I think that in their desire to explain this popularly, which is legitimate, they made a rather confused version of both the history and the mathematics. First, neither Weil nor Deligne are unheralded - they are about as famous and acknowledged as mathematicians can be, aside from a few universal geniuses such as Gauss and a few people who have solved a long-outstanding problem and were mentioned in the press a lot (e.g. Andrew Wiles). (Incidentally, Weil once joked that many generations from now, people will probably think that André Weil and Andrew Wiles, both professors from Princeton, were the same person).
Second, it's an exaggeration (to put it mildly) to say that Weil is the first one to think of a connection between numbers and geometry. I mean, this goes to Descartes at the latest. And such sophisticated tools as the fundamental group of a topological surface were introduced in the 19th century.
For anyone confused by the statement that the solutions to x^2+y^2=1 in complex numbers form a sphere: they obviously don't form a sphere in the usual sense, since the solutions are not bounded (since there is a root of every polynomial in complex numbers, you can find a corresponding y for every x, indeed usually two such y's). What's meant is that the Riemann surface defined by the graph of this function (i.e. y=sqrt(1-x^2), which is two-dimensional surface embedded into four-dimensional space, has genus 0, i.e. it is topologically equivalent to a sphere, which is to say it doesn't have any holes in it. I think you are supposed to prove that this is so by considering this surface on the complex plane minus the set [1,+inf], seeing that the function has two distinct continuous branches on the remaining set, each of which is equivalent to a sphere since it's a simple one-valued function over the complex plane, and then gluing the two spheres along the line [1,+inf] like you would glue two regular spheres along edges of cuts made in both. You end up with another sphere (from the point of view of topology).
I don't see any place in the article where they say that Weil was the first to form a connection between numbers and geometry. The only thing close they say:
Weil suggested that sentences written in the language of number theory could be translated into the language of geometry, and vice versa.
And since this is elaborated upon later on in the article I don't see anything wrong with the sentence.
My problem is not with this sentence, but with this passage from the second paragraph:
> ... number theory is the study of numbers [...]
> Geometry, on the other hand, studies shapes [...]
> But French mathematician André Weil had a penetrating
> insight that the two subjects are in fact closely related.
This pretty clearly says Weil came up with the idea of connecting number theory and geometry - or at least connecting them closely - which is very, very wrong.
Second, it's an exaggeration (to put it mildly) to say that Weil is the first one to think of a connection between numbers and geometry. I mean, this goes to Descartes at the latest. And such sophisticated tools as the fundamental group of a topological surface were introduced in the 19th century.
For anyone confused by the statement that the solutions to x^2+y^2=1 in complex numbers form a sphere: they obviously don't form a sphere in the usual sense, since the solutions are not bounded (since there is a root of every polynomial in complex numbers, you can find a corresponding y for every x, indeed usually two such y's). What's meant is that the Riemann surface defined by the graph of this function (i.e. y=sqrt(1-x^2), which is two-dimensional surface embedded into four-dimensional space, has genus 0, i.e. it is topologically equivalent to a sphere, which is to say it doesn't have any holes in it. I think you are supposed to prove that this is so by considering this surface on the complex plane minus the set [1,+inf], seeing that the function has two distinct continuous branches on the remaining set, each of which is equivalent to a sphere since it's a simple one-valued function over the complex plane, and then gluing the two spheres along the line [1,+inf] like you would glue two regular spheres along edges of cuts made in both. You end up with another sphere (from the point of view of topology).