Rank and Bias in Families of Hyperelliptic Curves
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1 Rank and Bias in Families of Hyerellitic Curves Trajan Hammonds 1 Ben Logsdon 2 thammond@andrew.cmu.edu bcl5@williams.edu Joint with Seoyoung Kim 3 and Steven J. Miller 2 1 Carnegie Mellon University 2 Williams College 3 Brown University Québec-Maine Number Theory Conference Université Laval, October 2018
2 Hyerellitic Curves Define a hyerellitic curve of genus g over Q(T ): X : y 2 = f (x, T ) = x 2g+1 +A 2g (T )x 2g + +A 1 (T )x +A 0 (T ).
3 Hyerellitic Curves Define a hyerellitic curve of genus g over Q(T ): X : y 2 = f (x, T ) = x 2g+1 +A 2g (T )x 2g + +A 1 (T )x +A 0 (T ). Let a X () = + 1 #X (F ). Then ( ) f (x, t) a X () = x()
4 Hyerellitic Curves Define a hyerellitic curve of genus g over Q(T ): X : y 2 = f (x, T ) = x 2g+1 +A 2g (T )x 2g + +A 1 (T )x +A 0 (T ). Let a X () = + 1 #X (F ). Then ( ) f (x, t) a X () = x() and its m-th ower sum A m,x () = t() a X () m.
5 Generalized Nagao s conjecture Generalized Nagao s Conjecture 1 lim 1 X X A 1,χ() log = rank J X (Q(T)). X
6 Generalized Nagao s conjecture Generalized Nagao s Conjecture 1 lim 1 X X A 1,χ() log = rank J X (Q(T)). X Goal: Construct families of hyerellitic curves with high rank.
7 Hyerellitic curves with moderately large rank over Q(T )
8 Moderate-Rank Family Theorem (HLKM, 2018) Assume the Generalized Nagao Conjecture and trivial Chow trace Jacobian. For any g 1, we can construct infinitely many genus g hyerellitic curves X over Q(T ) such that rank J X (Q(T)) = 4g + 2.
9 Moderate-Rank Family Theorem (HLKM, 2018) Assume the Generalized Nagao Conjecture and trivial Chow trace Jacobian. For any g 1, we can construct infinitely many genus g hyerellitic curves X over Q(T ) such that rank J X (Q(T)) = 4g + 2. Close to current record of 4g + 7.
10 Moderate-Rank Family Theorem (HLKM, 2018) Assume the Generalized Nagao Conjecture and trivial Chow trace Jacobian. For any g 1, we can construct infinitely many genus g hyerellitic curves X over Q(T ) such that rank J X (Q(T)) = 4g + 2. Close to current record of 4g + 7. No height matrix or basis comutation.
11 Moderate-Rank Family Theorem (HLKM, 2018) Assume the Generalized Nagao Conjecture and trivial Chow trace Jacobian. For any g 1, we can construct infinitely many genus g hyerellitic curves X over Q(T ) such that rank J X (Q(T)) = 4g + 2. Close to current record of 4g + 7. No height matrix or basis comutation. This generalizes a construction of Arms, Lozano-Robledo, and Miller in the ellitic surface case.
12 Idea of Construction Define a genus g curve X : y 2 = f (x, T ) = x 2g+1 T 2 + 2g(x)T h(x) g(x) = x 2g+1 + 2g i=0 a i x i h(x) = (A 1)x 2g+1 + 2g i=0 A i x i. The discriminant of the quadratic olynomial is D T (x) := g(x) 2 + x 2g+1 h(x).
13 Idea of Construction A 1,X () = ( ) f (x, t) t() x()
14 Idea of Construction A 1,X () = ( ) f (x, t) t() x() = ( ) x 2g+1 ( 1) x() D t (x) 0 + x() D t (x) 0 ( ) x 2g+1 ( 1)
15 Idea of Construction A 1,X () = ( ) f (x, t) t() x() = ( ) x 2g+1 ( 1) + x() x() D t (x) 0 = x() D t (x) 0 ( ) x x() ( x ) D t (x) 0 ( ) x 2g+1 ( 1)
16 Idea of Construction A 1,X () = ( ) f (x, t) t() x() = ( ) x 2g+1 ( 1) x() D t (x) 0 = x() D t (x) 0 ( ) x + x() D t (x) 0 ( ) x 2g+1 ( 1)
17 Idea of Construction A 1,X () = ( ) f (x, t) t() x() = ( ) x 2g+1 ( 1) x() D t (x) 0 = x() D t (x) 0 ( ) x ( Therefore, A 1,X () is D t (x). x + x() D t (x) 0 ( ) x 2g+1 ( 1) ) summed over the roots of
18 Idea of Construction A 1,X () = ( ) f (x, t) t() x() = ( ) x 2g+1 ( 1) x() D t (x) 0 = x() D t (x) 0 ( ) x ( Therefore, A 1,X () is x + x() D t (x) 0 ( ) x 2g+1 ( 1) ) summed over the roots of D t (x). To maximize the sum, we make each x a erfect square.
