aa r X i v : . [ m a t h . N T ] F e b ON THE AVERAGE OF p -SELMER RANK IN QUADRATIC TWISTFAMILIES OF ELLIPTIC CURVES OVER FUNCTION FIELD SUN WOO PARK AND NIUDUN WANG
Abstract.
We show that if the quadratic twist family of a given elliptic curve over F q [ t ]with Char( F q ) ≥ N é ron model has a multiplicative reduction awayfrom ∞ , then the average p -Selmer rank is p + 1 in large q -limit for almost all primes p . Introduction
Let F q be a finite field with Char( F q ) ≥
5, let C = P / F q and K = F q ( C ) = F q ( t ).Say E : y = x + A ( t ) x + B ( t ) is a non-isotrivial elliptic curve defined over F q [ t ]. Definethe canonical (naive) height of the elliptic curve as follows, where E ′ is any elliptic curveisomorphic to E of the form y = x + C ( t ) x + D ( t ). h ( E ) := inf E ′ ∼ = E (max { C, D } )Let E f be the quadratic twist of E by square-free polynomial f ( t ) ∈ F q [ t ]. E f : f ( t ) y = x + A ( t ) X + B ( t )Let M ( n, q ) be the set of square-free polynomials over F q such that h ( E f ) ≤ n .Poonen-Rains heuristic shows that the average p -Selmer rank of elliptic curves over a globalfield k is p + 1(see [Poo12].). It is natural to ask whether the same heuristic is reasonablefor the family of quadratic twists of a fixed elliptic curve. We denote by E n,p the average p -Selmer rank over those in the family of quadratic twists with canonical height at most n ,namely: E n,p = P f ∈ M ( n,q ) | Sel p E f || M ( n, q ) | In this paper, we show that under certain assumption on the quadratic twist family of thefixed elliptic curve E , the average size of p-Selmer groups is p + 1 in large q limit. Inparticular, we can assume for some large enough q such that the discriminant ∆ E of E splitsin F q [ t ], there exists a quadratic twist E of E with minimal height among the quadratictwist family. Theorem 1.1.
Let E be an elliptic curve defined over K = F q ( C ) = F q ( t ) such that thereexists at least one quadratic twist of E whose Néron model admits a multiplicative reductionaway from ∞ . Let E be the quadratic twist of E with minimal height among the family ofquadratic twists of E . Then for all primes p ≥ , and coprime to q and all local Tamagawafactors of E , we have the following equation. lim n →∞ lim q →∞ E n,p = lim n →∞ lim q →∞ P f ∈ M ( n,q ) | Sel p E f || M ( n, q ) | = p + 1 Date : February 4, 2021.
Remark 1.2.
The main theorem shows that for all but finitely many primes p , the average p -Selmer rank in the large q -limit over the family of quadratic twists of E is p + 1. Remark 1.3.
While writing the paper, we learned the contemporaneous results from AaronLandesman on a similar problem. Given a universal family of elliptic curves over F q [ t ] with q coprime to 6 n , the geometric average size of n -Selmer group of the universal family is equalto sum of divisors of n as q → ∞ . We refer to [Lan18] for more details.In subsequent sections, we will calculate lim n →∞ lim q →∞ E n,k,p as follows. Fix an ellipticcurve E over F q [ t ]. Let F d,E be the set of square-free polynomials f of degree d over ¯ F q suchthat f is coprime to ∆ E , the discriminant of E . Chris Hall’s construction of étale F l -lissesheaf over F d,E gives the average size of Sel p ( E f ) for a subfamily of quadratic twists of E . Wethen order the family of quadratic twists of E by the canonical height h ( E f ) which enablesus to calculate the average size of Sel p ( E f ) in large q -limit. Acknowledgements.
