arXiv subjects

Dipramit Majumdar

Publications and source records attributed to Dipramit Majumdar.

18 recordsLinked to original sources

arXiv2026

Explicit mock Heegner points and BSD formula on certain Mordell curves

For a natural number $a$, let $E_{2a}$ be the Mordell elliptic curve $X^3 +Y^3=2a$. We give an explicit construction of (mock) Heegner point on the Mordell curve $E_{2p}$ for a prime $p\equiv 4 \mod 9$ and $E_{2p^2}$ for a prime $p\equiv 7 \mod 9$, under the assumption that $2$ is not a cube modulo $p$. We also verify the explicit Gross-Zagier formula for these curves and go on to show that the BSD formula holds for these curves up to a $2$-adic unit. Using a result of Burungale-Flach, we show that the full BSD formula holds for the rank zero curve $E_{2p}$ for $p\equiv 7 \mod 9$ and $E_{2p^2}$ for $p\equiv 4 \mod 9$, whenever $2$ is not a cube modulo $p$.

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arXiv2026

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 $Γ_0(N)$ 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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arXiv2025

Hilbert's 10th Problem via Mordell curves

We show that for $5/6$-th of all primes $p$, Hilbert's 10-th Problem is unsolvable for $\mathbb{Q}(ζ_3, \sqrt[3]{p})$. We also show that there is an infinite set $S$ of square free integers such tha Hilbert's 10-th Problem is unsolvable over the number fields $\mathbb{Q}(ζ_3, \sqrt{D}, \sqrt[3]{p})$ for every $D \in S$ and every prime $p \equiv 2,5 \pmod{9}$. We use the CM elliptic curves $Y^2=X^3-432D^2$ associated to the cube sum problem, with $D$ varying in suitable congruence class, in our proof.

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arXiv2025

$\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks

We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b)^2$ with $a,b \in \mathbb{Z}$. These elliptic curves have a rational $3$-isogeny, say $φ$. We give an upper and a lower bound on the rank of the $φ$-Selmer group of $E_{a,b}$ over $K:=\mathbb{Q}(ζ_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $K$. Using our bounds on the Selmer groups, we construct infinitely many curves in this family with arbitrary large $3$-Selmer rank over $K$ and no non-trivial $K$-rational point of order $3$. We also show that for a positive proportion of natural numbers $n$, the curve $E_{n,n}/\mathbb{Q}$ has root number $-1$ and $3$-Selmer rank $=1$.

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arXiv2025

$3$-Selmer group, ideal class groups and cube sum problem

Consider a Mordell curve $E_a:y^2=x^3+a$ with $a \in \mathbb Z$. These curves have a rational $3$-isogeny, say $φ$. We give an upper and a lower bound on the rank of the $φ$-Selmer group of $E_a$ over $\mathbb Q(ζ_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $\mathbb Q(ζ_3)$. Using our bounds on the Selmer groups, we prove some cases of the rational cube sum problem. Further, using these bounds, we give explicit families of the Mordell curves to show that for a positive proportion of $E_a$, ${\rm Sel}^3(E_{a}/\mathbb Q)=0$ (respectively ${\rm Sel}^3(E_{a}/\mathbb Q)$ has $\mathbb F_3$-rank $1$).

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arXiv2024

Relative $p$-class groups and $p$-Selmer groups

Let $E$ be an elliptic curve with $j$-invariant $0$ or $1728$ and let $\widetilde{E}$ be a $k^{th}$ twist of $E$. We show that for any prime $p$ of good reduction of $\widetilde{E}$, a degree $k$ relative $p$-class group and the root number of $\widetilde{E}$ determines the dimension of the $p$-Selmer group of $\widetilde{E}$. As a consequence, we construct families of large rank $p$-class group. We also relate congruent number and cube sum problem with relative $p$-class group.

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arXiv2024

Binary Cubic Forms and Rational Cube Sum Problem

In this note, we use integral binary cubic forms to study the rational cube sum problem. We prove (unconditionally) that for any positive integer $d$, infinitely many primes in each of the residue classes $ 1 \pmod {9d}$ as well as $ -1 \pmod {9d}$, are sums of two rational cubes. Among other results, we prove that every non-zero residue class $a \pmod {q}$, for any prime $q$, contains infinitely many primes which are sums of two rational cubes. Further, for an arbitrary integer $N$, we show there are infinitely many primes $p$ in each of the residue classes $ 8 \pmod 9$ and $1 \pmod 9$, such that $Np$ is a sum of two rational cubes.

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arXiv2022

Cube sum problem for integers having exactly two distinct prime factors

Given an integer n>1, it is a classical Diophantine problem that whether n can be written as a sum of two rational cubes. The study of this problem, considering several special cases of n, has a copious history that can be traced back to the works of Sylvester, Satgé, Selmer etc. and up to the recent works of Alpöge-Bhargava-Shnidman. In this article, we consider the cube sum problem for cube-free integers n which has two distinct prime factors none of which is 3.

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arXiv2022

Cyclic Cubic Extensions of Q

In this article we explicitly describe irreducible trinomials X^3-aX+b which gives all the cyclic cubic extensions of Q. In doing so, we construct all integral points (x,y,z) with GCD(y,z)=1, of the curves X^2+3Y^2 = 4DZ^3 and X^2+27Y^2=4DZ^3 as D varies over cube-free positive integers. We parametrise these points using well known parametrisation of integral points (x,y,z) of the curve X^2+3Y^2=4Z^3 with GCD(y,z)=1. As an accidental byproduct of our result we show that there are infinitely many primes congruent to 1 or 8 modulo 9, can be expressed as sum of two rational cubes.

