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Kalyan Banerjee

Publications and source records attributed to Kalyan Banerjee.

At least 19 recordsLinked to original sources
arXiv2025

On the simultaneous $3$-divisibility of class numbers of quadruples of real quadratic fields

In this paper, we construct infinitely many quadruples of real quadratic fields whose class numbers are all divisible by $3$. To the best of our knowledge, this is the first result towards the divisibility of the class numbers of certain tuples of real quadratic fields. At the end, we give an application of this result to produce some elliptic curves having a $3$-torsion subgroup.

math.NT↗
arXiv2025

Class groups of imaginary biquadratic fields

We present two distinct families of imaginary biquadratic fields, each of which contains infinitely many members, with each member having large class groups. Construction of the first family involves elliptic curves and their quadratic twists, whereas to find the other family, we use a combination of elliptic and hyperelliptic curves. Two main results are used, one from Soleng and the other from Banerjee and Hoque.

math.NT↗
arXiv2025

Elliptic surfaces to class groups and Selmer groups

In this note, we connect the $n$-torsions of the Picard group of an elliptic surface to the $n$-divisibility of the class group of torsion fields for a given integer $n>1$. We also connect the $n$-divisibility of the Selmer group to that of the class group of torsion fields.

math.NT↗
arXiv2025

Bloch's conjecture on surfaces of general type with $p_g=q=0, K^2=3$ and with an involution

In this short note we prove that an involution on certain examples of surfaces of general type with $p_g=0=q, K^2=3$, acts as identity on the Chow group of zero cycles of the relevant surface. In particular we consider examples of such surfaces when the quotient is bi-rational to an Enriques surface or to a surface of Kodaira dimension one and show that the Bloch conjecture holds for such surfaces.

math.AG↗
arXiv2025

Beilinson's conjecture on K3 surfaces with an involution

In this note we prove that the Beilinson conjecture holds for certain examples of K3 surfaces over $\bar {\mathbb{Q}}$ equipped with an involution, when the quotient of the surface by the involution is the projective plane branched along a sextic.

math.AG↗
arXiv2022

Selmer group associated to the Chow group of certain codimension two cycles

Let $X$ be a surface with geometric genus and irregularity zero which is defined over a number field $K$. Let $\mathscr{X}$ denote a smooth spread of $X$ over the spectrum of a Zariski open subset in the spectrum of the ring of integers and $A^2$ stands for the group of algebraically trivial cycles on schemes modulo rational equivalence. If $j^*: A^2(\mathscr{X})\to A^2(X)$ be the flat pull-back corresponding to the embedding $j:X\hookrightarrow \mathscr{X}$ then we prove that $\im(j^*)(K)/A^2(\mathscr{X})(K)$ is a torsion group. Here $\im(j^*)(K)$, $A^2(\mathscr{X})(K)$ stand for the cycles fixed under the action of the absolute Galois group.

math.NT↗
arXiv2022

Zero cycles on Prym varieties

In this text we prove that if an abelian variety $A$ admits an embedding into the Jacobian of a smooth projective curve $C$, and if we consider $Θ_A$ to be the divisor $Θ_C\cap A$, where $Θ_C$ denotes the theta divisor of $J(C)$, then the embedding of $Θ_A$ into $A$ induces an injective push-forward homomorphism (under certain conditions) at the level of Chow groups. We show that this is the case for every Prym varietiy arising from an unramified double cover of smooth projective curves. As a consequence we prove that there does not exist a universal codimension two cycle on the product of a very general cubic threefold and the Prym variety associated to it. Hence we conclude that a very general cubic threefold is stably irrational.

math.AG↗
arXiv2021

Push-forwards of Chow groups of smooth ample divisors

We introduce a homological Lefschetz conjecture on (rational) Chow groups, which can be deduced from some well known conjectures, and illustrate it by a series of key examples. We then prove the injectivity of the push-forward morphism on Chow groups, induced by the closed embedding of the Theta divisor in it's Jacobian $J(C)$. Here $C$ is a smooth irreducible complex projective curve.

math.AG↗