### Lautman on Reciprocity

#### Posted by David Corfield

In ‘Nouvelles Recherches sur la Structure Dialectique des Mathématiques’ (1939), Albert Lautman discusses the use of analysis in number theory. He notes that some have felt uncomfortable with this use and have sought to eliminate it. But Lautman sees no metaphysical necessity for this ‘purification’. Rather than take arithmetic as metaphysically prior to analysis, instead he proposes that we consider them equally as realisations of the same ‘dialectical’ structures.

He gives the example of reciprocal entities. In arithmetic we have quadratic reciprocity, where the Legendre symbols are acting as a kind of inverse to each other.

$\left( \frac{p}{q} \right) \left( \frac{q}{p} \right) = (-1)^{\left(\frac{p-1}{2}\right) \left( \frac{q-1}{2}\right)}$

He goes on to note that it has been possible to generalise reciprocity in two different ways. First, to algebraic integers in any field. Second, to allow more general congruences, not just to a square, but to other powers. This has been achieved algebraically he notes, but he adds that Hecke has also provided analytic means of deriving general quadratic reciprocity results using theta functions.

Here we define

$\theta (\tau) = \sum_{m = - \infty}^{m = + \infty} e^{- \pi \tau m^2},$

noting that singular points are at $\tau = 2 i r$, $r$ a rational, but that for any such $r$, $\sqrt \tau \theta (\tau + 2 i r)$ takes a finite value. Then we have

$\theta (1/ \tau) = \sqrt{\tau} \theta (\tau).$

This reciprocity, he claims, is the mainspring of the transcendental proof of quadratic reciprocity. The same dialectical idea is manifesting itself in different branches. He continues

This dialectical idea of reciprocity between elements can be so clearly distinguished from its realisations in arithmetic and in in analysis that it is possible to find a certain number of other mathematical theories in which it realises itself similarly.

He then claims that the dimensions $m$ and $n - m$ in Poincaré duality associated to a manifold of dimension $n$ also stand similarly in an inverse relationship, and further that Weil has shown (1937) a relationship between reciprocity laws and duality theorems.

Do people recognise a useful thought here? Is there a ‘walking’ reciprocity realising itself in different situations?

Posted at August 14, 2008 3:48 PM UTC
## Re: Lautman on Reciprocity

Just when I had run out of things to think about :-). I found an extended commentary on Lautman, seemingly more philosophical, here.