The projective line minus three fractional 3 kinds of integral - PowerPoint PPT Presentation

The projective line minus three fractional points Bjorn Poonen The projective line minus three fractional 3 kinds of integral points points Darmon’s M -curves Campana’s orbifolds Almost integral points Counting points of Bjorn Poonen bounded height Counting functions Heuristics Theorems and University of California at Berkeley conjectures Consequences July 13, 2006 (grew out of discussions with many people at the Spring 2006 MSRI program on Rational and Integral Points on Higher-Dimensional Varieties, especially Fr´ ed´ eric Campana, Jordan Ellenberg, and Aaron Levin)

The projective line minus three fractional points Bjorn Poonen 3 kinds of integral points 1 3 kinds of integral points Darmon’s M -curves Darmon’s M -curves Campana’s orbifolds Campana’s orbifolds Almost integral points Almost integral points Counting points of bounded height Counting functions Heuristics Theorems and conjectures Counting points of bounded height Consequences 2 Counting functions Heuristics Theorems and conjectures Consequences

The projective line Motivation: a generalized Fermat equation minus three fractional points Let Bjorn Poonen ( x , y , z ) ∈ Z 3 : x 2 + y 3 = z 7 � � S ( Z ) := . 3 kinds of integral gcd( x , y , z ) = 1 points Darmon’s M -curves Then Campana’s orbifolds Almost integral points � 1 � Counting points of S ( Z ) → P 1 ( Q ) := Q ∪ bounded height 0 Counting functions Heuristics ( x , y , z ) �→ x 2 = 1 − y 3 � � Theorems and conjectures . Consequences z 7 z 7 induces a bijection   num( q ) is a square S ( Z )    q ∈ P 1 ( Q ) : sign ↔ num( q − 1) is a cube  . den( q ) is a 7 th power Darmon and Granville applied Faltings’ theorem to covers of P 1 ramified only over { 0 , 1 , ∞} to prove that the right hand side is finite, and hence deduce that S ( Z ) is finite.

� � The projective line Geometric interpretation minus three fractional points Define a Z -scheme Bjorn Poonen x 2 + y 3 = z 7 in A 3 � � S := − { (0 , 0 , 0) } . 3 kinds of integral points Then the morphism Darmon’s M -curves Campana’s orbifolds Almost integral points ( x , y , z ) S Counting points of � bounded height Counting functions Heuristics Theorems and x 2 / y 7 P 1 conjectures Consequences has multiple fibers above 0, 1, ∞ , having multiplicities 2, 3, 7, respectively. So S → P 1 factors through a stack ˜ P 1 := [ S / G m ] that looks like P 1 except that the points 0, 1, ∞ have been replaced by a 1 / 2-point, a 1 / 3-point, and a 1 / 7-point, respectively. Points in S ( Z ) map to ˜ P 1 ( Z ) ⊂ P 1 ( Z ) = P 1 ( Q ). Moral: Multiple fibers impose conditions on images of integral points.

The projective line Numerator with respect to a point minus three fractional points Bjorn Poonen We saw that a fiber of multiplicity 2 above 0 ∈ P 1 ( Q ) 3 kinds of integral imposes the condition that num( q ) be a square. points Darmon’s M -curves What condition is imposed, say, by a fiber of Campana’s orbifolds multiplicity 2 above the point 3 / 5 ∈ P 1 ( Q )? Almost integral points Counting points of bounded height Answer: The value of num 3 / 5 ( a / b ) := | 5 a − 3 b | should Counting functions be a square. Heuristics Theorems and conjectures In general: Consequences Definition (Numerator with respect to the point c / d ) For c / d ∈ P 1 ( Q ), define num c / d ( a / b ) := | ad − bc | . Examples If c ∈ Z , then num c ( a / b ) = num( a / b − c ). num ∞ ( q ) = den( q ).

The projective line Darmon’s M-curves minus three fractional points M-curve data: Bjorn Poonen points P 1 , . . . , P N ∈ P 1 ( Q ), with multiplicities m 1 , . . . , m N ∈ { 2 , 3 , . . . } ∪ {∞} . 3 kinds of integral points An M-curve may be denoted formally by P 1 − ∆, where Darmon’s M -curves Campana’s orbifolds Almost integral points N � 1 − 1 � � Counting points of ∆ := [ P i ] . bounded height m i Counting functions i =1 Heuristics Theorems and (It is really a kind of stack.) conjectures Consequences Define the Euler characteristic χ ( P 1 − ∆) := χ ( P 1 ) − deg ∆ ∞ � 1 − 1 � � = 2 − . m i i =1 Definition (Integral points in Darmon’s sense) ( P 1 − ∆)( Z ) := { q ∈ P 1 ( Q ) : num P i ( q ) is an m i -th power ∀ i } Note: “ ∞ -th power” means unit (i.e., ± 1).

