The Iwasawa theory of modular forms over the \(p\)-adic cyclotomic extension of the rationals is a very interesting field, which has been studied by many authors since the work of Mazur, Swinnerton-Dyer and Manin in the early 1970s. The flavour of this theory depends sharply on the choice of the prime \(p\). If we neglect the finitely many bad primes dividing the level of the form, then the key distinction is between “ordinary” primes (where the Hecke eigenvalue at \(p\) is a unit) and “non-ordinary”, or “supersingular”, primes.
In the former case, one can define a \(p\)-adic \(L\)-function as a measure on the cyclotomic Galois group (using the tools of modular symbols), and a corresponding Selmer group which is a cotorsion module over the Iwasawa algebra, and the “main conjecture” asserts that the \(p\)-adic \(L\)-function generates the characteristic ideal of the dual of the Selmer group. One inclusion is known by the work of K. Kato [Cohomologies \(p\)-adiques et applications arithmétiques. III. Astérisque 295, 117–290 (2004; Zbl 1142.11336)] and (since this paper was written) the opposite inclusion has been proved by Skinner and Urban (unpublished).
The current paper is devoted to the much more subtle non-ordinary (supersingular) case. Here there are two \(p\)-adic \(L\)-functions attached to \(f\), and these are not bounded measures but elements of the larger algebra of distributions; and the Selmer group attached to \(f\) is not torsion. R. Pollack [Duke Math. J. 118, No. 3, 523–558 (2003; Zbl 1074.11061)] showed that if the Hecke eigenvalue of \(f\) at \(p\) is zero, then one can modify the \(L\)-functions to define two bounded measures (the so-called “plus” and “minus” \(L\)-functions). When the form \(f\) corresponds to an elliptic curve, S.-I. Kobayashi has shown [Invent. Math. 152, No. 1, 1–36 (2003; Zbl 1047.11105)] how to modify the definition of the Selmer group to give corresponding plus and minus Selmer groups, using Honda theory applied to the formal group of the elliptic curve; these are cotorsion, and it is natural to conjecture that the characteristic ideals are generated by the corresponding \(L\)-functions (and Kato’s construction proves one inclusion, as in the ordinary situation). However, Kobayashi’s construction does not apply to modular forms of weight \(\geq 2\), so a new approach is needed, and this is the innovation of the paper under review.
The author uses B. Perrin-Riou’s exponential map [Invent. Math. 115, No. 1, 81–149 (1994; Zbl 0838.11071)] for the local Galois representation attached to \(f\) in order to define two homomorphisms of Iwasawa modules, the plus and minus Coleman maps, from the Iwasawa cohomology of \(f\) into the Iwasawa algebra. These take the place of the Coleman maps defined using Honda theory in Kobayashi’s work. As in the special case studied by Kobayashi, when applied to the Kato zeta element, these Coleman maps produce Pollack’s plus and minus \(L\)-functions; while on the other hand, the kernels of these Coleman maps furnish the local conditions needed to define modified Selmer groups. Again, the results of Kato on the zeta element show that the plus and minus \(L\)-functions lie in the characteristic ideals of the corresponding Selmer groups, proving half of the Iwasawa main conjecture in this context.
The final part of the paper (Section 7) is devoted to the case of modular forms of CM type, where there is an abundant supply of primes to which the method applies. In this case, the author shows that the plus and minus main conjectures for such forms follow from Rubin’s work on two-variable main conjectures for imaginary quadratic fields, as was shown in the elliptic curve case by R. Pollack and K. Rubin [Ann. Math. (2) 159, No. 1, 447–464 (2004; Zbl 1082.11035)].
This paper represents an important advance in the field, and has contributed to rekindling interest in Iwasawa theory for non-ordinary primes, a subject that has been somewhat neglected in recent years.