arXiv subjects

Tathagata Mandal

Publications and source records attributed to Tathagata Mandal.

7 recordsLinked to original sources

arXiv2026

Towards a generalized Maeda conjecture for modular forms with quadratic nebentypus

We establish a lower bound for the number of non-CM Galois orbits of newforms in $S_k(N,Ψ)$ with non-trivial quadratic nebentypus $Ψ$ for sufficiently large weights. Extending the work of Dieulefait, Pacetti, and Tsaknias in the trivial nebentypus setting, we analyze the restrictions imposed by the quadratic character on local inertial types and determine the number of admissible Galois orbits of such types. We further prove that Atkin-Li pseudo-eigenvalues are Galois equivariant and hence, up to a natural equivalence relation, define a global Galois invariant. Together with existence results for newforms having prescribed local behavior, these invariants yield a lower bound for the number of non-CM Galois orbits by counting compatible pairs of local-global invariants. Finally, computations in small weights show that this lower bound is not always attained, indicating that certain local equivalences are not realized globally by Galois conjugation over the coefficient field of the modular form.

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arXiv2024

On the change of epsilon factors for symmetric square transfers under twisting and applications

Let us consider the symmetric square transfer of the automorphic representation $π$ associated to a modular form $f \in S_k(N,ε)$. In this article, we study the variation of the epsilon factor of ${\mathrm{sym}}^2(π)$ under twisting in terms of the local Weil-Deligne representation at each prime $p$. As an application, we detect the possible types of the symmetric square transfer of the local representation at $p$. Furthermore, as the conductor of ${\mathrm{sym}}^2(π)$ is involved in the variation number, we compute it in terms of $N$.

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arXiv2023

Contribution of symmetric power transfers to the cuspidal cohomology of ${\rm GL_n}$

Let $π$ be a cuspidal automorphic representation of ${\mathrm {GL}}_2(\mathbb{A}_\mathbb{Q})$. Newton and Thorne have proved that for every $n\geq 1$, the symmetric power lifting ${\mathrm {sym}^n(π)}$ is automorphic if $π$ is attached to a non-CM Hecke eigenform. In this article, we establish an asymptotic estimate of the number of cuspidal automorphic representations of ${\mathrm {GL}}_{n+1}(\mathbb{A}_\mathbb{Q})$ which contribute to the cuspidal cohomology of ${\mathrm {GL}}_{n+1}$ and are obtained by symmetric $n$th transfer of cuspidal representations of ${\mathrm {GL}}_2(\mathbb{A}_\mathbb{Q})$. Here we fix the weight and vary the level. This generalises the previous works done for ${\mathrm {GL}}_3$ and ${\mathrm {GL}}_4$.

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arXiv2023

Two properties of symmetric cube transfers of modular forms

In this article, we study two important properties of ${\rm{sym}}^3$ transfers of the automorphic representation $π$ associated to a modular form. First we compute the conductor of ${\rm{sym}}^3(π)$. Then we detect the types of local automorphic representations at bad primes by the variation of the epsilon factors of symmetric cube transfer of the representation $π$ attached to a cusp form $f$. Here we twist the modular forms by a specific quadratic character. From this variation number, for each prime $p$, we classify all possible types of symmetric cube transfers of the local representations $π_p$. For ${\rm{sym}}^3$ transfer, the most difficult prime is $p=3$.

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arXiv2020

Multiplicities in Selmer groups and root numbers for Artin twists

Let $K/F$ be a finite Galois extension of number fields and $σ$ be an absolutely irreducible, self-dual representation of $\mathrm{Gal}(K/F)$. Let $p$ be an odd prime and consider two elliptic curves $E_1, E_2$ with good, ordinary reduction at primes above $p$ and equivalent mod-$p$ Galois representations. In this article, we study the variation of the parity of the multiplicities of $σ$ in the representation space associated to the $p^\infty$-Selmer group of $E_i$ over $K$. We also compare the root numbers for the twist of $E_i/F$ by $σ$ and show that the $p$-parity conjecture holds for the twist of $E_1/F$ by $σ$ if and only if it holds for the twist of $E_2/F$ by $σ$. We also express Mazur-Rubin-Nekovář's arithmetic local constants in terms of certain local Iwasawa invariants.

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arXiv2018

Supercuspidal ramifications and traces of adjoint lifts at good primes

In this paper, we write down the local Brauer classes of the endomorphism algebras of motives attached to non-CM primitive Hecke eigenforms for all supercuspidal primes in terms of traces of adjoint lifts at auxiliary primes. We give an alternative proof of the result for odd primes $p$ obtained in [MR3391026] and write down the ramification formulas for odd unramified supercuspidal primes of level zero also removing a mild hypothesis of [MR3391026]. We also give a complete description of ramifications for $p=2$. The philosophy of adjoint lifts help us to determine the local Brauer classes at non dihedral primes by using results similar to [MR2770587]. We provide some numerical examples using {Sage} and {LMFDB} supporting some of our theorems.

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arXiv2018

A note on quadratic twisting of epsilon factors for modular forms with arbitrary nebentypus

In this article, we investigate the variance of local $\varepsilon$-factor for a modular form with arbitrary nebentypus with respect to twisting by a quadratic character. We detect the type of the supercuspidal representation from that. For modular forms with trivial nebentypus, similar results are proved by Pacetti. Our method however is completely different from that of Pacetti and we use representation theory crucially. For ramified principal series (with $\infdiv{p}{N}$ and $p$ odd) and unramified supercuspidal representations of level zero, we relate these numbers with the Morita's $p$-adic Gamma function.

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