arXiv subjects

Manami Roy

Publications and source records attributed to Manami Roy.

31 records · Page 2Linked to original sources

arXiv2023

Dimension formulas for Siegel modular forms of level $4$

We prove several dimension formulas for spaces of scalar-valued Siegel modular forms of degree $2$ with respect to certain congruence subgroups of level $4$. In case of cusp forms, all modular forms considered originate from cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A})$ whose local component at $p=2$ admits non-zero fixed vectors under the principal congruence subgroup of level $2$. Using known dimension formulas combined with dimensions of spaces of fixed vectors in local representations at $p=2$, we obtain formulas for the number of relevant automorphic representations. These in turn lead to new dimension formulas, in particular for Siegel modular forms with respect to the Klingen congruence subgroup of level $4$.

math.NT↗

arXiv2022

Generalized Ramanujan-Sato Series Arising from Modular Forms

Motivated by work of Chan, Chan, and Liu, we obtain a new general theorem which produces Ramanujan-Sato series for $1/π$. We then use it to construct explicit examples related to non-compact arithmetic triangle groups, as classified by Takeuchi. Some of our examples are new, and some reproduce existing examples.

math.NT↗

arXiv2022

Gamma-rays from the circumgalactic medium of M31

We discuss the production of $γ$-rays from cosmic rays (CR) in the circumgalactic medium (CGM) of Andromeda (M31) in light of the recent detection of $γ$-rays from an annular region of $\sim 5.5-120$ kpc away from the M31 disc. We consider the CRs accelerated as a result of the star-formation in the M31 disk, which are lifted to the CGM by advection due to outflow and CR diffusion. The advection time scale due to bulk flow of gas triggered by star formation activity in the M31 disc is comparable ($\sim$ Gyr) to the diffusion time scale with diffusion coefficient $\ge10^{29}$ cm$^2$ s$^{-1}$ for the propagation of CR protons with energy $\sim 412$ GeV that are responsible for the highest energy photons observed. We show that a leptonic origin of the $γ$-rays from cosmic ray (CR) electrons has difficulties, as the inverse Compton time scale ($\sim$Myr) is much lower than advection time scale ($\sim$Gyr) to reach $120$ kpc. Invoking CR electrons accelerated by accretion shocks in the CGM at $\sim100-120$ kpc does not help since it would lead to diffuse X-ray features that are not observed. We, therefore, study the production of $γ$-rays via hadronic interaction between CR protons and CGM gas with the help of numerical two-fluid (thermal + CR) hydrodynamical simulation. We find that a combination of these mechanisms, that are related to the star formation processes in M31 in the last $\sim $ Gyr, along with diffusion and hadronic interaction, can explain the observed flux from the CGM of M31.

astro-ph.GA↗

arXiv2022

The completed standard $L$-function of modular forms on $G_2$

The goal of this paper is to provide a complete and refined study of the standard $L$-functions $L(π,\operatorname{Std},s)$ for certain non-generic cuspidal automorphic representations $π$ of $G_2(\mathbb{A})$. For a cuspidal automorphic representation $π$ of $G_2(\mathbb{A})$ that corresponds to a modular form $φ$ of level one and of even weight on $G_2$, we explicitly define the completed standard $L$-function, $Λ(π,\operatorname{Std},s)$. Assuming that a certain Fourier coefficient of $φ$ is nonzero, we prove the functional equation $Λ(π,\operatorname{Std},s) = Λ(π,\operatorname{Std},1-s)$. Our proof proceeds via a careful analysis of a Rankin-Selberg integral that is due to an earlier work of Gurevich and Segal.

math.NT↗

arXiv2022

Constraints on cosmic rays in the Milky Way circumgalactic medium from OVIII observations

We constrain the cosmic ray (CR) population in the circumgalactic medium (CGM) of Milky Way by comparing the observations of absorption lines of OVIII ion with predictions from analytical models of CGM : precipitation (PP) and isothermal (IT) model. For a CGM in hydrostatic equilibrium, the introduction of CR suppresses thermal pressure, and affects the OVIII ion abundance. We explore the allowances given to the ratio of CR pressure to thermal pressure ($\rm{P}_{\rm{CR}}/\rm{P}_{\rm{th}}=η$), with varying boundary conditions, CGM mass content, photoionization by extragalactic ultraviolet background and temperature fluctuations. We find that the allowed maximum values of $η$ are : $η\lesssim10$ in the PP model and $η\lesssim6$ in the IT model. We also explore the spatial variation of $η$ : rising ($η=Ax$) or declining ($η=A/x$) with radius, where A is the normalization of the profiles. In particular, the models with declining ratio of CR to thermal pressure fare better than those with rising ratio with suitable temperature fluctuation (larger $σ_{\rm ln T}$ for PP and lower for IT). The declining profiles allow $A\lesssim8$ and $A\lesssim10$ in the case of IT and PP models, respectively, thereby accommodating a large value of $η\,(\simeq 200)$ in the central region, but not in the outer regions. These limits, combined with the limits derived from $γ$-ray and radio background, can be useful for building models of Milky Way CGM including CR population. However, the larger amount of CR can be packed in cold phase which may be one way to circumvent these constraints.

astro-ph.GA↗

arXiv2022

Representations attached to elliptic curves with a non-trivial odd torsion point

We give a classification of the cuspidal automorphic representations attached to rational elliptic curves with a non-trivial torsion point of odd order. Such elliptic curves are parameterizable, and in this paper, we find the necessary and sufficient conditions on the parameters to determine when split or non-split multiplicative reduction occurs. Using this and the known results on when additive reduction occurs for these parametrized curves, we classify the automorphic representations in terms of the parameters.

