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On the $K_4$ group of modular curves
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abstract
We construct elements in the $K_4$ group of modular curves using the polylogarithmic complexes of weight 3 defined by Goncharov and De Jeu. The construction is uniform in the level and relies on new modular units arising as cross-ratios of division values of the Weierstrass $\wp$ function. These units provide explicit triangulations of the $3$-term relations in $K_2$, which in turn give rise to elements in $K_4$. Based on numerical computations and on results of Wang, we conjecture that these elements are proportional to the Beilinson elements defined via Eisenstein symbols.
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Bernoulli determinants and cuspidal subgroups
The order of the rational cuspidal class group of X_1(N) is given by an explicit product over even Dirichlet characters involving generalized Bernoulli numbers B_{2,χ}, valid for all N ≥ 5.
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