ABSTRACT
A lot of modern geometry has been developed in order to deal with problems of a more arithmetical nature that are usually easily formulated in terms of numbers or solutions of polynomial equations in integers or rational numbers. Diophantes of Alexandria (± 250AD) gathered his mathematical ideas in the thirteen volumes of the “Arithmetics”, seven of which have been rediscovered. The basic problem of Diophantine geometry, as we now call it and as we have mentioned in CH.I §2, deals with a polynomial f ∈ ℤ [x1,...,xn]; the question is to find a method of deciding whether the equation f = 0 has a solution in ℤ, or in Q, to decide whether there are infinitely or only finitely many solutions and to describe all of these. Even when some equations define “relatively easy” geometric objects, e.g., rational surfaces, the Diophantine problems cannot easily be solved. A classical example is obtained by looking at
axp + byq + czr = d,
where a, b, c, d are nonzero in Q and p, q, r are positive integers such that https://www.w3.org/1998/Math/MathML">1p+1q+1r≥1https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315274515/64311f82-81f6-4c52-89ea-d39dfd59d7fa/content/inline-math658.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (note: there are only finitely many such p, q, r). pqr The case p = q = r = 2 may be treated by classical methods. Even in the case p = q = r = 3, e.g., x3 + y3+ z3 = d one needs heavy 308modern mathematical technology (K-theory, Chow groups,...) in order to understand the diophantine problems here, not withstanding the fact that Ryley (± 1825) already knew that there is a "simple” solution https://www.w3.org/1998/Math/MathML">d=(d3-3632d2+34d+36)3+(−d3-35d+3632d2+34d+36)3+(33d2+35d32d2+34d+36)3https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315274515/64311f82-81f6-4c52-89ea-d39dfd59d7fa/content/math371.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>