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Chandrashekhar Khare

Publications and source records attributed to Chandrashekhar Khare.

At least 37 records · Page 2Linked to original sources
arXiv2008

Functoriality and the Inverse Galois problem II: groups of type B_n and G_2

For every finite field F and every positive integer r, there exists a finite extension F' of F such that either SO(2r+1,F') or its simple derived group can be realized as a Galois group over Q. If the characteristic of F is 3 or 5 (mod 8), then we can guarantee that the derived group of SO(2r+1,F') can be realized. Likewise, for every finite field F, there exists a finite extension F' of F such that the finite simple group G_2(F') can be realized a Galois group over Q. The proof uses automorphic forms to construct Galois representations which cut out Galois extensions of the desired type.

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arXiv2004

On Serre's reciprocity conjecture for 2-dimensional mod p representations of the Galois group of Q

We first prove the existence of minimally ramified p-adic lifts of 2-dimensional mod p representations, that are odd and irreducible, of the absolute Galois group of Q,in many cases. This is predicted by Serre's conjecture that such representations arise from newforms of optimal level and weight. Using these minimal lifts, and arguments using compatible systems, we prove some cases of Serre's conjectures in low levels and weights. For instance we prove that there are no irreducible (p,p) type group schemes over the rational integers. We prove that a Galois representation as above of Artin conductor 1 and Serre weight 12 arises from the Ramanujan Delta-function. In the last part of the paper we present arguments that reduce Serre's conjecture on modularity of Galois representations as above to proving modularity lifting theorems of the type pioneered by Wiles. While these modularity lifting results are not known as yet they might be relatively accessible because of the basic method of Wiles and Taylor-Wiles and recent developments in the p-adic Langlands programmeinitiated by Breuil.

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arXiv2004

Mod $\ell$ representations of arithmetic fundamental groups: Part I (An analog of Serre's conjecture for function fields)

We formulate for function fields an analog of Serre's conjecture on the modularity of 2-dimensional irreducible mod l representations of the absolute Galois group of Q: our analog is not restricted to 2-dimensional represntations. While the original conjecture of Serre is wide open at the moment, in this paper we prove the function field analog of Serre's conjecture only assuming largeness of image of the representation and some assumptions on the ramification of the representation (for example if the mod l representation is unramified and has full image, l is odd and the dimension and characteristic of the representation are coprime).

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arXiv2004

Mod $\ell$ representations of arithmetic fundamental groups II (A conjecture of A.J. de Jong)

As a sequel to our proof of the analog of Serre's conjecture for function fields in Part I of this work, we study in this paper the deformation rings of $n$-dimensional mod $\ell$ representations $ρ$ of the arithmetic fundamental group $π_1(X)$ where $X$ is a geometrically irreducible, smooth curve over a finite field $k$ of characteristic $p$ ($\neq \ell$). We are able to show in many cases that the resulting rings are finite flat over $\BZ_\ell$. The proof principally uses a lifting result of the authors in Part I of this two-part work, Taylor-Wiles systems and the result of Lafforgue. This implies a conjecture of A.J. ~de Jong for representations with coefficients in power series rings over finite fields of characteristic $\ell$, that have this mod $\ell$ representation as their reduction.

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arXiv2004

Transcendental l-adic Galois representations

We consider continuous representations of the Galois group G of a number field K taking values in the completion C of an algebraic closure A of the field of l-adic numbers. We give a construction of irreducible representations of G in GL(2,C) which cannot be conjugated into GL(2,A) but for which there exist density-one sets of primes with Frobenius traces in A. We also show that any such representation must be ramified at infinitely many primes of K.

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arXiv2003

Reciprocity law for compatible systems of abelian mod $p$ Galois representations

The main result of the paper is a reciprocity law which proves that compatible systems of semisimple, abelian mod $p$ representations (of arbitrary dimension) of absolute Galois groups of number fields, arise from Hecke characters. In the last section of the paper analogs for Galois groups of function fields of these results are explored, and a question is raised whose answer will require developments in transcendence theory in characteristic $p$.