math.NT↗

arXiv2022

Local data of rational elliptic curves with non-trivial torsion

By Mazur's Torsion Theorem, there are fourteen possibilities for the non-trivial torsion subgroup $T$ of a rational elliptic curve. For each $T$, such that $E$ may have additive reduction at a prime $p$, we consider a parameterized family $E_T$ of elliptic curves with the property that they parameterize all elliptic curves $E/\mathbb{Q}$ which contain $T$ in their torsion subgroup. Using these parameterized families, we explicitly classify the Kodaira-Néron type, the conductor exponent, and the local Tamagawa number at each prime $p$ where $E/\mathbb{Q}$ has additive reduction. As a consequence, we find all rational elliptic curves with a $2$-torsion or a $3$-torsion point that have global Tamagawa number $1$.

math.NT↗

arXiv2021

Classical and adelic Eisenstein series

We carry out "Hecke summation" for the classical Eisenstein series $E_k$ in an adelic setting. The connection between classical and adelic functions is made by explicit calculations of local and global intertwining operators and Whittaker functions. In the process we determine the automorphic representations generated by the $E_k$, in particular for $k=2$, where the representation is neither a pure tensor nor has finite length. We also consider Eisenstein series of weight $2$ with level, and Eisenstein series with character.

math.NT↗

arXiv2021

Congruences for dimensions of spaces of Siegel cusp forms and $4$-core partitions

Using the relationship between Siegel cusp forms of degree $2$ and cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$, we derive some congruences involving dimensions of spaces of Siegel cusp forms of degree $2$ and the class number of $\mathbb{Q}(\sqrt{-p})$. We also obtain some congruences between the $4$-core partition function $c_4(n)$ and dimensions of spaces of Siegel cusp forms of degree $2$.

math.NT↗

arXiv2021

Paramodular forms coming from elliptic curves

There is a lifting from a non-CM elliptic curve $E/\mathbb{Q}$ to a paramodular form $f$ of degree $2$ and weight $3$ given by the symmetric cube map. We find the level of $f$ in an explicit way in terms of the coefficients of the Weierstrass equation of $E$. In order to compute the paramodular level, we use the available description of the local representations of $\mathrm{GL}(2,\mathbb{Q}_p)$ attached to $E$ for $p \ge 5$ and determine the local representation of $\mathrm{GL}(2,\mathbb{Q}_3)$ attached to $E$.

math.NT↗

arXiv2021

A panoramic view of the circumgalactic medium in the photoionized precipitation model

We consider a model of the circumgalactic medium (CGM) in which feedback maintains a constant ratio of cooling time to freefall time throughout the halo, so that the entire CGM is marginally unstable to multiphase condensation. This 'precipitation model' is motivated by observations of multiphase gas in the cores of galaxy clusters and the halos of massive ellipticals. We derive from the model density and temperature profiles for the CGM around galaxies with masses similar to the Milky Way. After taking into consideration the geometrical position of our solar system in the Milky Way, we show that the CGM model is consistent with observed OVI, OVII, and OVIII column densities only if temperature fluctuations with a log-normal dispersion $σ_{\ln T} \sim 0.6$-$1.0$ are included. We show that OVI column densities observed around star-forming galaxies require systematically greater values of $σ_{\ln T}$ than around passive galaxies, implying a connection between star formation in the disk and the state of the CGM. Photoionization by an extra-galactic UV background does not significantly change these CGM features for galaxies like the Milky Way but has much greater and significant effects on the CGM of lower-mass galaxies.

astro-ph.GA↗

arXiv2021

On counting cuspidal automorphic representations for $\mathrm{GSp}(4)$

We find the number $s_k(p,Ω)$ of cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$ with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight $k\ge 3$, and the non-archimedean component at $p$ is an Iwahori-spherical representation of type $Ω$ and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for $s_k(p,Ω)$ generalizes to the vector-valued case and a finite number of ramified places.

math.NT↗

arXiv2020

Gamma-ray and radio background constraints on cosmic rays in Milky Way circumgalactic medium

We study the interaction of cosmic rays (CRs) with the diffuse circumgalactic gas of Milky Way (MW) galaxy that results in hadronic $γ-$ray emission and radio synchrotron emission. We aim to constrain the CR population in our circumgalactic medium (CGM) with the help of observed isotropic $γ$-ray background (IGRB), its anisotropy and radio continuum. We modify different models of CGM gas in hydrostatic equilibrium discussed in literature by including a cosmic ray population, parametrized by $η\equiv P_{\rm CR}/P_{\rm th}$. For the simplest isothermal model, while the IGRB intensity allows $η\lesssim 3$, the anisotropy resulting from the Solar system's off-center position in MW rules out all values of $η$. For the precipitation model, in which the cooling of the CGM gas is regulated with an optimum ratio of cooling time to free-fall time,while the observed IGRB intensity allows $η\lesssim 230$, the observed anisotropy allows only very large values of $η$, of order $\gtrsim 100$. The radio continuum limits $η\lesssim 400$ for precipitation model and does not constrain isothermal model, however these constraints are mitigated by synchrotron loss time being comparable to CR diffusion time scales. These bounds are relevant for current numerical simulations that indicate a significant CR population in CGM of galaxies of MW mass.

astro-ph.GA↗