $M$ to have a free direct summand over $A$ of positive rank. It is in terms of the lengths of $Ψ_A(M)$ and the torsion part of $\mathfrak{p}/\mathfrak{p}^2$. It generalizes results of Wiles, Lenstra, and Diamond, that deal with the case when the codimension of $\mathfrak{p}$ is zero. Number theoretic applications include integral (non-minimal) $R=\mathbb T$ theorems in situations of positive defect conditional on certain standard conjectures. Here $R$ is a deformation ring parametrizing certain Galois representations and $\mathbb T$ is a Hecke algebra. An example is a modularity lifting for 2-dimensional $\ell$-adic Galois representations over an imaginary quadratic field. The proofs combine our commutative algebra results with a generalization due to Calegari and Geraghty of the patching method of Wiles and Taylor--Wiles and level raising arguments that go back to Ribet. The results provide new evidence in favor of the intriguing, and as yet fledgling, torsion analog of the classical Langlands correspondence. We also prove unconditional integral $R=\mathbb T$ results for Hecke algebras $\mathbb T$ acting on weight one cohomology of Shimura curves over $\mathbb Q$. This leads to a torsion Jacquet--Langlands correspondence comparing integral Hecke algebras acting on weight one cohomology of Shimura curves and modular curves. In this case the cohomology has abundant torsion and so our correspondence cannot be deduced by means of the classical Jacquet--Langlands correspondence.

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arXiv2023

Congruence modules in higher codimension and zeta lines in Galois cohomology

This work builds on earlier work of the first three authors where a notion of congruence modules in higher codimension is introduced. The main new results are a criterion for detecting regularity of local rings in terms of congruence modules, and a more refined version of a result tracking the change of congruence modules under deformation is proved. Number theoretic applications include the construction of canonical lines in certain Galois cohomology groups arising from adjoint motives of Hilbert modular forms.

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arXiv2022

Splitting fields of $X^n-X-1$ (particularly for $n=5$), prime decomposition and modular forms

We study the splitting fields of the family of polynomials $f_n(X)= X^n-X-1$. This family of polynomials has been much studied in the literature and has some remarkable properties. Serre related the function on primes $N_p(f_n)$, for a fixed $n \leq 4$ and $p$ a varying prime, which counts the number of roots of $f_n(X)$ in $\mathbb F_p$ to coefficients of modular forms. We study the case $n=5$, and relate $N_p(f_5)$ to mod $5$ modular forms over $\mathbb Q$, and to characteristic 0, parallel weight 1 Hilbert modular forms over $\mathbb Q(\sqrt{19 \cdot 151})$.

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arXiv2022

Freeness of Hecke modules at non-minimal levels

We build on the results of [6] to show that the homology groups $\mathrm{H}_{r_1+r_2}(Y_0(\mathcal{N}_Σ),\mathcal{O})_{\mathfrak{m}_Σ}$ of arithmetic manifolds are free over certain deformation rings $R_Σ$, when there are enough geometric characteristic 0 representations. Hitherto we had proved that the homology group has a nonzero free $R_Σ$-direct summand. The new ingredient is a commutative algebra argument involving congruence modules defined in higher codimension in [6].

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arXiv2022

Wiles defect for modules and criteria for freeness

F. Diamond proved a numerical criterion for modules over local rings to be free modules over complete intersection rings. We formulate a refinement of these results using the notion of Wiles defect. A key step in the proof is a formula that expresses the Wiles defect of a module in terms of the Wiles defect of the underlying ring.

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arXiv2021

Modularity of $\operatorname{PGL}_2(\mathbb{F}_p)$-representations over totally real fields

We study an analogue of Serre's modularity conjecture for projective representations $\overlineρ: \operatorname{Gal}(\overline{K} / K) \rightarrow \operatorname{PGL}_2(k)$, where $K$ is a totally real number field. We prove new cases of this conjecture when $k = \mathbb{F}_5$ by using the automorphy lifting theorems over CM fields established in previous work of the authors.

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arXiv2021

Wiles defect of Hecke algebras via local-global arguments

We continue our study of the Wiles defect of deformation rings $R$ and Hecke rings $T$ (at a newform $f$) acting on the cohomology of Shimura curves. The Wiles defect at an augmentation $λ_f:T \to O$ measures the failure of $R,T$ to be complete intersections locally at $λ_f$. In situations we study here the Taylor-Wiles-Kisin patching method gives an isomorphism $R=T$ without the rings being complete intersections. Using novel arguments in commutative algebra and patching, we generalize significantly and give different proofs of our earlier results that compute the Wiles defect at $λ_f: R=T \to O$, and explain in an a priori manner why the answer is a sum of local defects. As a curious application of our work we give a new and more robust approach to the result of Ribet--Takahashi that computes change of degrees of optimal parametrizations of elliptic curves by Shimura curves as we vary the Shimura curve.

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arXiv2021

A local Langlands parameterization for generic supercuspidal representations of $p$-adic $G_2$

We construct a Langlands parameterization of supercuspidal representations of $G_2$ over a $p$-adic field. More precisely, for any finite extension $K / \QQ_p$ we will construct a bijection \[ \CL_g : \CA^0_g(G_2,K) \rightarrow \CG^0(G_2,K) \] from the set of generic supercuspidal representations of $G_2(K)$ to the set of irreducible continuous homomorphisms $ρ: W_K \to G_2(\CC)$ with $W_K$ the Weil group of $K$. The construction of the map is simply a matter of assembling arguments that are already in the literature, together with a previously unpublished theorem of G. Savin on exceptional theta correspondences, included as an appendix. The proof that the map is a bijection is arithmetic in nature, and specifically uses automorphy lifting theorems. These can be applied thanks to a recent result of Hundley and Liu on automorphic descent from $GL(7)$ to $G_2$.

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arXiv2020

Abelian varieties with isogenous reductions

If A and B are abelian varieties over a number field K such that there are non-trivial geometric homomorphisms of abelian varieties between reductions of A and B at most primes of K, then there exists a non-trivial (geometric) homomorphism from A to B defined over an algebraic closure of K.

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arXiv2015

Automorphy of some residually S_5 Galois representations

We prove automorphy lifting theorems for 2-dimensional Galois representations of absolute Galois groups of totally real fields when the residual representation is of "exceptional" type. This exceptional case is when we are in characteristic 5, the residual representation has projective image PGL_2(F_5), and the fixed field of its kernel contains a primitive 5th root of unity. In this case the Taylor-Wiles patching method does not suffice and we prove our theorem by combining patching with the technique of automorphy by p-adic approximation.

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