Rank and Bias in Families of Hyperelliptic Curves Trajan Hammonds 1 - PowerPoint PPT Presentation

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Rank and Bias in Families of Hyperelliptic 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 1

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Hyperelliptic Curves Define a hyperelliptic curve of genus g over Q ( T ) : X : y 2 = f ( x , T ) = x 2 g + 1 + A 2 g ( T ) x 2 g + · · · + A 1 ( T ) x + A 0 ( T ) . 2

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Hyperelliptic Curves Define a hyperelliptic curve of genus g over Q ( T ) : X : y 2 = f ( x , T ) = x 2 g + 1 + A 2 g ( T ) x 2 g + · · · + A 1 ( T ) x + A 0 ( T ) . Let a X ( p ) = p + 1 − # X ( F p ) . Then � f ( x , t ) � � a X ( p ) = − p x ( p ) 3

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Hyperelliptic Curves Define a hyperelliptic curve of genus g over Q ( T ) : X : y 2 = f ( x , T ) = x 2 g + 1 + A 2 g ( T ) x 2 g + · · · + A 1 ( T ) x + A 0 ( T ) . Let a X ( p ) = p + 1 − # X ( F p ) . Then � f ( x , t ) � � a X ( p ) = − p x ( p ) and its m -th power sum � a X ( p ) m . A m , X ( p ) = t ( p ) 4

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Generalized Nagao’s conjecture Generalized Nagao’s Conjecture 1 − 1 � lim pA 1 ,χ ( p ) log p = rank J X ( Q ( T )) . X X →∞ p ≤ X 5

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Generalized Nagao’s conjecture Generalized Nagao’s Conjecture 1 − 1 � lim pA 1 ,χ ( p ) log p = rank J X ( Q ( T )) . X X →∞ p ≤ X Goal: Construct families of hyperelliptic curves with high rank. 6

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Hyperelliptic curves with moderately large rank over Q ( T ) 7

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements 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 hyperelliptic curves X over Q ( T ) such that rank J X ( Q ( T )) = 4g + 2 . 8

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements 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 hyperelliptic curves X over Q ( T ) such that rank J X ( Q ( T )) = 4g + 2 . Close to current record of 4 g + 7. 9

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements 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 hyperelliptic curves X over Q ( T ) such that rank J X ( Q ( T )) = 4g + 2 . Close to current record of 4 g + 7. No height matrix or basis computation. 10

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements 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 hyperelliptic curves X over Q ( T ) such that rank J X ( Q ( T )) = 4g + 2 . Close to current record of 4 g + 7. No height matrix or basis computation. This generalizes a construction of Arms, Lozano-Robledo, and Miller in the elliptic surface case. 11

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Idea of Construction Define a genus g curve X : y 2 = f ( x , T ) = x 2 g + 1 T 2 + 2 g ( x ) T − h ( x ) 2 g g ( x ) = x 2 g + 1 + � a i x i i = 0 2 g h ( x ) = ( A − 1 ) x 2 g + 1 + � A i x i . i = 0 The discriminant of the quadratic polynomial is D T ( x ) := g ( x ) 2 + x 2 g + 1 h ( x ) . 12

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Idea of Construction � f ( x , t ) � � � − A 1 , X ( p ) = p t ( p ) x ( p ) 13

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Idea of Construction � f ( x , t ) � � � − A 1 , X ( p ) = p t ( p ) x ( p ) � x 2 g + 1 � x 2 g + 1 � � � � = ( p − 1 ) + ( − 1 ) p p x ( p ) x ( p ) D t ( x ) ≡ 0 D t ( x ) �≡ 0 14

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Idea of Construction � f ( x , t ) � � � − A 1 , X ( p ) = p t ( p ) x ( p ) � x 2 g + 1 � x 2 g + 1 � � � � = ( p − 1 ) + ( − 1 ) p p x ( p ) x ( p ) D t ( x ) ≡ 0 D t ( x ) �≡ 0 � x � � x � � � = p − p p x ( p ) x ( p ) D t ( x ) ≡ 0 15

