Cusp forms without complex multiplication as $p$-adic limits
Dalen Dockery
TL;DR
This work extends the phenomenon of $p$-adic limits for cusp forms to non-CM, one-dimensional spaces with trivial character and genus-zero level, showing that a carefully constructed weakly holomorphic form $F$ interacts with Hecke operators $T_k(p^m)$ to yield a $p$-adic limit to the unique normalized cusp form $g$ in $S_k(N)$. The author builds two modular families, $F_m$ and $oldsymbol{\phi}_n$, and leverages Zagier duality to relate their coefficients, enabling explicit $p$-adic valuation formulas for $C(p^m)$ and for the difference $ rac{Fig|T_k(p^m)}{C(p^m)}-g$ that grow linearly with $m$ at slope $(k-1)$. The main theorem provides precise conditions under which $v_p(C(p^m))=(k-1)m-v_p(C(p))$ and $v_pig( rac{Fig|T_k(p^m)}{C(p^m)}-gig)=(k-1)m-v_p(C(p))$, resulting in the $p$-adic limit $ rac{Fig|T_k(p^m)}{C(p^m)} o g$ as $m o ty$. Together with Hanson–Jameson, these results substantially broaden the class of cusp forms known to admit $p$-adic limit representations, while highlighting weight-$2$ and non-genus-zero obstacles as avenues for future work.
Abstract
In 2016, Ahlgren and Samart used the theory of holomorphic modular forms to obtain lower bounds on $p$-adic valuations related to the Fourier coefficients of three cusp forms. In particular, their work strengthened a previous result of El-Guindy and Ono which expresses a cusp form as a $p$-adic limit of weakly holomorphic modular forms. Subsequently, Hanson and Jameson extended Ahlgren and Samart's result to all one-dimensional cusp form spaces of trivial character and having a normalized form that has complex multiplication. Here we prove analogous $p$-adic limits for several one-dimensional cusp form spaces of trivial character but whose normalized form does not have complex multiplication.