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arXiv2003

Constructing semisimple p-adic Galois representations with prescribed properties

In this paper we show that two dimensional (mod p) Galois representations satisfying mild hypotheses can be lifted to p-adic Galois representations ramified at infinitely many primes such that the characteristic polynomials of Frobenius at a density one set of unramified primes are defined over the rational integers. In particular we show that often one can "density one compatibly" lift mod p and mod q Galois representations.

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arXiv2002

Reduction of homomorphisms mod p and algebraicity

{Let $K$ be a number field, and $A_1,A_2$ abelian varieties over $K$. Let $P$ (resp. $Q$) be a non-torsion point in $ A_1(K)$ (resp. $A_2(K)$) such that for almost all places $v$ of $K$, the order of $Q$ mod $v$ divides the order of $P$ mod $v$. Then we prove (under some conditions on $A_i, i=1,2$) that there is a homomorphism $j$ from $A_1$ to $A_2$ such that $j(P) = Q$. We formulate and extend such a result for any subgroups of $A_i(K), i=1,2$. These results in particular extend the work of Corrales and Schoof from elliptic curves to abelian varieties.

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arXiv2002

Finiteness of Selmer groups and deformation rings

We prove the finiteness of Selmer groups attached to lifts of certain 2-dimensional mod p representations of the absolute Galois group of Q. The mod p representation can be either even or odd. The lifts considered are the ones that were proven to exist by the seond named author.

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arXiv2002

On isomorphisms between deformation rings and Hecke rings

We give a different approach to the well-known modularity lifting results of Wiles and Taylor. Instead of Taylor-Wiles systems we use a Galois cohomological discovery of R. Ramakrishna. This paper is 2 years older than math.NT/0210296 where another approach to modularity lifting results was presented.

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arXiv2002

Hasse invariant and group cohomology

Let p be a prime number. The Hasse invariant is a modular form modulo p that is often used to produce congruences between modular forms of different weights. We show how to produce such congruences between forms of weights 2 and p+1, in terms of group cohomology. We also show how our method works in the contexts of quadratic imaginary fields (where there is no Hasse invariant available) and Hilbert modular forms over totally real fields of even degree.

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arXiv2002

F-split Galois representations are potentially abelian

In this note we relate the property of a semisimple l-adic Galois representation being "F-split" for a number field F to its having abelian image. By F-split we mean that the characteristic polynomials of Frobenii all split in F.

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arXiv2002

Mod pq Galois representations and Serre's conjecture

Motives and automorphic forms of arithmetic type give rise to Galois representations that occur in {\it compatible families}. These compatible families are of p-adic representations with p varying. By reducing such a family mod p one obtains compatible families of mod p representations. While the representations that occur in such a p-adic or mod p family are strongly correlated, in a sense each member of the family reveals a new face of the motive. In recent celebrated work of Wiles playing off a pair of Galois representations in different characteristics has been crucial. In this paper we investigate when a pair of mod p and mod q representations of the absolute Galois group of a number field K simultaneously arises from an {\it automorphic motive}: we do this in the 1-dimensional (Section 2) and 2-dimensional (Section 3: this time assuming $K={\mathbb Q}$) cases. In Section 3 we formulate a mod pq version of Serre's conjecture refining in part a question of Barry Mazur and Ken Ribet.

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arXiv2002

Compatible systems of mod p Galois representations II

We prove a reciprocity law for one-dimensional compatible systems of mod p representations of absolute Galois groups of number fields. We prove that these arise from Hecke characters, and in particular recover by purely algebraic means the classical result (that is usually proven using deep results of Waldschmidt in transcendental number theory) that one-dimensional compatible p-adic systems arise from Hecke characters.

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arXiv2002

Belyi parametrisations of elliptic curves and congruence defects

In this note we consider questions about parametrisations of elliptic curves defined over number fields by quotients of the upper half-plane by finite index subgroups of SL_2(Z). We ask if we can choose such a parametrisation of an elliptic curve to reflect the field over which it is defined: this can be viewed as a generalisation of the Shimura-Taniyama-Weil conjecture.

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