# Watson Transformation Squared reference request

George Leo Watson (1909-1988) wrote in a conversational manner, it is difficult to see when he switches from the trivial to the incredibly subtle, and at whiles he left out utterly elementary consequences of his results. Saturday night (January 22, for my future biographers) I proved such a corollary, I am asking here for an existing reference.

What many current authors (Wai Kiu Chan, Andrew G. Earnest, Byeong-Kweon Oh) call a Watson transformation, and denote $\lambda_m,$ is what Watson called the $m$-mapping in the 1962 article

For current terminology (based on the Timothy O'Meara book, Introduction to Quadratic Forms), see, for example, Chan and Oh or Chan and Earnest

Now, usually $\lambda_m^2(f)$ has lower discriminant than $f$ itself, this is discussed in Theorem 2 on page 580. Sometimes, though, $\lambda_m^2(f)$ takes us back to the same discriminant as $f,$ and if that happens, on the level of equivalence classes of forms, $\lambda_m^2$ is the identity. In this case, $\lambda_m$ (the same notation is used for the mapping in both directions) gives a bijection of classes between the genus containing $f$ and the genus containing $\lambda_m(f).$

So, if $\lambda_m^2$ is the identity, and if $m$ and $M$ are as in formulas (2.4) and (2.5) on Watson's page 578, $N$ is as in Lemma 2(ii) on his page 581 (so $NM =MN = mI$), $A$ is the Gram (or Hessian, or second partials) matrix as in (2.1) on page 578, finally given some automorph $Q$ of $f$ with $\det Q = \pm 1$ and $Q' A Q = A,$ ( $Q'$ means the transpose) then the Gram matrix of $\lambda_m(f)$ is just $$B = \frac{1}{m} \; M' A M .$$ Also an integral automorph of $\lambda_m(f)$ is just $$R = \frac{1}{m} \; N Q M.$$ Finally, and this all needed proof, the correct expression of $\lambda_m$ in the reverse direction is $$A = \frac{1}{m} \; N' B N,$$ and then we pull back automorphs with $$Q = \frac{1}{m} \; M R N.$$

The punchline is that the matching genera have the same mass, my quadratic forms being positive definite. As Watson had some involvement with the mass formula, it is difficult to see why this special case was not explored to its conclusion. However, Watson or some later author may have written this down somewhere, so I am asking for references.

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