# Schottky Implies Poincare in any genus at least 4

# Schottky Implies Poincare in any genus at least 4

The Schottky problem is the problem of finding holomorphic equations on the Siegel upper half plane of degree g ( at least 4) which cut out the space of Jacobi varieties in the Siegel upper half plane. The Poincare problem is the problem of cutting out the same space but in a neighborhood of the diagonal matrices. Poincare solved his problem for all g at least 4. The Schottky problem was completely solved by Schottky only in the case g=4. In this talk we shall show for all genus at least 4 how to write a set of (g-3)(g-2)/2 equations ( of Schottky type ) which vanish on the space of Jacobi varieties and reduce to the Poincare equations near the diagonal matrices. The main ideas of the proof will involve the Riemann theta formula, The Schottky Jung proportionalities, and the transformation theory of theta constants. These ideas will be reviewed in the talk.