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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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Abstract Let $$X$$ be a quasi-projective variety over a number field, admitting (after passage to $$\mathbb {C}$$) a geometric variation of Hodge structure whose period mapping has zero-dimensional fibers. Then the integral points of $$X$$ are sparse: the number of such points of height $$\leq B$$ grows slower than any positive power of $$B$$. For example, homogeneous integral polynomials in a fixed number of variables and degree, with discriminant divisible only by a fixed set of primes, are sparse when considered up to integral linear substitutions.more » « less
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We propose a relationship between the cohomology of arithmetic groups, and the motivic cohomology of certain (Langlands-)attached motives. The motivic cohomology group in question is that related, by Beilinson’s conjecture, to the adjoint L-function at s=1. We present evidence for the conjecture using the theory of periods of automorphic forms, and using analytic torsion.more » « less
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