### Recovering l-adic representations, by C. S. Rajan

We consider the problem of
recovering l-adic representations from a
knowledge of the character values at the Frobenius elements associated
to l-adic representations constructed algebraically out of the original
representations. These results generalize earlier results
in of the author concerning refinements of strong multiplicity one for
$l$-adic represntations, and a result of Ramakrishnan
recovering modular forms from a knowledge of the
squares of the Hecke eigenvalues.
For example, we show that if the characters of some tensor or
symmetric powers of two absolutely irreducible l-adic representation
with the algebraic envelope of the image being connected, agree
at the Frobenius elements corresponding to a set of places of
positive upper density, then the representations are twists of each
other by a finite order character.

C. S. Rajan <rajan@math.tifr.res.in>