19 Idea of Construction Key Idea Make the roots of D t (x) distinct nonzero erfect squares. Choose roots ρ 2 i of D t (x) so that D t (x) = A 4g+2 i=1 ( x ρ 2 i ).
20 Idea of Construction Key Idea Make the roots of D t (x) distinct nonzero erfect squares. Choose roots ρ 2 i of D t (x) so that D t (x) = A Equate coefficients in D t (x) = A 4g+2 i=1 4g+2 i=1 ( x ρ 2 i ). ( x ρ 2 i ) = g(x) 2 + x 2g+1 h(x).
21 Idea of Construction Key Idea Make the roots of D t (x) distinct nonzero erfect squares. Choose roots ρ 2 i of D t (x) so that D t (x) = A Equate coefficients in D t (x) = A 4g+2 i=1 4g+2 i=1 ( x ρ 2 i ). ( x ρ 2 i ) = g(x) 2 + x 2g+1 h(x). Solve the nonlinear system for the coefficients of g, h.
22 Idea of the Construction A 1,χ ()
23 Idea of the Construction A 1,χ () = x mod D t (x) 0 ( ) x 2g+1
24 Idea of the Construction A 1,χ () = x mod D t (x) 0 ( ) x 2g+1 = (# of erfect-square roots of D t (x)) = (4g + 2).
25 Idea of the Construction A 1,χ () = x mod D t (x) 0 ( ) x 2g+1 = (# of erfect-square roots of D t (x)) = (4g + 2). Then by the Generalized Nagao Conjecture lim X 1 X X 1 (4g + 2) log = 4g + 2 = rank J X (Q(T )).
26 Future Work Find a linearly indeendent basis. Generalizing another technique in Arms, Lozano-Robledo, and Miller.
27 Bias Conjecture
28 Bias Conjecture Michel s Theorem For one-arameter families of ellitic curves E, the second moment A 2,E () is A 2,E () = 2 + O ( 3/2).
29 Bias Conjecture Michel s Theorem For one-arameter families of ellitic curves E, the second moment A 2,E () is A 2,E () = 2 + O ( 3/2). Bias Conjecture (Miller) The largest lower order term in the second moment exansion that does not average to 0 is on average negative.
30 Bias Conjecture Michel s Theorem For one-arameter families of ellitic curves E, the second moment A 2,E () is A 2,E () = 2 + O ( 3/2). Bias Conjecture (Miller) The largest lower order term in the second moment exansion that does not average to 0 is on average negative. Goal: Find as many hyerellitic families with as much bias as ossible.
31 The Bias Family Theorem (HLKM 2018) Consider X : y 2 = x n + x h T k. If gcd(k, n h, 1) = 1, then (gcd(n h, 1) 1)( 2 ) h even A 2,X () = gcd(n h, 1)( 2 ) h odd ( ) 0 otherwise
32 Calculations Part 1: k-periodicity A 2,X () = t,x,y() x n + x h t k y n + y h t k
33 Calculations Part 1: k-periodicity A 2,X () = t,x,y() = t,x,y() x n + x h t k y n + y h t k (t n x n ) + (t h x h )t k (t n y n ) + (t h y h )t k
34 Calculations Part 1: k-periodicity A 2,X () = t,x,y() = t,x,y() = t,x,y() x n + x h t k y n + y h t k (t n x n ) + (t h x h )t k (t n y n ) + (t h y h )t k x n + x h t (k+(n h)) y n + y h t (k+(n h))
35 Calculations Part 1: k-periodicity A 2,X () = t,x,y() = t,x,y() = t,x,y() x n + x h t k y n + y h t k (t n x n ) + (t h x h )t k (t n y n ) + (t h y h )t k x n + x h t (k+(n h)) y n + y h t (k+(n h)) The second moment is eriodic in k with eriod (n h).
36 Calculations Part 2 A 2,X () = t,x,y() x n + x h t k y n + y h t k
37 Calculations Part 2 A 2,X () = t,x,y() = t,x,y() x n + x h t k y n + y h t k x n + x h t m y n + y h t m (m n h k)
38 Calculations Part 2 A 2,X () = t,x,y() = t,x,y() = t,x,y() x n + x h t k y n + y h t k x n + x h t m y n + y h t m x n + x h t y n + y h t (m n h k) (Frobenius)
39 Calculations Part 2 A 2,X () = t,x,y() = t,x,y() = t,x,y() x n + x h t k y n + y h t k x n + x h t m y n + y h t m x n + x h t y n + y h t gcd(n h, k, 1) = 1 (m n h k) (Frobenius)
40 Calculations Part 2 A 2,X () = t,x,y() = t,x,y() = t,x,y() x n + x h t k y n + y h t k x n + x h t m y n + y h t m x n + x h t y n + y h t gcd(n h, k, 1) = 1 (m n h k) (Frobenius) Thus, this reduces to calculating the second moment of y 2 = x n + x h T, which is straightforward.
41 We thank our advisors Steven J. Miller and Seoyoung Kim, Williams College, the Finnerty Fund, the SMALL REU and the National Science Foundation (grants DMS and DMS ).
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