Number Theory Seminar
Thursday, September 30
Milner 317, 1:00 PM
Title: Elliptic sequences and continued fractions
Abstract: The symmetric sequence ..., 2, 1, 1, 1, 1, 2, 3, 7, 23, 59, ..., defined by A_{h-2}A_{h+2} = A_{h-1}A_{h+1} + A_{h}^2 arises from the curve V^2 - V = U^3 + 3U^2 + 2U by reporting the denominators of the points M+hS, with M=(-1,1) and S=(0,0). Just so, the sequence ..., 2, 1, 1, 1, 1, 1, 1, 2, 3, 4, 8, 17, 50, ..., given by the recursion B_{h-3}B_{h+3} = B_{h-2}B_{h+2} + B_{h}^2 arises from adding multiples of the class of the divisor at infinity on the Jacobian of the curve Y^2 = (X^3-4X+1)^2 + 4(X-2) of genus 2 to the class of the divisor given by [(phi,0),(\bar{phi},0)]; here, no doubt to the joy of adherents to the cult of Fibonacci, phi is the golden ratio.
Of course, the real surprise is that the stated recursions produce sequences of integers. I will give a lowbrow explanation of all this by showing that certain continued fraction expansions in quadratic function fields both yield these examples and allow one to identify the curves giving rise to such examples.
Prerequisites: notwithstanding the big words in the abstract, almost none.
Title: Folded continued fractions
Abstract: Denote the sum \sum_{h=0}^{\infty} 2^{-2^h} by sigma. Remarkably, the continued fraction expansion of sigma is highly patterned and all its partial quotients are very small, so sigma is only poorly approximable. I'll deal with the `why is this so' of the matter by introducing and explaining continued fraction expansions, by then expanding the formal power series S = X\sum_{h=0}^{\infty} X^{-2^h}, finding that after a single 1, all its partial quotients are +/-X, recognising the signs +/- as arising from folding a piece of paper in half many times, and ....
Title: Paperfolding, automata, and rational functions
Abstract: The act of folding a sheet of paper in half, and iterating the operation, places in that sheet a sequence of creases appearing as valleys or ridges. Coding these as 1 and 0 respectively yields a sequence (f_h), the paper folding sequence, with generating function f(X) = \sum_{h >= 1}f_h X^h, the paperfolding function. It turns out to be easy to notice that f(X) satisfies a functional equation of a kind first studied by Mahler nearly seventy years ago. Moreover, viewed as defined over F_2, the field of two elements, the paperfolding function is algebraic --- it satisfies a polynomial equation over F_2(X). It's also easy to see that the paperfolding sequence is `automatic'; it is generated by binary substitutions. These phenomena are not unique to paperfolding. They are shared by the good reductions modulo a prime p of arbitrary diagonals of arbitrary rationals functions in many variables, equivalently by the reductions modulo p of a wide class of series in one variable satisfying linear differential equations with polynomial coefficients. I will tell the necessary stories to explain all this [and will show some relevant pictures from Michael Crichton's novel Jurassic Park. Audience members should bring note paper along, not to write on, of course, but to fold].