BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Anwesh Ray (Univ of British Columbia) DTSTART:20211101T220000Z DTEND:20211101T225000Z DTSTAMP:20260423T022923Z UID:UCLA_NTS/36 DESCRIPTION:Title: Iwasawa theory and congruences for the symmetric square of a modular for m\nby Anwesh Ray (Univ of British Columbia) as part of UCLA Number The ory Seminar\n\n\nAbstract\nI will report on joint work with R. Sujatha and V. Vatsal. Two\n$p$-ordinary Hecke-eigenforms are are congruent at a prim e $\\varpi|p$ if\nall but finitely many of their Fourier coefficients are congruent modulo\n$\\varpi$. R. Greenberg and V. Vatsal showed in 2000 tha t the\nIwasawa-invariants of congruent modular forms are related. As a res ult\, if\n$\\mu$-invariant vanishes and the main conjecture holds for a gi ven\nHecke-eigenform\, then the same is true for a congruent Hecke-eigenfo rm.\nThis involves studying the behavior of Selmer groups and p-adic L-fun ctions\nwith respect to congruences. We generalize these results to symmet ric\nsquare representations.\n\n The main task at hand is that the p-adic L-functions for the symmetric\nsquare exhibit congruences. In this setting \, the normalized L-values for\n$sym^2(f)$ can be expressed in terms of th e Petersson inner product of $f$\nwith a nearly holomorphic function. This function is expressed as the\nproduct of a theta function and an Eisentei n series. The ordinary\nholomorphic projection of this function is shown t o have nice properties.\nThe Petersson inner product is modified and relat ed to an abstractly\ndefined algebraic pairing due to Hida\, and the two p airing are related up\nto a "canonical period". Under further hypotheses\, it is shown that this\ncanonical period is suitably well behaved. For thi s\, we assume a certain\nversion of Ihara's lemma\, which is known in cert ain cases.\n\n With these preparations\, we are able to show that normaliz ed L-values for\nthe symmetric square behave well with respect to congruen ce\, and hence\, the\np-adic L-functions too. It follows that the analytic Iwasawa invariants for congruent Hecke-eigencuspforms are related. Such r esults for the algebraic Iwasawa invariants follow from work of R. Greenbe rg and V. Vatsal. Just as in the classicial case\, the results have implic ations to the main conjecture. If time permits\, we will introduce the rol e of the fine-Selmer\ngroup and discuss a condition for the vanishing on t he $\\mu$-invariant that\ncan be stated purely in terms of the residual re presentation.\n LOCATION:https://researchseminars.org/talk/UCLA_NTS/36/ END:VEVENT END:VCALENDAR