Question

In his cellebrated 1984 paper "Higher regulators and values of L-functions", Beilinson proved (among many other exciting things) that the value at the non-critical point $s=2$ of the Rankin L-function $L(f\otimes g,s)$ of the convolution of two eigenforms of weight $2$ (say of the same level $N$) is related to the image under the complex regulator map of a certain diagonal element $\Delta$ in the $K_1$ of the surface $S=X_1(N)\times X_1(N)$.

This happens in the pretty short section 6 of the above mentioned paper. However, I have not been able to spot the precise relationship between $L(f\otimes g,2)$ and $\mathrm{reg}(\Delta)$. I have also read several of the surveys on the subject existing in the literature, like Tony Scholl's "Integral elements in $K$-theory and products of modular curves" and a few others, again without luck: those articles rather focus on other aspects of the story. (To the best of my knowledge, of course, I'll be happy to be corrected.)

My question is: is there any place in the literature where such a formula is precisely stated? By this I mean a formula where all the involved quantities are explicitly written down.

With my collaborators I am working on $p$-adic analogues of this formula, that's why the interest. We can redo Beilinson's computations in order to write down the formula (as it surely is a formal consequence of the ideas of his paper, once everything is written down carefully enough), but I wonder whether this is already done explicitly elsewhere.

Answer 1

Dear Victor, I think an explicit formula has been worked out by S. Baba and R. Sreekantan in the following article :

MR2064735 (2005c:11073) Baba, Srinath ; Sreekantan, Ramesh. An analogue of circular units for products of elliptic curves. Proc. Edinb. Math. Soc. (2) 47 (2004), no. 1, 35--51.

They consider the case where $f$ and $g$ are associated to elliptic curves of the same conductor $N$. They obtain an explicit formula giving $L'(f \otimes g,1)$ (which relates to $L(f \otimes g,2)$ using the functional equation) as the regulator of an explicit element in the motivic cohomology of the product of modular curves $X_0(N) \times X_0(N)$.

The hypothesis that $f$ and $g$ come from elliptic curves is not very restrictive, in the sense that their method probably also works for arbitrary eigenforms of weight 2 and level $\Gamma_0(N)$. It is also possible in principle to work with $X_1(N)$ instead of $X_0(N)$, but as usual this may require some work, in particular this would probably require to adapt the construction of the motivic cohomology elements.