Iwasawa Theory and Modular Forms in Number Fields

Iwasawa theory investigates the variation of arithmetic invariants—such as class groups and Selmer groups—along infinite towers of number fields, most classically the cyclotomic Z_p-extension of the rational numbers. By viewing Selmer groups as modules over the Iwasawa algebra of the Galois group of these extensions, one obtains deep structural insights and formulates main conjectures equating characteristic ideals of these modules with suitably constructed p-adic L-functions. Modular forms, in turn, give rise to Galois representations whose arithmetic behaviour is reflected in L-values and in the structure of associated Selmer groups. The confluence of these subjects has led to striking results: proofs of special cases of the main conjecture for GL₂, explicit reciprocity laws linking Euler systems to values of p-adic L-functions, and refinements of classical conjectures of Birch and Swinnerton-Dyer and of Bloch and Kato. Recent extensions encompass Hilbert and Bianchi modular forms over totally real and imaginary quadratic fields, non-abelian generalisations of the cyclotomic tower, and the study of µ- and λ-invariants governing fine growth in these settings. Concrete applications include the control of Shafarevich–Tate groups, predictions for ranks of elliptic curves over number fields, and the modularity of higher-dimensional abelian varieties.

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Technical terms

Iwasawa algebra: The completed group algebra Z_p[[Gal(F_∞/F)]] of the Galois group of a Z_p-extension F_∞/F, serving as the coefficient ring for Selmer modules.

Selmer group: A subgroup of Galois cohomology defined by local conditions on a p-adic Galois representation, encoding arithmetic invariants such as Mordell–Weil groups or Tate–Shafarevich groups.

p-adic L-function: A p-adic analytic function interpolating special values of complex L-functions at arithmetic points, conjecturally linked to Selmer modules via main conjectures.

Modular form: A holomorphic function on the upper half-plane satisfying transformation laws under a subgroup of SL₂(Z), whose Fourier coefficients encode arithmetic data and give rise to Galois representations.

Euler system: A compatible family of cohomology classes in Galois representations over varying fields, used to bound Selmer groups and to establish explicit reciprocity laws.

References

1. Special values of anticyclotomic Rankin–Selberg $L$-functions. Documenta Mathematica (2014).
2. Rankin–Eisenstein classes in Coleman families. Research in the Mathematical Sciences (2016).
3. P‐adic L‐functions of Bianchi modular forms. Proceedings of the London Mathematical Society (2017).