arXiv subjects

Jyoti Prakash Saha

Publications and source records attributed to Jyoti Prakash Saha.

25 records · Page 2Linked to original sources

arXiv2018

The density of ramified primes

Let $F$ be a number field, $\mathcal{O}$ be a domain with fraction field $\mathcal{K}$ of characteristic zero and $ρ: \mathrm{Gal}(\overline F/F) \to \mathrm{GL}_n(\mathcal{O})$ be a representation such that $ρ\otimes\overline{\mathcal{K}}$ is semisimple. If $\mathcal{O}$ admits a finite monomorphism from a power series ring with coefficients in a $p$-adic integer ring (resp. $\mathcal{O}$ is an affinoid algebra over a $p$-adic number field) and $ρ$ is continuous with respect to the maximal ideal adic topology (resp. the Banach algebra topology), then we prove that the set of ramified primes of $ρ$ is of density zero. If $\mathcal{O}$ is a complete local Noetherian ring over $\mathbb{Z}_p$ with finite residue field of characteristic $p$, $ρ$ is continuous with respect to the maximal ideal adic topology and the kernels of pure specializations of $ρ$ form a Zariski-dense subset of $\mathrm{Spec} \mathcal{O}$, then we show that the set of ramified primes of $ρ$ is of density zero. These results are analogues, in the context of big Galois representations, of a result of Khare and Rajan, and are proved relying on their result.

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arXiv2018

On Frobenius semisimplicity in Hida families

Let $p\geq 5$ be a prime and $\ell\neq p$ be a prime not dividing the tame level of a $p$-ordinary Hida family. We prove that the actions of the Frobenius element at $\ell$ on the Galois representations attached to almost all arithmetic specializations are semisimple and non-scalar. If $k_f$ denotes the weight of a cusp form $f(z)= \sum_{n\geq 1} a_\ell(f) e^{2πi n z}$, then the inequality $$|a_\ell(f) | \leq 2 \ell^{(k_f-1)/2},$$ predicted by the Ramanujan conjecture, is a strict inequality for almost all members $f$ of the family.

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arXiv2018

Distinguishing pure representations by normalized traces

Given two pure representations of the absolute Galois group of an $\ell$-adic number field with coefficients in $\overline{\mathbb{Q}}_p$ (with $\ell\neq p$), we show that the Frobenius-semisimplifications of the associated Weil--Deligne representations are twists of each other by an integral power of certain unramified character if they have equal normalized traces. This is an analogue of a recent result of Patankar and Rajan in the context of local Galois representations.

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arXiv2016

Variation of Weyl modules in $p$-adic families

Given a Weil-Deligne representation with coefficients in a domain, we prove the rigidity of the structures of the Frobenius-semisimplifications of the Weyl modules associated to its pure specializations. Moreover, we show that the structures of the Frobenius-semisimplifications of the Weyl modules attached to a collection of pure representations are rigid if these pure representations lift to Weil-Deligne representations over domains containing a domain $\mathscr{O}$ and a pseudorepresentation over $\mathscr{O}$ parametrizes the traces of these lifts.

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arXiv2015

Conductors in $p$-adic families

Given a Weil-Deligne representation of the Weil group of an $\ell$-adic number field with coefficients in a domain $\mathscr{O}$, we show that its pure specializations have the same conductor. More generally, we prove that the conductors of a collection of pure representations are equal if they lift to Weil-Deligne representations over domains containing $\mathscr{O}$ and the traces of these lifts are parametrized by a pseudorepresentation over $\mathscr{O}$.

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arXiv2014

Purity for families of Galois representations

We formulate a notion of purity for $p$-adic big Galois representations and pseudorepresentations of Weil groups of $\ell$-adic number fields for $\ell\neq p$. This is obtained by showing that all powers of the monodromy of any big Galois representation stay "as large as possible" under pure specializations. The role of purity for families in the study of the variation of local Euler factors, local automorphic types along irreducible components, the intersection points of irreducible components of $p$-adic families of automorphic Galois representations is illustrated using the examples of Hida families and eigenvarieties. Moreover, using purity for families, we improve a part of the local Langlands correspondence for $\mathrm{GL}_n$ in families formulated by Emerton and Helm.

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arXiv2014

Congruences of algebraic $L$-functions of motives

We develop a framework to investigate conjectures on congruences between the algebraic part of special values of $L$-functions of congruent motives. We show that algebraic local Euler factors satisfy precise interpolation properties in $p$-adic families of motives and that algebraic $p$-adic $L$-functions exist in quite large generality for $p$-adic families of automorphic motives. We formulate two conjectures refining (and correcting) the currently existing formulation of the Equivariant Tamagawa Number Conjecture with coefficients in Hecke algebras and pointing out the links between conjectures on special values and completed cohomology.

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