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Debargha Banerjee

Publications and source records attributed to Debargha Banerjee.

At least 19 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 $S_k(N,\Psi)$ with non-trivial quadratic nebentypus $\Psi$ 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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arXiv2026

A note on extensions of $p$-adic representations of $\mathrm{GL}_2(\mathbb{Q}_p)$

We compute extension groups in the category of duals of $p$-adic Banach space representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. Focusing on representations arising from the $p$-adic local Langlands correspondence for generic Galois representations, we classify these extensions completely. These results are then applied to prove the vanishing of extensions between the duals of reducible representations and supercuspidal isotypic components of the \`etale cohomology of the finite level Drinfeld spaces.

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arXiv2025

An algorithm to compute upper bounds of dimensions for Siegel Modular Forms of Prime Level and Arbitrary Nebentypus

We describe an algorithmic method to determine the image of restriction maps for Siegel modular forms with \textit{arbitrary} characters and arbitrary weight. A program has been implemented in the mathematical software \texttt{Java} to compute the Fourier expansion of the image of these restriction maps for Siegel modular forms of genus two. This approach allows us to compute an upper bound for the space of Siegel modular forms with {\it non-trivial} character (which has not been previously known) and arbitrary weights (including low weight $k \leq 4$).

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arXiv2024

Two results on Eisenstein part of homology and cohomology groups of Bianchi modular groups

We explicitly write down the {\it Eisenstein cycles} in the first homology groups of quotients of the hyperbolic three spaces as linear combinations of Cremona symbols (a generalization of Manin symbols) for imaginary quadratic fields. They generate the Eisenstein part of the homology groups. We also study the Eisenstein part of the cohomology groups. As an application, we find an asymptotic dimension formula (level aspect) for the cuspidal cohomology groups of congruence subgroups of the form $\Ga_1(N)$ inside the full {\it Bianchi groups}.

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arXiv2023

Modular forms with non-vanishing central values and linear independence of Fourier coefficients

In this article, we are interested in modular forms with non-vanishing central critical values and linear independence of Fourier coefficients of modular forms. The main ingredient is a generalization of a theorem due to VanderKam to modular symbols of higher weights. We prove that for sufficiently large primes $p$, Hecke operators $T_1, T_2, \ldots, T_D$ act linearly independently on the winding elements inside the space of weight $2k$ cuspidal modular symbol $\mathbb{S}_{2k}(\Gamma_0(p))$ with $k\geq 1$ for $D^2\ll p$. This gives a bound on the number of newforms with non-vanishing arithmetic $L$-functions at their central critical points and linear independence on the reductions of these modular forms for prime modulo $l\not=p$.

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arXiv2023

On the existence of Siegel modular forms with extra twists

In this paper, we study Siegel modular forms with extra twists. We provide conditions on the level and genus of the forms that is necessary for the existence of extra twists for Siegel modular forms. We also give explicit examples of Siegel modular forms with extra twists that are different from the complex conjugation

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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 $\pi$ associated to a modular form. First we compute the conductor of ${\rm{sym}}^3(\pi)$. 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 $\pi$ 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 $\pi_p$. For ${\rm{sym}}^3$ transfer, the most difficult prime is $p=3$.

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arXiv2022

The intersection matrices of $X_0(p^r)$ and some applications

We compute intersection matrices for modular curves of the form $X_0(p^r)$ with $r \in \{3,4\}$ and as an application, we compute an asymptotic expression for the Arakelov self-intersection number of the relative dualizing sheaf of Edixhoven's minimal regular model for the modular curve $X_0(p^r)$ over $\qq$ with $r$ as above. This computation will be useful to understand an effective version of the Bogolomov conjecture for the stable models of modular curves $X_0(p^r)$ with $r \in \{3,4\}$ and obtain a bound on the stable Faltings height for those curves.

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arXiv2022

The Heisenberg covering of the Fermat curve

For $N$ integer $\ge1$, K. Murty and D. Ramakrishnan defined the $N$-th Heisenberg curve, as the compactified quotient $X'_N$ of the upper half-plane by a certain non-congruence subgroup of the modular group. They ask whether the Manin-Drinfeld principle holds, namely if the divisors supported on the cusps of those curves are torsion in the Jacobian. We give a model over ${\bf Z}[\mu_N,1/N]$ of the $N$-th Heisenberg curve as covering of the $N$-th Fermat curve. We show that the Manin-Drinfeld principle holds for $N=3$, but not for $N=5$. We show that the description by generator and relations due to Rohrlich of the cuspidal subgroup of the Fermat curve is explained by the Heisenberg covering, together with a higher covering of a similar nature. The curves $X_N$ and the classical modular curves $X(n)$, for $n$ even integer, both dominate $X(2)$, which produces a morphism between jacobians $J_N\rightarrow J(n)$. We prove that the latter has image $0$ or an elliptic curve of $j$-invariant $0$. In passing, we give a description of the homology of $X'_{N}$.

