Interview with Professor Eyal Goren
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Interview with Professor Eyal Goren Michael McBreen
The Delta-Epsilon interviewed Professor Eyal Goren from the Department of Mathematics and Statistics early this spring, asking about his research as an arithmetic geometer, but also about what made him choose his path. This is what he had to say. √ could pick Q[ 2], and in this field there’s a ring of integers, some of which are units. The question is how to find these In the large, I’m an arithmetic geometer. My research com- units, and that turns out to be one of the major probbines number theory, which in essence studies integers and lems of this type of algebraic number theory. Some of the their various generalizations like algebraic integers (num- strongest tools we have come from arithmetic geometry. bers that satisfy monic polynomial equations with integer coefficients), with algebraic geometry, which studies manifolds or varieties defined as the solutions to systems of polynomial equations in several variables. Variety Given a set of polynomial equations Pi (x1 , x2 , x3 , . . . , xn ) = 0 in a field K, the associated variety is the set of points (x1 , x2 , . . . , xn ) ∈ Kn which satisfy the equations. Varieties generally have the structure of a manifold away from a smaller singular locus, where they may have jagged edges or self-intersections. What research are you currently working on?
In arithmetic geometry, you might take a polynomial equation with integer coefficients, reduce the coefficients mod p and ask for solutions in characteristic p. This brings another dimension to the picture. The same equation gives a variety in characteristic p for every p, a complex variety when you look at complex solutions, and so on. Arithmetic Figure 1: Eyal Goren geometry, in the large, makes use of this extra dimension to study problems that arise in number theory. The main idea in my research is that you take some variety over the complex numbers, which is defined by integer polynomials so that you can look at its reduction Can you picture varieties in characteristic p? mod p for various primes p. When you do this you get all Yes, but it’s not clear what the picture means. It gives these (sometimes singular) varieties, and we think of them an intuition or a way of organizing your thoughts, rather as a single geometric object. We take a function f on the than any solid meaning. But still, if you have an equation variety which makes sense arithmetically, perhaps also defor a line, you like to draw a line on the board because fined with Z coefficients, and we evaluate it at some point things behave rather similarly to usual geometry in many x. Suppose f (x) = ab where a and b are algebraic integers, respects. Somehow, this whole geometric intuition makes and gcd(a, b) = 1 (one can make this precise). We want to arithmetic geometry work, and I enjoy very much trans- know if f (x) is a unit. lating questions about numbers into geometric questions. There’s an analogue of prime numbers called prime ideals, and for our ring of integers one can make sense of the My own research is deeply concerned with constructing units. Pick a polynomial, say a monic polynomial with statement “p appears in the denominator of f (x)”. You integer coefficients, so that a root would be an algebraic in- don’t have unique factorization into primes, but if you teger. If its free coefficient is 1 or -1, the root would in fact think of f (x) as generating a principal ideal, there is a be a unit. In other words, one can construct a ring whose unique factorization into prime ideals. Hence, the ideal elements are algebraic integers and that element would be generated by f (x) in the ring of algebraic integers of this invertible in that ring. It’s not hard to see, because the field can be decomposed as a product of powers of prime free coefficient is the product of the roots of the polyno- ideals. If an ideal p appears in the denominator of f (x), mial, which are all algebraic integers. If it’s 1, then the roots are invertible. That’s not so hard, but the game is there’s another way to think about it, which is to say that really played differently: you first pick the extension of Q f (x) = ∞ mod p. If it’s in the numerator, then f (x) = 0 in which you want the number to lie in, for instance you mod p. So the picture is that we have a variety given by The δelta-ǫpsilon
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