We compute the regulator of the Beilinson–Deninger–Scholl elements in terms of special values of $L$ -functions of modular forms. The main tool is the Rogers–Zudilin method.

We construct a variant of Karoubi’s relative Chern character for smooth separated schemes over the ring of integers in a $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$-adic field, and prove a comparison with the rigid syntomic regulator. For smooth projective schemes, we further relate the relative Chern character to the étale $p$-adic regulator via the Bloch–Kato exponential map. This reproves a result of Huber and Kings for the spectrum of the ring of integers, and generalizes it to all smooth projective schemes as above.

We construct families of hyperelliptic curves over ${\mathbb Q}$ of arbitrary genus g with (at least) g integral elements in K2. We also verify the Beilinson conjectures about K2 numerically for several curves with g = 2, 3, 4 and 5. The first few sections of the paper also provide an elementary introduction to the Beilinson conjectures for K2 of curves.