부산대학교 도서관

In this paper, Hatley and Kundu conduct a quantitative exploration ofthe Iwasawa $\lambda$-invariants within families of modular formsassociated with absolutely irreducible, modular Galois representationsof weight 2. \par For each representation $\overline \rho$, there exists an associatedmodular newform $g$ that gives rise to $\overline \rho$. The set$\Cal{H}(g)$ is defined as the collection of all weight 2 newforms $f$with level prime to $p$ such that the associated Galoisrepresentations match $\overline \rho$. An important observation,stemming from earlier work by Greenberg, Vatsal, and others, isdiscussed: when $\mu(g)$ equals zero, the same holds true for $\mu(f)$for all $f$ in $\Cal{H}(g)$. However, this equivalence does not extendto $\lambda(g)$ and $\lambda(f)$. Instead, $\lambda(f)$ can be computedbased on $\lambda(g)$ and local information about the representations.\par The main goals of the paper are to examine subsets of positiveintegers $M$, representing levels of newforms in $\Cal{H}(g)$, withspecific arithmetic properties. The paper provides explicit results,including the identification of positive densities in $M$ where$\lambda(f)$ either equals or exceeds $\lambda(g)$. Importantly, theseresults are formulated in a manner dependent solely on the prime $p$.\par In conclusion, Hatley and Kundu's paper significantly contributes tothe field by offering a quantitative analysis of $\lambda$-invariantsin families of modular forms. The explicit results and clearpresentation make it accessible to researchers familiar with thesubject. The paper builds upon known results and extends its reach tonon-ordinary cases. This review suggests that mathematicians interestedin Galois representations, modular forms, and the quantitative aspectsof $\lambda$-invariants would find this paper to be a valuable additionto the existing literature.