The authors would like to sincerely appreciate Jordan Ellenberg forintroducing the problem, patiently explaining various mathematical backgrounds, and givingconstructive comments and suggestions on possible ways to approach the problem. Theauthors would like to thank Chris Hall for explaining via email about the constructionof étale F p -lisse sheaf in his paper [Hal06]. The authors would also like to thank AaronLandesman and Soumya Sankar for helpful and insightful discussions.2. Monodromy Group
In this section, we briefly discuss the main machinery used to prove Theorem 1.1. Thissection follows closely to chapter 2 and 3 of [Ell14]. Throughout this paper, we denote by X ¯ F q the base change X × F q ¯ F q where X is a scheme over F q .We start with a brief exposition on the moduli space of a family of quadratic twists of E by polynomials g ∈ F q [ t ] of degree n such that ( g, ∆ E ) = 1. A polynomial g ( x ) = a x n + a x n − + · · · + a n of degree n corresponds to a point in the affine space A n +1 withcoordinates ( a , a , a , . . . , a n ). Note that the square-free polynomials are parameterized bythe set of points on A n +1 where Disc( g ) does not vanish, while ( g, ∆ E ) = 1 amounts to( a , a , a , . . . , a n ) not on the zero locus given by the resultant of g and ∆ E . Thus thosesquare free polynomials g ∈ F q [ t ] with ( g, ∆ E ) = 1 are parameterized by an open subschemeof A n +1 , denoted by F n . It is reasonable to expect that it suffices to compute the average p -Selmer rank on the elliptic curves parameterized by the open subscheme F n . We will explainin later sections how we can bound the average p -Selmer rank on those quadratic twistsparametrized by the complement of F n .Suppose there exists an étale cover X → F n such that the number of F q -points on thegeometric fiber of X at f ∈ F n ( F q ) equals to the size of Sel p E f . Then we have the followingequation. | X ( F q ) | = X f ∈ F n ( F q ) | Sel p E f | On the other hand, the Grothendieck-Lefschetz trace formula gives an explicit equationof the number of F q -points on X ¯ F q ([Mil13].) | X ( F q ) | = X i ( − i Tr Frob q | H i ´ et ; c ( X ¯ F q , Q l ) | N THE AVERAGE OF p -SELMER RANK IN QUADRATIC TWIST FAMILIES OF ELLIPTIC CURVES OVER FUNCTION FIELD3 F n is an open subscheme of A n +1 implies that | F n ( F q ) | = q n +1 − O( q n ). The leading termof F n ( F q ) is q n +1 . Hence, computing the average size of Sel p in large q -limit amounts tocomputing the following equation.lim q →∞ P f ∈ F n ( F q ) | Sel p E f || F n ( F q ) | = lim q →∞ q − ( n +1) | X ( F q ) | The Weil bounds imply that the eigenvalues of the Frobenius action Frob q on H i ´ et ; c ( X ¯ F q ; Q l )have absolute value bounded by q n +1 − i . Thus for a fixed n , if q becomes sufficiently large,any cohomology term other than H et ; c ( X ¯ F q ; Q l ) vanishes. Note that H et ; c ( X ¯ F q ; Q l ) is the Q l vector space spanned by the irreducible components of X ¯ F q . Hence, the following observationholds.lim q →∞ q − ( n +1) | X ( F q ) | = X rational over F q (2.1)Let f ∈ F n be a fixed basepoint. we have the following short exact sequence.1 −→ π (( F n ) ¯ F q , f ) −→ π ( F n , f ) −→ Gal( ¯ F q / F q ) −→ f ∈ F n , the geometric fiber of X at f is an F p -vector space X f . Observe that π ( F n , f ) acts linearly on X f . Hence we can define the monodromy group of the cover X → F n as the image of π ( F n , f ) in GL( X f ) and the geometric monodromy group as theimage of π (( F n ) ¯ F q , f ). Denote by Γ the monodromy group, and denote by Γ the geometricmonodromy group. This gives us another short exact sequence, where [ q ] corresponds tothe class in Γ / Γ corresponding to the image of the Frobenius Frob q ∈ Gal(¯ F q / F q ), and [ q ] Z corresponds to a subgroup of Γ / Γ generated by [ q ].1 −→ Γ −→ Γ −→ [ q ] Z −→ Lemma 2.1.