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Idea of Construction � f ( x , t ) � � � − A 1 , X ( p ) = p t ( p ) x ( p ) � x 2 g + 1 � x 2 g + 1 � � � � = ( p − 1 ) + ( − 1 ) p p x ( p ) x ( p ) D t ( x ) ≡ 0 D t ( x ) �≡ 0 � x � � = p p x ( p ) D t ( x ) ≡ 0 16

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Idea of Construction � f ( x , t ) � � � − A 1 , X ( p ) = p t ( p ) x ( p ) � x 2 g + 1 � x 2 g + 1 � � � � = ( p − 1 ) + ( − 1 ) p p x ( p ) x ( p ) D t ( x ) ≡ 0 D t ( x ) �≡ 0 � x � � = p p x ( p ) D t ( x ) ≡ 0 � � x Therefore, − A 1 , X ( p ) is p summed over the roots of p D t ( x ) . 17

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Idea of Construction � f ( x , t ) � � � − A 1 , X ( p ) = p t ( p ) x ( p ) � x 2 g + 1 � x 2 g + 1 � � � � = ( p − 1 ) + ( − 1 ) p p x ( p ) x ( p ) D t ( x ) ≡ 0 D t ( x ) �≡ 0 � x � � = p p x ( p ) D t ( x ) ≡ 0 � � x Therefore, − A 1 , X ( p ) is p summed over the roots of p D t ( x ) . To maximize the sum, we make each x a perfect square. 18

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Idea of Construction Key Idea Make the roots of D t ( x ) distinct nonzero perfect squares. Choose roots ρ 2 i of D t ( x ) so that 4 g + 2 � x − ρ 2 � � D t ( x ) = A . i i = 1 19

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Idea of Construction Key Idea Make the roots of D t ( x ) distinct nonzero perfect squares. Choose roots ρ 2 i of D t ( x ) so that 4 g + 2 � x − ρ 2 � � D t ( x ) = A . i i = 1 Equate coefficients in 4 g + 2 = g ( x ) 2 + x 2 g + 1 h ( x ) . � x − ρ 2 � � D t ( x ) = A i i = 1 20

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Idea of Construction Key Idea Make the roots of D t ( x ) distinct nonzero perfect squares. Choose roots ρ 2 i of D t ( x ) so that 4 g + 2 � x − ρ 2 � � D t ( x ) = A . i i = 1 Equate coefficients in 4 g + 2 = g ( x ) 2 + x 2 g + 1 h ( x ) . � x − ρ 2 � � D t ( x ) = A i i = 1 Solve the nonlinear system for the coefficients of g , h . 21

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Idea of the Construction − A 1 ,χ ( p ) 22

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Idea of the Construction � x 2 g + 1 � � − A 1 ,χ ( p ) = p p x mod p D t ( x ) ≡ 0 23

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Idea of the Construction � x 2 g + 1 � � − A 1 ,χ ( p ) = p p x mod p D t ( x ) ≡ 0 = p · ( # of perfect-square roots of D t ( x )) = p · ( 4 g + 2 ) . 24

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Idea of the Construction � x 2 g + 1 � � − A 1 ,χ ( p ) = p p x mod p D t ( x ) ≡ 0 = p · ( # of perfect-square roots of D t ( x )) = p · ( 4 g + 2 ) . Then by the Generalized Nagao Conjecture 1 1 � p · p · ( 4 g + 2 ) log p = 4 g + 2 = rank J X ( Q ( T )) . lim X X →∞ p ≤ X 25

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Future Work Find a linearly independent basis. Generalizing another technique in Arms, Lozano-Robledo, and Miller. 26

Background Hyperelliptic curves with moderately large rank over Q ( T ) Bias Conjecture Acknowledgements Bias Conjecture 27