There is something in common between

  • cluster variables in the theory of cluster algebras,

  • L-functions in number theory,

namely the fact that both map direct sums to products, just like determinants do.

This kind of behavior has been considered by Kapranov in "Analogies between the Langlands correspondence and topological quantum field theory".

A typical relation between cluster variables looks like $ X X' = Y + Z $, where $X$ and $X'$ are cluster variables and $Y$ and $Z$ are monomials (products of cluster variables).

Are there any known relations of this shape between L-functions ?


It seems that relations of this kind exist in the context of the famous article of Gross and Zagier "Heegner points and derivatives of L-series", Invent. Math. Let me try to explain.

Let $E$ be an elliptic curve over $\mathbf{Q}$ and let $K$ be an imaginary quadratic number field $K$ of discriminant $d$.

Then one has the classical L-function $L_E$ of $E$ and the L-function $L_{E,d}$ of the quadratic twist of $E$ by $d$.

Gross and Zagier introduced L-functions $L_{E,A}$ attached to pairs $(E,A)$ where $A$ is an ideal class in the number field $K$.

Then $L_E L_{E,d} = \sum_A L_{E,A}$

where the sum runs over the ideal class group of $K$. So the number of terms is the class number of $K$, which can be greater than 2.

More generally, the elliptic curve could be replaced by a modular form.

  • $\begingroup$ Is $d_{FG}=d_{F}+d_{G}$ (where $d_{F}$ is the degree of $F$ as an element of the Selberg class) off-topic? $\endgroup$ – Sylvain JULIEN May 19 '15 at 20:30
  • $\begingroup$ Well, maybe the degree of each term of the right hand side has to match the sum of degrees of the two factors of the left hand side. The degree may be a grading in the usual algebraic sense. $\endgroup$ – F. C. May 22 '15 at 10:11

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