Effective computations of HasseWeil zeta functions Edgar Costa - PowerPoint PPT Presentation

Effective computations of Hasse–Weil zeta functions Edgar Costa ICERM/Dartmouth College 20th October 2015 ICERM 1 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Variation of N´ eron-Severi ranks of K3 surfaces Edgar Costa ICERM/Dartmouth College 20th October 2015 ICERM 2 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Setup k a number field; X a K3 surface over k ; p a prime of k where X has good reduction X p ; NS( • ) := Pic( • ) / Pic 0 ( • ), the N´ eron-Severi group; � a Z lattice geometrically associated to the surface • ρ ( • ) := rk(NS( • )), the arithmetic/geometric Picard number of • . � ρ ( • ) the rank of lattice mentioned above. Question How are the geometric Picard numbers, ρ ( X ) and ρ ( X p ), related? How does the geometric Picard number behaves under reduction modulo p ? 3 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Some observations Question How are the geometric Picard numbers, ρ ( X ) and ρ ( X p ), related? How does the geometric Picard number behaves under reduction modulo p ? There is a natural specialization homomorphism s p : NS( X ) ֒ → NS( X p ) . Thus, ρ ( X ) ≤ ρ ( X p ). 1 ≤ ρ ( X ) ≤ 20 2 ≤ ρ ( X p ) ≤ 22, and ρ ( X p ) is always even. Computing ρ ( X ) is a hard problem. We can compute ρ ( X p ) by counting points on X p . 4 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Computing ρ ( X p ) � ∞ � � F q i � # X p 1 � T i Z X p ( T ) := exp = (1 − T ) P 2 ( X , T )(1 − q 2 T ) i i =1 where 1 − T Frob p | H 2 � � P 2 ( X , T ) := det ∈ Z [ t ] which have reciprocal roots of absolute value q . Theorem (Tate Conjecture) [Tate], [Charles], [Pera] and [Maulik] X an abelian surface or a K3 surface then: ρ ( X p ) = ord T =1 / q P 2 ( T ) ρ ( X p ) = � ζ ord T = ζ/ q P 2 ( T ), where ζ runs over all roots of unity. (Artin-Tate Conjecture) P 2 ( T ) � disc(NS( X p )) mod Q × 2 5 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Partial answer Question How are the geometric Picard numbers, ρ ( X ) and ρ ( X p ), related? Theorem [Charles] Charles constructed a function η ( X ) ≥ 0 a and proved that for all p of good reduction we have min q ρ ( X q ) = ρ ( X ) + η ( X ) ≤ ρ ( X p ) Equality occurs infinitely often. Furthermore, over some finite extension of k , the set of such primes has density one. a Depends on the Hodge structure underlying the transcendental lattice and its endomorphism algebra. 6 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Picard Jumps Let � � Π jump ( X ) := p : ρ ( X ) + η ( X ) < ρ ( X p ) � � = p : min q ρ ( X q ) < ρ ( X p ) Question What can we say about Π jump ( X )? What about γ ( X , B ) := # {� p � ≤ B : p ∈ Π jump ( X ) } as B → ∞ ? # {� p � ≤ B } 7 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Why? Information about Π jump ( X ) � Geometric statements Theorem [Bogomolov-Hassett-Tschinkel] and [Li-Liedtke] If #Π jump ( X ) = + ∞ or η ( X ) > 0 then X has infinitely many rational curves. 8 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Product of Elliptic Curves Let X ≃ Kummer( E 1 × E 2 ) and E i elliptic curve over Q . � 0 E 1 �∼ E 2 ; ρ ( X ) = 18 + rk(Hom( E 1 , E 2 )) = 18 + rk(End( E 1 )) E 1 ∼ E 2 � � X ρ X γ ( X , B ) square of CM 20 1 / 2 + o (1) CM theory log log B log B < • < C log B square of non-CM 19 [Elkies] B B 1 / 4 CM times CM 18 1 / 4 + o (1) CM theory CM times non-CM 18 ? non-CM times non-CM 18 ≫ 0 [Charles] Remark p ∈ Π jump ( X ) depends uniquely on the pair ( a E 1 ( p ) , a E 2 ( p )) 9 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