The geometric irreducible components of X are in bijection with the orbits ofthe geometric monodromy group on X f .Proof. Note that X ¯ F q is étale over ( F n ) ¯ F q because X is étale over F n . Hence, π (( F n ) ¯ F q ) actson X ¯ F q . The group action preserves each irreducible component of X ¯ F q since each componentis étale over ( F n ) ¯ F q , hence preserved by the functoriality of π . Therefore, under the actionof π (( F n ) ¯ F q ), each orbit of the geometric monodromy group on X f would lie inside oneirreducible component of X ¯ F q . On the other hand, π ( X ¯ F q ) acts transitively on the geometricfibers of f within an irreducible geometric component, which yields the bijection. (cid:3) Lemma 2.2.
The action of the Frobenius on the geometric components is given by the actionof [ q ] on the orbits of Γ . This comes directly from the bijection between the geometric irreducible components of X and the orbits of Γ . Therefore, in order to compute the number of geometrically irreduciblecomponents of X , it suffices to understand the geometric monodromy group Γ and computethe number of Γ -orbits on X f which are fixed by [ q ]. Therefore, equation (2.1) can berewritten as follows. SUN WOO PARK AND NIUDUN WANG lim q →∞ q − ( n +1) | X ( F q ) | = fixed by [ q ] (2.2)3. Construction of Moduli Space
Cohomology Groups of Néron Models.
In this subsection, we prove several claimswhich will help us with constructing the desired moduli space discussed in remark 2.3.Let q be a power of prime q = q k such that q is not divisible by 2 and 3. Fix an algebraicclosure F q → ¯ F q . Let C/ F q = P / F q , and let K = F q ( C ) = F q ( t ) be its function field.Fix an elliptic curve E over K , and let E f be the quadratic twist of the elliptic curve by f ∈ Conf n ( F q ). Let E → C be the Néron model for the elliptic curve E . For any prime p that is invertible in K , the multiplication by p map on E f ( K ) extends uniquely to an isogeny × p : E → E . Define E p to be the kernel of × p .We state a result from [Ces13], which states that the first cohomology group of E p andSel p are isomorphic under certain arithmetic conditions. First, we state the following lemmafrom Appendix B of [Ces13]. Lemma 3.1.
Let S be a connected Noetherian normal scheme of dimension ≤ . Let ¯ K be the function field of S , and for every point s ∈ S , let k ( s ) be the residue field of s . Let A → Spec ¯ K and B → Spec ¯ K be abelian varieties, and let A → S and B → S be their Néronmodels. Suppose φ : A → B is a ¯ K -isogeny of abelian varieties. Denote by ˜ φ : A → B themap induced on Néron models over S . If A has semiabelian reduction at all the nongeneric s ∈ S with Char( k ( s )) | deg φ , then ˜ φ : A → B is flat.Proof.
See [Ces13] lemma B.4. (cid:3)
In particular, the lemma above shows that the map × p : E → E is flat if q is coprimeto p . Using this implication, we can state the following application of Proposition 5.4 from[Ces13]. Theorem 3.2.
Let φ : A → B be a morphism of abelian varieties. Let A and B be theirNéron models. Denote by A [ φ ] the kernel of φ : A → B , and denote by A [ φ ] the kernel of φ : A → B . Let
Sel φ A be the φ -Selmer group of A . Suppose the morphism φ : A → B is flat.If deg φ is coprime to any local Tamagawa factors of A and B , and does not divide deg φ ,then the following equation holds inside H ( K, A [ φ ]) . H ( S, A [ φ ]) = Sel φ A Proof.
See [Ces13] proposition 5.4. (cid:3)
More specifically, the theorem above implies that when p is coprime to 2, q , and the localTamagawa factors of E , then there exists an isomorphism between H ( C, E p ) and Sel p E .In fact, under certain conditions on E , we can extend the results of Theorem 3.2 to thefamily of quadratic twists of E . Let E f be the quadratic twist of the elliptic curve by square-free polynomials f ∈ F q [ t ]. Denote by E f → C the Néron model for the elliptic curve E f .Let E f,p be the kernel of multiplication by p map over E f . Corollary 3.3.
