MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 7 down vote accepted

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.

share|cite|improve this answer
Merci beaucoup, François! – Victor Rotger Apr 18 '12 at 12:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.