Counting points on curves
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1 Counting points on curves Edgar Costa Dartmouth College Québec-Maine Number Theory conference, 8th October 2016 (joint work with David Harvey, UNSW) 1 / 22 Edgar Costa (Dartmouth College) Counting points on curves
2 Setup p = prime C P n F p = algebraic curve defined over F p The Hasse Weil zeta function of C is the generating function ( #C(F p a)t a ) Z C (t) := exp a a>0 2 / 22 Edgar Costa (Dartmouth College) Counting points on curves
3 Example Let p = 5 and C be genus 3 curve given in P 2 by x 4 + xy 3 + 2y 4 z 4 = 0 3 / 22 Edgar Costa (Dartmouth College) Counting points on curves
4 Example Let p = 5 and C be genus 3 curve given in P 2 by x 4 + xy 3 + 2y 4 z 4 = 0 By naive point enumeration, we find that #C(F p ) = 9, #C(F p 4) = 581, #C(F p 2) = 29, #C(F p 5) = 3309, #C(F p 3) = 156,... 3 / 22 Edgar Costa (Dartmouth College) Counting points on curves
5 Example Let p = 5 and C be genus 3 curve given in P 2 by x 4 + xy 3 + 2y 4 z 4 = 0 By naive point enumeration, we find that #C(F p ) = 9, #C(F p 4) = 581, #C(F p 2) = 29, #C(F p 5) = 3309, #C(F p 3) = 156,... thus: Z C (t) = 1 + 9t + 55t t t t t / 22 Edgar Costa (Dartmouth College) Counting points on curves
6 Example Let p = 5 and C be genus 3 curve given in P 2 by x 4 + xy 3 + 2y 4 z 4 = 0 By naive point enumeration, we find that thus: #C(F p ) = 9, #C(F p 4) = 581, #C(F p 2) = 29, #C(F p 5) = 3309, #C(F p 3) = 156,... Z C (t) = 1 + 9t + 55t t t t t 6 + = 1 + 3t + 6t2 + 19t t t t 6 (1 t)(1 5t) 3 / 22 Edgar Costa (Dartmouth College) Counting points on curves
7 Typical algorithms Naively computing is not practical! #C(F p ), #C(F p 2),, #C(F p g ) Looping over F p g takes at least O(p g ) time. 4 / 22 Edgar Costa (Dartmouth College) Counting points on curves
8 Typical algorithms Naively computing is not practical! #C(F p ), #C(F p 2),, #C(F p g ) Looping over F p g takes at least O(p g ) time. One can them compute the last line with variety of methods. l-adic cohomology p-adic cohomology 4 / 22 Edgar Costa (Dartmouth College) Counting points on curves
9 New method Today I will overview a new elementary method to compute L C (t). 5 / 22 Edgar Costa (Dartmouth College) Counting points on curves
10 New method Today I will overview a new elementary method to compute L C (t). Simple Practical Cohomology free! 5 / 22 Edgar Costa (Dartmouth College) Counting points on curves
11 New method - Input/Output Input: F (x, y, z) homogeneous polynomial of degree d Output: L C (t), where C is the desingularization of the zero locus of F (x, y, z) in P 2 6 / 22 Edgar Costa (Dartmouth College) Counting points on curves
12 New method - Input/Output Input: F (x, y, z) homogeneous polynomial of degree d Output: L C (t), where C is the desingularization of the zero locus of F (x, y, z) in P 2 Assumption: The polynomials F (0, y, z), F (x, 0, y) and F (x, y, 0) have no repeated factors. Geometrically, the curve given by F intersects the coordinate axes (x = 0, y = 0, and z = 0) transversally. 6 / 22 Edgar Costa (Dartmouth College) Counting points on curves
13 New method - Ingredients C curve of genus g that is the desingularization of {F = 0} P 2 Goal: compute #C(F p a) 7 / 22 Edgar Costa (Dartmouth College) Counting points on curves
14 New method - Ingredients C curve of genus g that is the desingularization of {F = 0} P 2 Goal: compute #C(F p a) Approach: Compute #{F (x, y, z) = 0 : (x, y, z) P 2 (F p a), xyz 0} Compute #{F (x, y, z) = 0 : (x, y, z) P 2 (F p a), xyz = 0} Resolve the singularities of F (x, y, z) = 0 (over F p ) Altogether, we can deduce #C(F p a). 7 / 22 Edgar Costa (Dartmouth College) Counting points on curves
15 New method - Ingredients C curve of genus g that is the desingularization of {F = 0} P 2 Goal: compute #C(F p a) Approach: Compute #{F (x, y, z) = 0 : (x, y, z) P 2 (F p a), xyz 0} Compute #{F (x, y, z) = 0 : (x, y, z) P 2 (F p a), xyz = 0} Resolve the singularities of F (x, y, z) = 0 (over F p ) Altogether, we can deduce #C(F p a). 7 / 22 Edgar Costa (Dartmouth College) Counting points on curves
