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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 $\mu=0$ for all but finitely many primes $p$ of good ordinary reduction., Comment: Comments welcome! arXiv admin note: text overlap with arXiv:2405.05871

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., Comment: Comments welcome!

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., Comment: 11 pages, 1 table. Comments welcome!