부산대학교 도서관

In this paper, Hatley and Kundu conduct a quantitative exploration of the Iwasawa $\lambda$-invariants within families of modular forms associated with absolutely irreducible, modular Galois representations of weight 2. \par For each representation $\overline \rho$, there exists an associated modular 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 Galois representations match $\overline \rho$. An important observation, stemming from earlier work by Greenberg, Vatsal, and others, is discussed: when $\mu(g)$ equals zero, the same holds true for $\mu(f)$ for all $f$ in $\Cal{H}(g)$. However, this equivalence does not extend to $\lambda(g)$ and $\lambda(f)$. Instead, $\lambda(f)$ can be computed based on $\lambda(g)$ and local information about the representations. \par The main goals of the paper are to examine subsets of positive integers $M$, representing levels of newforms in $\Cal{H}(g)$, with specific 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, these results are formulated in a manner dependent solely on the prime $p$. \par In conclusion, Hatley and Kundu's paper significantly contributes to the field by offering a quantitative analysis of $\lambda$-invariants in families of modular forms. The explicit results and clear presentation make it accessible to researchers familiar with the subject. The paper builds upon known results and extends its reach to non-ordinary cases. This review suggests that mathematicians interested in Galois representations, modular forms, and the quantitative aspects of $\lambda$-invariants would find this paper to be a valuable addition to the existing literature.