Let
E/K : y = x + Ax + B be an elliptic curve, and let ∆ E be thediscriminant of E . Suppose that no prime factors π of ∆ E satisfy the condition that π | A N THE AVERAGE OF p -SELMER RANK IN QUADRATIC TWIST FAMILIES OF ELLIPTIC CURVES OVER FUNCTION FIELD5 and π | B . Let p be a prime such that p is coprime to , , q and all the local Tamagawafactors of E . Then the following isomorphism holds for any square-free polynomial f ∈ F q [ t ] . H ( C, E f,p ) = Sel p E f Proof.
For an explicit calculation of local Tamagawa factors using Tate’s algorithm, see[Sil91] Chapter 4 Section 9. Say f = ( π . . . π s ) g , where J = { π . . . π s } are prime factorsof ∆ E and ( g, ∆ E ) = 1, We first note that the conditions on E imply that the discriminantof the twist E f is equal to f ∆ E . Tate’s algorithm shows E f has additive reduction on allprimes π dividing g . Therefore, all local Tamagawa factors arising from such π ’s are at most4. On the other hand, the additive reductions π i ∈ J of E will stay as additive reductionsof E f , while the multiplicative reductions π j ∈ J will all become additive reductions of E f .For all the other primes ρ | ∆ E but not in J , v ρ (∆ E ) and v ρ (∆ E f ) are the same. Therefore,for any ρ | ∆ E , no matter weather ρ divides f or not, we will have the local Tamagawa factorof E f at ρ either equals to the local Tamagawa factor of E or equals to 1,2,3. Then we canapply theorem 3.2. to E f . (cid:3) Remark 3.4.
For such an elliptic curve E in the above Corollary, we can actually concludethat if the Néron model of E itself has no multiplicative reduction away from ∞ , there is noquadratic twists of E whose Néron model admits a multiplicative reduction away from ∞ .This follows exactly from the fact that additive reductions π i ∈ J of E will stay as additivereductions of E f .The theorem hence implies that for all but finitely many primes p , the following isomor-phism holds. H ( C, E f,p ) = Sel p E f Since E f,p is a smooth commutative group scheme, we know that H ( C, E f,p ) is isomorphicto H et ( C, E f,p ). (See [Poo10] Remark 6.6.3)We now examine whether there is a way to explicitly compute the size of H et ( C, E f,p ).Chris Hall gives an explicit computation of the étale cohomology groups of E f,p over C ¯ F q under certain conditions on the size of the Galois group Gal( K ( E [ p ]) /K ). Definition 3.5.
Let
E/K be an elliptic curve. The geometric Galois group H p is thesubgroup of the Galois group Gal( K ( E [ p ]) /K ) whose fixed field is ( K ( E [ p ]) ∩ ¯ F q ) /K givenby adjoining a primitive p th root of unity. We say that E has big monodromy at p if thegeometric Galois group contains SL ( F p ).In fact, for any prime p ≥
5, any twist E f has big monodromy at p if and only if E hasbig monodromy at p . This is because SL ( F p ) does not have index 2 subgroups, so K ( E f [ p ])and k ( √ f ) are geometrically disjoint extensions of K . Theorem 3.6.
Let C/ F q be a proper smooth geometrically connected curve, and let K be itsfunction field. Let E/K be a non-isotrivial elliptic curve. Then there exists a constant c ( K ) such that E has big monodromy at p for any p ≥ c ( K ) and p coprime to Char( K ) . Theconstant c ( K ) is defined as follows. c ( K ) := 2 + max { l | l is prime ,
112 ( l − (6 + 3 e + 4 e )) ≤ genus( C ) } SUN WOO PARK AND NIUDUN WANG
Here, e and e are constants defined as follows. e = l ≡ − e = l ≡ − Proof.
See [CH05] theorem 1.1. (cid:3)
In particular, let C = P and K be the function field of C . Fix a non-isotrivial ellipticcurve E/K . Let { E f } be the family of quadratic twists of E , where f is any square-freepolynomial over F q . The theorem above implies that E/K has big monodromy at p for anyprime p ≥
15 and coprime to q . Hence, for any prime p ≥
15 and coprime to q , the twist E f /K also has big monodromy at p . Under the aforementioned conditions on p , we can givean explicit calculation of the étale cohomology group H ( C F q , E f,p ) for any E f . Lemma 3.7.
