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  1. ; (, Compositio Mathematica)
    Several authors have studied homomorphisms from first homology groups of modular curves to$$K_2(X)$$, with$$X$$either a cyclotomic ring or a modular curve. These maps send Manin symbols in the homology groups to Steinberg symbols of cyclotomic or Siegel units. We give a new construction of these maps and a direct proof of their Hecke equivariance, analogous to the construction of Siegel units using the universal elliptic curve. Our main tool is a$$1$$-cocycle from$$\mathrm {GL}_2(\mathbb {Z})$$to the second$$K$$-group of the function field of a suitable group scheme over$$X$$, from which the maps of interest arise by specialization. 
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  2. (, Journal of the European Mathematical Society)
    In a groundbreaking paper, T. Fukaya and K. Kato proved a slight weakening of a conjecture of the author under an assumption that a Kubota–Leopoldtp-adicL-function has no multiple zeros. This article describes a refinement of their method that sheds light on the role of thep-adicL-function. 
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  3. ; ; ; ; (, Forum of Mathematics, Sigma)
    Abstract Given a profinite group G of finite p -cohomological dimension and a pro- p quotient H of G by a closed normal subgroup N , we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H . We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H , we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H . We apply our study to prove lower bounds on the p -ranks of class groups of certain nonabelian extensions of $$\mathbb {Q}$$ and to give a new proof of the vanishing of Massey triple products in Galois cohomology. 
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