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arXiv2022

On characteristic ideal of Selmer group associated to Artin representations

Selmer group for an Artin representation over totally real fields was studied by Greenberg and Vatsal. In this paper we study the Selmer groups for an Artin representation over a totally complex field. We establish an algebraic function of the characteristic ideal of the Selmer group associated to Artin representation over the cyclotomic $\Z_p$- extension of the rational numbers under certain mild hypotheses and construct several examples to illustrate our result. We also prove that in this situation $μ$-invariant of the dual Selmer group is independent of the choice of the lattice.

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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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arXiv2021

Fruit Diophantine Equation

We show that the Diophantine equation given by X^3+ XYZ = Y^2+Z^2+5 has no integral solution. As a consequence, we show that the family of elliptic curve given by the Weierstrass equations Y^2-kXY = X^3 - (k^2+5) has no integral point.

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arXiv2019

$p^r$-Selmer companion modular forms

The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. Given two elliptic curves $E_1$ and $E_2$ over a number field $K$, Mazur-Rubin\cite{mr} have defined them to be {\it $n$-Selmer companion} if for every quadratic twist $χ$ of $K$, the $n$-Selmer groups of $E_1^χ$ and $E_2^χ$ over $K$ are isomorphic. Given a prime $p$, they have given sufficient conditions for two elliptic curves to be $p^r$-Selmer companion in terms of mod-$p^r$ congruences between the curves. We discuss an analogue of this for Bloch-Kato $p^r$-Selmer group of modular forms. We compare the Bloch-Kato Selmer groups of a modular form respectively with the Greenberg Selmer group when the modular form is $p$-ordinary and with the signed Selmer group of Lei-Loeffler-Zerbes when the modular form is non-ordinary at $p$. We also indicate the corresponding results over $\Q_\cyc$ and its relation with the well known congruence results of the special values of the corresponding $L$-functions due to Vatsal.

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arXiv2017

p-adic Asai transfer

Let $K/Q$ be a real quadratic field. Given an automorphic representation $π$ for $GL_{2}/K$, let $As^{\pm}(π)$ denote the plus/minus Asai transfer of $π$ to an automorphic representation for $GL_{4}/Q$. In this paper, we construct a rigid analytic map from the universal eigenvariety of $GL_{2}/K$ to the universal eigenvariety of $GL_{4}/Q$, which at nice classical points interpolate this Asai transfer.

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arXiv2015

Functional equation for the Selmer group of nearly ordinary Hida deformation of Hilbert modular forms

We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension of a totally real number field. Further, we use this result to establish a duality or algebraic `functional equation' for the `big' Selmer groups associated to the corresponding nearly ordinary Hida deformation. The multivariable cyclotomic Iwasawa main conjecture for nearly ordinary Hida family of Hilbert modular forms is not established yet and this can be thought of as an evidence to the validity of this Iwasawa main conjecture. We also prove a functional equation for the `big' Selmer group associated to an ordinary Hida family of elliptic modular forms over the $\mathbb{Z}_{p}^{2}$ extension of an imaginary quadratic field.

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arXiv2015

l-Class groups of cyclic extensions of prime degree l

Let K/F be a cyclic extension of prime degree l over a number field F. If F has class number coprime to l, we study the structure of the l-Sylow subgroup of the class group of K. In particular, when F contains the l-th roots of unity, we obtain bounds for the F_ rank of the l-Sylow subgroup of K using genus theory. We obtain some results valid for general l. Following that, we obtain more complete results for l=5 and F =Q(ζ_5). The rank of the 5-class group of K is expressed in terms of power residue symbols. We compare our results with tables obtained using SAGE (the latter is under GRH). We obtain explicit results in several cases. Using these results, and duality theory, we deduce results on the 5-class numbers of fields of the form Q(n^1/5).

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arXiv2014

Geometry of the eigencurve at critical Eisenstein series of weight 2

In this paper we show that the critical Eisenstein series of weight 2, E_{2}^{crit_{p}}, is smooth in the eigencurve C(l), where l is a prime. We also show that E_{2}^{crit_{p},ord_{l}} is smooth in the full eigencurve C^{full}(l) and E_{2}^{crit_{p},ord_{l_{1}},ord_{l_{2}}} is non-smooth in the full eigencurve C^{full}(l_{1}l_{2}). Further, we show that, E_{2}^{crit_{p}}, is étale over the weight space in the eigencurve C(l). As a consequence, we show that level lowering conjecture of Paulin fails to hold at E_{2}^{crit_{p},ord_{l}}.

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arXiv2014

Endoscopic transfer between eigenvarieties for definite unitary groups

In this paper, we extend the endoscopic transfer of definite unitary group U(n), which sends a pair of automorphic forms of U(m),U(n) to an automorphic form of U(m+n), to finite slope p-adic automorphic forms for definite unitary groups by constructing a rigid analytic map between eigenvarieties E_m \times E_n \to E_{m+n}, which at classical points interpolates endoscopic transfer map.

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