16 New method - Trace formula Let B l = {u N 3 : i u i = l}. For u B dn, let (F n ) u be the coefficient of x u 0 y u 1 z u 2 in F n. (M s ) v,u := ( F s(p 1)) pv u for v, u B ds. 8 / 22 Edgar Costa (Dartmouth College) Counting points on curves
17 New method - Trace formula Let B l = {u N 3 : i u i = l}. For u B dn, let (F n ) u be the coefficient of x u 0 y u 1 z u 2 in F n. (M s ) v,u := ( F s(p 1)) pv u for v, u B ds. M 0, M 1,..., M λ #C(F p a) mod p λ a 8 / 22 Edgar Costa (Dartmouth College) Counting points on curves
18 New method - Trace formula Let B l = {u N 3 : i u i = l}. For u B dn, let (F n ) u be the coefficient of x u 0 y u 1 z u 2 in F n. (M s ) v,u := ( F s(p 1)) pv u for v, u B ds. M 0, M 1,..., M λ #C(F p a) mod p λ a Harvey (2015) For p not small: #{F (x, y, z) = 0 : (x, y, z) P 2 (F p a), xyz 0} λ ( ) = (p a 1) 2 λ Tr(Ms a ) mod p λ s s=0 8 / 22 Edgar Costa (Dartmouth College) Counting points on curves
19 M 1 for p = 7, d = 5 9 / 22 Edgar Costa (Dartmouth College) Counting points on curves
20 M 1 for p = 13, d = 5 10 / 22 Edgar Costa (Dartmouth College) Counting points on curves
21 M 1 for p = 53, d = 5 11 / 22 Edgar Costa (Dartmouth College) Counting points on curves
22 M 3 for p = 53, d = 5 12 / 22 Edgar Costa (Dartmouth College) Counting points on curves
23 M 3 for p = 199, d = 5 13 / 22 Edgar Costa (Dartmouth College) Counting points on curves
24 New method - Key idea There are relations between neighbouring coefficients of F n. We can derive these relations from: or any other derivative. F n+1 = F F n x F n+1 = (n + 1) x F F n 14 / 22 Edgar Costa (Dartmouth College) Counting points on curves
25 Example, d = 4 and n = 4 The coefficient (F n+1 ) (12,3,5) is known if given F n on: 15 / 22 Edgar Costa (Dartmouth College) Counting points on curves
26 Example, d = 4 and n = 4 Indeed, there are 2 independent equations involving the red dots 16 / 22 Edgar Costa (Dartmouth College) Counting points on curves
27 Example, d = 4 and n = 4 Combining enough relations we can move a larger triangle. 17 / 22 Edgar Costa (Dartmouth College) Counting points on curves
28 Example, d = 4 and n = 4 Combining enough relations we can move a larger triangle. 18 / 22 Edgar Costa (Dartmouth College) Counting points on curves
29 Example, d = 4 and n = 4 Combining enough relations we can move a larger triangle. 19 / 22 Edgar Costa (Dartmouth College) Counting points on curves
30 New method - Altogether Start with a triangle at one of the vertices, where the coefficients of F s(p 1) are trivial to compute. Move it around until we have computed by all the target coefficients. Assemble M s in O(p) time 20 / 22 Edgar Costa (Dartmouth College) Counting points on curves
31 Toy implementation in sage Setup: d = 6 and genus = 5 p = 1009 : 7 minutes p = 1999 : 14 minutes 21 / 22 Edgar Costa (Dartmouth College) Counting points on curves
32 Toy implementation in sage Setup: d = 6 and genus = 5 p = 1009 : 7 minutes p = 1999 : 14 minutes Tuitman s algorithm in Magma: p = 1009 : 8 minutes p = 1999 : 24 minutes Disclaimer: this is NOT a fair comparison. 21 / 22 Edgar Costa (Dartmouth College) Counting points on curves
33 Future Short term: Finish C++ implementation precision bounds Long term, perhaps consider the two obvious improvements: reduce the running time to p 1/2+o(1). average polynomial time 22 / 22 Edgar Costa (Dartmouth College) Counting points on curves
34 Future Short term: Finish C++ implementation precision bounds Long term, perhaps consider the two obvious improvements: reduce the running time to p 1/2+o(1). average polynomial time Thank you! 22 / 22 Edgar Costa (Dartmouth College) Counting points on curves
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