arXiv subjects

Adithya Chakravarthy

Publications and source records attributed to Adithya Chakravarthy.

4 recordsLinked to original sources

arXiv2024

The Iwasawa $\mu$-invariants of Elliptic Curves over $\mathbb{Q}$

In this paper, we discuss a longstanding conjecture of Greenberg in the Iwasawa theory of elliptic curves. Greenberg's conjecture states that if $E/\mathbb{Q}$ is an elliptic curve with good ordinary reduction at $p$, and $E[p]$ is irreducible as a Galois module, then the Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$ extension of $\mathbb{Q}$ has $\mu$-invariant zero. We prove that if $E$ is an elliptic curve over $\mathbb{Q}$, then we have $\mu \leq 1$ for all but finitely many primes $p$ of good ordinary reduction.

math.NT↗

arXiv2024

The Iwasawa $\mu$-invariant of certain elliptic curves of analytic rank zero

This paper is about the Iwasawa theory of elliptic curves over the cyclotomic $\mathbb{Z}_p$-extension $\mathbb{Q}^{\text{cyc}}$ of $\mathbb{Q}$. We discuss a deep conjecture of Greenberg that if $E/\mathbb{Q}$ is an elliptic curve with good ordinary reduction at $p$, and $E[p]$ is irreducible as a Galois module, then the Selmer group of $E$ over $\mathbb{Q}^{\text{cyc}}$ has $\mu$-invariant zero. We prove new cases of Greenberg's conjecture for some elliptic curves of analytic rank $0$. The proof involves studying the $p$-adic $L$-function of $E$. The crucial input is a new technique using the Rankin-Selberg method.

math.NT↗

arXiv2024

The non-$p$-part of the fine Selmer group in a $\mathbf{Z}_p$-extension

Fix two distinct primes $p$ and $\ell$. Let $A$ be an abelian variety over $\mathbf{Q}(\zeta_{\ell})$, the cyclotomic field of $\ell$-th roots of unity. Suppose that $A(\mathbf{Q}(\zeta_{\ell}))[\ell] \neq 0$. We show that there exists a number field $L$ and a $\mathbf{Z}_p$ extension $L_{\infty}/L$ where the $\ell$-primary fine Selmer group of $A$ grows arbitrarily quickly. This is a fine Selmer group analogue of a theorem of Washington which says that there are certain (non-cyclotomic) $\mathbf{Z}_p$-extensions where the $\ell$-part of the class group can grow arbitrarily quickly. We also prove this for a wide class of non-commutative $p$-adic Lie extensions. Finally, we include several examples to illustrate this theorem.

math.NT↗

arXiv2023

Crank equidistribution and $(k,j)$-overlined partitions

In a paper published in 2023, Wagner introduced and studied Jacobi forms with complex multiplication, and gave several applications. One such application was in constructing a new doubly-infinite family of partition-theoretic objects, called $(k,j)$-coloured overpartitions and labelled by $\overline{p}_{k,j}$, and using the Jacobi forms to construct crank functions which explain the Ramanujan-type congruences satisfied by $\overline{p}_{k,j}$. In this note, we give an asymptotic formula for the number of $(k,j)$-coloured overpartitions and prove that any crank constructed by Wagner is asymptotically equidistributed on arithmetic progressions, following several recent papers in the literature.

math.NT↗