Let C/ F q be a proper smooth geometrically connected curve, and let K be itsfunction field. Fix an elliptic curve E/K . Let p be a prime such that E has big monodromyat p . Then for any square-free polynomial f over F q , the étale cohomology groups of E f,p over C ¯ F q are F p -vector spaces with the following dimensions. dim F p (H i ´ et ( C ¯ F q , E f,p )) = deg( M f ) + 2 deg( A f ) − C ) −
1) if i = 10 otherwise Here, M f and A f are the divisors of multiplicative and additive reduction of E f,p → C .Proof. See [Hal06] lemma 5.2. (cid:3)
In particular, if C = P , then the dimension of H ( C F q , E f,p ) as an F p -vector space isdeg( M f ) + 2 deg( A f ) + 4 for any twist E f . Remark 3.8.
The Weil pairing on E f [ p ] induces a non-degenerate skew-symmetric pairingon E f,p . Hence the Weil pairing induces a non-degenerate symmetric pairing on H et ( C ¯ F q , E f,p )as follows. H et ( C ¯ F q , E f,p ) × H et ( C ¯ F q , E f,p ) → H et ( C ¯ F q , E f,p ⊗ E f,p ) → H et ( C ¯ F q , F p (1))The first map comes from the cup product of cohomology classes and Poincaré duality, whilethe second map comes from the induced Weil pairing E f,p × E f,p → F p (1). Note that thefollowing isomorphism holds. (See [Mil13] Chapter 14.) H et ( C ¯ F q , F p (1)) ∼ = F p Therefore, the Weil pairing on E f [ p ] induces a non-degenerate symmetric bilinear pairing offirst étale cohomology groups.H et ( C ¯ F q , E f,p ) × H et ( C ¯ F q , E f,p ) → F p Using the above lemma and the Leray spectral sequence, we can derive a relation betweenétale cohomology groups of E f,p over C and those over C ¯ F q . N THE AVERAGE OF p -SELMER RANK IN QUADRATIC TWIST FAMILIES OF ELLIPTIC CURVES OVER FUNCTION FIELD7 Theorem 3.9 (Leray Spectral Sequence) . Let φ : Y → X be a morphism of schemes, andlet F be a sheaf on Y et . Then there exists the following spectral sequence. E r,s = H r ´ et ( X, R s φ ∗ F ) ⇒ H r + s ´ et ( Y, F ) Proof.
See [Mil13] theorem 12.7. (cid:3)
Lemma 3.10.
Fix an elliptic curve
E/K : y = x + Ax + B such that no prime factors π of ∆ E satisfy the condition that π | A and π | B . Suppose p is a prime satisfying theseconditions.(1) p ≥ (2) ( p, Char( K )) = 1 (3) p does not divide any local Tamagawa factors of E .Then for any quadratic twist E f , the following isomorphism exists. H et ( C ¯ F q , E f,p ) Gal(¯ F q / F q ) ∼ = Sel p E f Proof.