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arXiv2022

Ordinary local representations and $\Ext$ groups

We can associate an admissible unitary representation $\Pi(\rho_p)$ of $\GL_2(\Q_p)$ with every local Galois representation $\rho_p$ by the $p$-adic local Langlands correspondence. If $\rho_p$ is ordinary, we prove local and global vanishing results for $\Ext$ functors with respect to these representations.

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arXiv2022

Towards a mod-$p$ Lubin-Tate theory for $\GL_2$ over totally real fields

We show that the conjectural mod $p$ local Langlands correspondence can be realised in the mod $p$ cohomology of the Lubin-Tate towers. The proof utilizes a well known conjecture of Buzzard-Diamond-Jarvis \cite[Conj. 4.9]{BDJ10}, a study of completed cohomology of the ordinary and supersingular locus of the Shimura curves for a totally real field $F$ and of mod $l(\neq p)$ local Langlands correspondence as given by Emerton-Helm \cite{EmertonHelm14}. %And then we connect the completed cohomlgy with the cohomology of Lubin-Tate towers. In the case of modular curves a similar theorem was obtained by Chojecki \cite{Cho15}.

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arXiv2022

The Eisenstein cycles and Manin Drinfeld properties

Consider a subgroup of finite index of modular group. We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian of the corresponding modular curve. By BelyI theorem, such a criterion would apply to any curve over a number field. Our main tool is the explicit description, in terms of modular symbols, of what we call Eisenstein cycles. The latter are representations of relative homology classes over which integration of any holomorphic differential forms vanishes. Our approach relies in an essential way on the specific case , where we can consider convenient generalized Jacobians instead of Jacobian. The Eisenstein classes are the real part of certain homology classes with complex coefficients. The imaginary part of those classes are related to the scattering constants attached to Eisenstein series. Finally, we illustrate our theory by considering Fermat curves.

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arXiv2021

Cuspidal subgroups associated with non-rational Eisenstein maximal ideals

In this paper, we are interested in the generalization of Ramanujan-like Eisenstein congruences (congruences between cusp forms and Eisenstein series) for congruence subgroups of the form $\Gamma_0(N)$ with $N \in \mathbb{N}$. We determine the possible primes that can produce Eisenstein congruences. We provide several examples of Eisenstein congruences to substantiate our method. Ribet conjectured (\cite[p. 360]{MR3540618}) about these congruences for the square-free level $N$. Yoo proved the conjecture. For general $N$, Yoo proved a generalization of the conjecture, under some hypotheses, provided that those ideals are {\it rational}. We show that the generalization of Ribet's conjecture for certain non-square-free levels $N$ is true even for {\it non-rational} Eisenstein maximal ideals.

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arXiv2021

A generalization of Mazur's theorem (Ogg's conjecture) for number fields

In this article, we prove a generalization of a theorem (Ogg's conjecture) due to Bary Mazur for arbitrary $N\in \N$ and for {\it number fields}. The main new observation is a modification of a theorem due to Glenn Stevens for the congruence subgroups of the form $\Ga_0(N)$ for any $N \in \N$. This in turn help us to determine the relevant part of the cuspidal subgroups without dependence on Shimura subgroups.

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arXiv2020

Semi-stable models of modular Curves $X_0(p^2)$ and some arithmetic applications

In this paper, we compute the semi-stable models of modular curves $X_0(p^2)$ for odd primes $p > 3$ and compute the Arakelov self-intersection numbers of the relative dualising sheaves for these models. We give two arithmetic applications of our computations. In particular, we give an effective version of the Bogomolov conjecture following the strategy outlined by Zhang and find the stable Faltings heights of the arithmetic surfaces corresponding to these modular curves.

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arXiv2019

Arakelov Self-intersection numbers of minimal regular models of modular curves $X_0(p^2)$

We compute an asymptotic expression for the Arakelov self-intersection number of the relative dualizing sheaf of Edixhoven's minimal regular model for the modular curve $X_0(p^2)$ over $\mathbb{Q}$. The computation of the self-intersection numbers are used to prove effective Bogolomov conjecture for the semi-stable models of modular curves $X_0(p^2)$ and obtain a bound on the stable Faltings height for those curves in a companion article arXiv:1802.06968.

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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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