Iwasawa theory and congruences for the symmetric square of a modular form

Abstract: I will report on joint work with R. Sujatha and V. Vatsal. Two $p$-ordinary Hecke-eigenforms are are congruent at a prime $\varpi|p$ if all but finitely many of their Fourier coefficients are congruent modulo $\varpi$. R. Greenberg and V. Vatsal showed in 2000 that the Iwasawa-invariants of congruent modular forms are related. As a result, if $\mu$-invariant vanishes and the main conjecture holds for a given Hecke-eigenform, then the same is true for a congruent Hecke-eigenform. This involves studying the behavior of Selmer groups and p-adic L-functions with respect to congruences. We generalize these results to symmetric square representations.

The main task at hand is that the p-adic L-functions for the symmetric square exhibit congruences. In this setting, the normalized L-values for $sym^2(f)$ can be expressed in terms of the Petersson inner product of $f$ with a nearly holomorphic function. This function is expressed as the product of a theta function and an Eisentein series. The ordinary holomorphic projection of this function is shown to have nice properties. The Petersson inner product is modified and related to an abstractly defined algebraic pairing due to Hida, and the two pairing are related up to a "canonical period". Under further hypotheses, it is shown that this canonical period is suitably well behaved. For this, we assume a certain version of Ihara's lemma, which is known in certain cases.

With these preparations, we are able to show that normalized L-values for the symmetric square behave well with respect to congruence, and hence, the p-adic L-functions too. It follows that the analytic Iwasawa invariants for congruent Hecke-eigencuspforms are related. Such results for the algebraic Iwasawa invariants follow from work of R. Greenberg and V. Vatsal. Just as in the classicial case, the results have implications to the main conjecture. If time permits, we will introduce the role of the fine-Selmer group and discuss a condition for the vanishing on the $\mu$-invariant that can be stated purely in terms of the residual representation.

number theory

Audience: researchers in the topic

UCLA Number Theory Seminar