The simplest case Let X ≃ Kummer( E 1 × E 2 ) and E i elliptic curve over Q . � 0 E 1 �∼ E 2 ; ρ ( X ) = 18 + rk(Hom( E 1 , E 2 )) = 18 + rk(End( E 1 )) E 1 ∼ E 2 � � ρ P ( p ∈ Π jump ( X )) γ ( X , B ) X X square of CM 20 1 / 2 1 / 2 √ ∼ 1 / √ p square of non-CM 19 c / B CM times CM 18 1 / 4 1 / 4 √ ∼ 1 / √ p CM times non-CM 18 c / B √ ∼ 1 / √ p non-CM times non-CM 18 c / B Remark p ∈ Π jump ( X ) depends uniquely on the pair ( a E 1 ( p ) , a E 2 ( p )) 10 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Sato-Tate for Abelian Surfaces Let A be an abelian surface. Let G ⊂ Sp 4 be the “smallest” group such that for each p of good reduction there is an g ∈ G such that det(1 − Tg ) = det(1 − T / √ q Frob p | H 1 ( A )) . Then ST A ⊂ USp 4 is the maximal compact subgroup of G . Conjecture [Sato-Tate] The conjugacy classes of Frob p / √ q | H 1 are equidistributed with respect to the Haar measure on ST A . Theorem [Fit´ e–Kedlaya–Rotger–Sutherland] The Galois structure on End( A ) ⊗ R determines ST A . Only 52 groups up to conjugacy can be realized as ST A Example, if End( A ) = Z , then ST A = USp 4 . 11 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Sato-Tate for Abelian Surfaces Conjecture [Sato-Tate] The conjugacy classes of Frob p / √ q | H 1 are equidistributed with respect to the Haar measure on ST A . 1 − T / √ q Frob p | H 1 � := 1 + a 1 T + a 2 T 2 + a 1 T 3 + T 4 , then � If det ( a 1 , a 2 ) ∼ (Tr( M ) , Tr( M ∧ M )) with M ∈ ST A . Question � 1 − T / q Frob p | H 2 � What about det ? H 2 = H 1 ∧ H 1 , thus 1 − T / q Frob p | H 2 � = ( T − 1) 2 ψ p ( T ) � det 1 − 2 a 2 ) T 2 + (2 − a 2 ) T 3 + T 4 ψ p ( T ) := 1 + (2 − a 2 ) T + (2 + a 2 12 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Abelian Surfaces Let X ≃ Kummer( A ), A an abelian surface over Q . End( A ) � ρ ( A ) and ST A ρ ( X ) = 16 + ρ ( A ) � � A ρ A γ ( A , B ) as B → ∞ √ the generic case 1 c / B √ non-CM times non-CM (non Galois) 2 c / B non-CM times non-CM (Galois) or simple RM 2 1/2 √ CM times non-CM 2 c / B simple CM 2 3/4 CM times CM 2 1/4 √ simple QM or square of non-CM 3 c / B square of CM 4 1/2 Remark Easy to verify the conjectures numerically! 13 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

What about K3 surfaces? Let X be a K3 surface. In this case H 1 ( X ) is trivial and dim H 2 ( X ) = 22. In general, we need to compute det(1 − T / q Frob p | H 2 ( X )) to deduce ρ ( X p )! The definition of ST X follows closely the one for an abelian surface. We just need to replace det(1 − T / √ q Frob p | H 1 ( A )) by det(1 − T / q Frob p | H 2 ( X )) . ST X ⊂ O 22 − ρ ( X ) and ST 0 X ⊂ SO 22 − ρ ( X ) . Very little is known about what subgroups of O 21 can show up as ST X ! 14 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Kummer surface If X ≃ Kummer( A ), where A is an abelian surface, then ρ ( X ) = 16 + ρ ( A ) and ST X = ST A / {± 1 } ⊂ SO 5 ∼ = USp 4 / {± 1 } . 15 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Computing ρ ( X ) Let T X be the orthogonal complement of NS( X top C ) in H 2 ( X top C , Q ). Let E X be the endomorphism algebra of T X that respects the Hodge structure E X is a totally real field or a CM-field. Theorem [Charles] � ρ ( X ) if E X is a CM or dim E X ( T X ) is even, ρ ( X p ) ≥ ρ ( X ) + [ E X : Q ] if E X is a totally real or dim E X ( T X ) is odd. Further, assume that we are in the second case, then exist infinitely many pairs ( p , q ) such that the equality holds and disc(NS( X p )) �≡ disc(NS( X q )) mod Q × 2 16 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Numerical experiments for ρ ( X ) = 1 X a quartic K3 surface with ρ ( X ) = 1 and E X = Q . √ γ ( X , B ) ∼ c X / B , B → ∞ Prob(p ∈ Π jump (X)) ∼ 1 / √ p Heuristics for the 1 / √ p behaviour?!? k 1 2 3 4 5 0.002 0.988 -0.016 2.866 -0.218 E � Tr(Frob p / p ) k � -0.008 1.016 -0.065 3.096 -0.658 E O [ Tr ( M ) k ] 0 1 0 3 0 17 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Numerical experiments for ρ ( X ) = 2 Now ρ ( X ) = ρ ( X ) = 2 and E X = Q or CM No obvious trend . . . Is it related to the splitting of primes in a quadratic extension over Q ? Hint: Q ( √ D X ). k 1 2 3 4 5 0.022 1.009 0.0573 2.959 0.136 E � Tr(Frob p / p ) k � 0.001 0.990 -0.039 2.941 -0.454 0.009 1.013 0.055 3.069 0.015 E O [ Tr ( M ) k ] 0 1 0 3 0 ∼ 9000 CPU hours per example. 18 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces

Discriminant of a K3 surface Let X be a quartic K3 surface over Q and D X be its discriminant. Theorem [Elsenhans-Jahnel] The functional equation of the Frobenius action on H 2 ( X ) has the plus sign if and only if D X is square mod p . Theorem If ρ ( X ) = 2 r then ρ ( X p ) ≥ 2 r + 2 at all primes p such that D X is not square mod p . Corollary If D X is not a square, ρ ( X ) = ρ ( X ) = 2 r and η ( X ) = 0, then � �� �� p : p is inert in Q ⊂ Π jump ( X ) , D X lim inf B →∞ γ ( X , B ) ≥ 1 / 2 , and X has infinitely many rational curves. 19 / 24 Edgar Costa Variation of N´ eron-Severi ranks of K3 surfaces