The projective line Campana’s orbifolds: motivation minus three fractional points Bjorn Poonen 3 kinds of integral points Suppose π : S → P 1 is such that the fiber above 0 Darmon’s M -curves Campana’s orbifolds consists of two irreducible components, one of Almost integral points Counting points of multiplicity 2 and one of multiplicity 5. bounded height Counting functions If s ∈ S ( Z ), then π ( s ) is again restricted: its numerator Heuristics is of the form u 2 v 5 . Theorems and conjectures Consequences Equivalently, in the prime factorization of num( π ( s )), every exponent is a nonnegative integer combination of 2 and 5. In particular (but not equivalently), num( π ( s )) is a squareful integer, i.e., p e 1 1 · · · p e r with all e i ≥ 2. r

More generally: The projective line minus three fractional points Definition Bjorn Poonen An integer a is called m -powerful if in its prime factorization all (nonzero) exponents are ≥ m . 3 kinds of integral points An integer a is called ∞ -powerful if a = ± 1. Darmon’s M -curves Campana’s orbifolds Almost integral points Definition (Integral points in Campana’s sense) Counting points of bounded height For an M-curve P 1 − ∆, define Counting functions Heuristics Theorems and conjectures ( P 1 − ∆) C ( Z ) := { q ∈ P 1 ( Q ) : num P i ( q ) is m i -powerful ∀ i } Consequences Example Let ∆ = 1 2 [0] + 1 2 [3] + [ ∞ ]. Then   a is squareful, a ( P 1 − ∆) C ( Z ) =   b ∈ P 1 ( Q ) : a − 3 b is squareful, and b = 1   = { a ∈ Z : a , a − 3 are both squareful }

The projective line Almost integral points minus three fractional points Definition (Height and penalty) Bjorn Poonen For an M -curve P 1 − ∆ and q = a / b ∈ P 1 ( Q ), define 3 kinds of integral points Darmon’s M -curves H ( q ) := max ( | a | , | b | ) Campana’s orbifolds Almost integral points N Counting points of p 1 − 1 bounded height � � mi . penalty P 1 − ∆ ( q ) := Counting functions Heuristics i =1 p such that Theorems and conjectures m i ∤ v p (num Pi ( q )) Consequences Remark: If ∆ consists of whole points, then log(penalty) is the “truncated counting function” in Vojta’s “more general abc conjecture”. Fix a real number r ∈ [0 , deg ∆] (“tolerance level”). Definition (Almost integral points) ( P 1 − ∆ + r )( Z ) := q ∈ P 1 ( Q ) : penalty P 1 − ∆ ( q ) ≤ H ( q ) r � � Also define χ ( P 1 − ∆ + r ) := χ ( P 1 − ∆) + r .

The projective line Counting points of bounded height minus three fractional points Bjorn Poonen 3 kinds of integral points We will study when the set of integral points (in each of Darmon’s M -curves the three senses) is finite. Campana’s orbifolds Almost integral points When it is infinite, we will measure it by counting Counting points of bounded height points of bounded height. Counting functions Heuristics Theorems and conjectures Definition (Counting functions) Consequences ( P 1 − ∆)( Z ) ≤ B := q ∈ ( P 1 − ∆)( Z ) : H ( q ) ≤ B � � . ( P 1 − ∆) C ( Z ) ≤ B := q ∈ ( P 1 − ∆) C ( Z ) : H ( q ) ≤ B � � . ( P 1 − ∆ + r )( Z ) ≤ B := q ∈ ( P 1 − ∆ + r )( Z ) : H ( q ) ≤ B � � .

The projective line Heuristics for Darmon’s M-curves minus three fractional points Bjorn Poonen ( P 1 − ∆)( Z ) #( P 1 − ∆)( Z ) ≤ B ∆ 3 kinds of integral � a ∼ B 2 points � 0 b : gcd( a , b ) = 1 Darmon’s M -curves � a b : b is m th power 1 − 1 ∼ B · B 1 / m � � � Campana’s orbifolds [ ∞ ] m Almost integral points � a � � � 1 − 1 � ∼ B χ ? [ P i ] b : · · · Counting points of m i bounded height Counting functions Heuristics Theorems and conjectures Heuristic: Consequences 1 − 1 � � In the case ∆ = [ ∞ ], the probability that a m point satisfies the condition at ∞ is ∼ B · B 1 / m 1 = B 1 − 1 / m . B 2 If conditions at different points are independent, the count for ∆ = � � � 1 − 1 [ P i ] should be m i � 1 � � 1 � ∼ B 2 = B χ . · · · B 1 − 1 / m 1 B 1 − 1 / m N

The projective line Heuristics for Darmon’s M-curves minus three fractional points Bjorn Poonen ( P 1 − ∆)( Z ) #( P 1 − ∆)( Z ) ≤ B ∆ 3 kinds of integral � a ∼ B 2 points � 0 b : gcd( a , b ) = 1 Darmon’s M -curves � a b : b is m th power 1 − 1 ∼ B · B 1 / m � � � Campana’s orbifolds [ ∞ ] m Almost integral points � a � � � 1 − 1 � ∼ B χ ? [ P i ] b : · · · Counting points of m i bounded height Counting functions Heuristics Theorems and conjectures Heuristic: Consequences 1 − 1 � � In the case ∆ = [ ∞ ], the probability that a m point satisfies the condition at ∞ is ∼ B · B 1 / m 1 = B 1 − 1 / m . B 2 If conditions at different points are independent, the count for ∆ = � � � 1 − 1 [ P i ] should be m i � 1 � � 1 � ∼ B 2 = B χ . · · · B 1 − 1 / m 1 B 1 − 1 / m N