Consider the morphism of schemes φ : C → Spec( F q ). By the Leray spectral sequence,the following spectral sequence exists. E r,s = H r ( F q , H s ´ et ( C ¯ F q , E f,p )) ⇒ H r + s ´ et ( C, E f,p )By lemma 3.7, the cohomology group H r ( F q , H s ´ et ( C ¯ F q , E f,p )) is trivial whenever s = 1. Hence,the entries of the E page of the spectral sequence are given as follows. r r ( F q , H et ( C ¯ F q , E f,p )) 0 · · · ( F q , H et ( C ¯ F q , E f,p )) 0 · · ·
01 0 H ( F q , H et ( C ¯ F q , E f,p )) 0 · · ·
00 0 H ( F q , H et ( C ¯ F q , E f,p )) 0 · · ·
00 1 2 · · · s The Leray spectral sequence implies the following isomorphism.H ( F q , H et ( C ¯ F q , E f,p )) ∼ = H et ( C, E f,p )Recall the following isomorphism.H et ( C, E f,p ) ∼ = H ( C, E f,p )Note that the 0-th cohomology group is precisely the fixed subgroup of H et ( C ¯ F q , E f,p ) byGal(¯ F q / F q ). H et ( C ¯ F q , E f,p ) Gal(¯ F q / F q ) ∼ = H ( F q , H et ( C ¯ F q , E f,p )) SUN WOO PARK AND NIUDUN WANG
Corollary 3.3 implies the following isomorphism holds for all but finitely many p .H ( C, E f,p ) ∼ = Sel p E f Hence, for all but finitely many p , the following isomorphism holds.H et ( C ¯ F q , E f,p ) Gal(¯ F q / F q ) ∼ = Sel p E f (cid:3) Construction of Étale F p -lisse Sheaf. In this subsection, we follow through ChrisHall’s construction of étale F p -lisse sheaf over a subset of square-free polynomials f of fixeddegree over F q . The construction will help us calculate the average size of Sel p ( E f ) for asubfamily of quadratic twists of E .As before, let C = P over F q , and let K be the function field of C . Fix a non-isotrivialelliptic curve E/K : y = x + Ax + B . We recall the construction of the space F n over ¯ F q ,as mentioned in section 2. F n = { f ∈ ¯ F q [ t ] | f is square-free , deg f = n, ( f, ∆ E ) = 1 } As constructed in [Hal06] (See Chapter 5.3), consider the étale F p -lisse sheaf τ n,p,E → F n whose geometric fiber over f ∈ F n ( F q ) is H (Conf n ¯ F q , E f,p ). Note that Chris Hall’s constructionof τ n,p,E is an F p -analogue of Katz’s construction of étale ¯ Q p -lisse sheaves using middleconvolutions. We refer to [Kat98] proposition 5.2.1. for a detailed explanation on theconstruction of the étale ¯ Q p -lisse sheaves.We state the following theorem by Chris Hall, which gives an explicit computation of thegeometric monodromy group of τ n,p,E → F n . Theorem 3.11.
Let E be an elliptic curve over K such that there exists at least one quadratictwist of E whose Néron model admits a multiplicative reduction away from ∞ . Let p be aprime such that E has big monodromy at p , i.e. p ≥ . Let O (H et (Conf n ¯ F q , E f,p )) be theorthogonal group of H et (Conf n ¯ F q , E f,p ) which preserves the non-degenerate symmetric bilinearpairing µ . (See Remark 3.7 for the construction of µ .)Then the geometric monodromy group of τ n,p,E → F n is isomorphic to a subgroup of the or-thogonal group O (H et (Conf n ¯ F q , E f,p )) of index at most and is not isomorphic to SO (H et (Conf n ¯ F q , E f,p )) .Proof. See [Hal06] Theorem 5.3 (cid:3)
Note that the commutator subgroup of O (H et (Conf n ¯ F q , E f,p )) is of index 4. Hence, in orderto understand the number of orbits of the desired geometric monodromy group, it suffices tounderstand the number of orbits of both O (H et (Conf n ¯ F q , E f,p )) and its commutator subgroup.We finish this section with the following lemma, which describes the number of the orbits ofthe aforementioned two groups. Lemma 3.12.
Let V be an F p -vector space of dimension d where char( F p ) = 2 . Given anon-degenerate symmetric bilinear pairing µ : V × V → F p , let O ( V ) be the orthogonal groupof V . Then the number of orbits of O ( V ) is p +1 , and the number of orbits of the commutatorsubgroup [ O ( V ) , O ( V )] is p + 1 if d ≥ .Proof. We first show that the number of orbits of O ( V ) on V is p + 1 for any d . It sufficesto show that the orthogonal group acts transitively on the set of nonzero vectors of a givennorm. Suppose v, w ∈ V are two non-zero vectors of the same norm. Then there exists an N THE AVERAGE OF p -SELMER RANK IN QUADRATIC TWIST FAMILIES OF ELLIPTIC CURVES OVER FUNCTION FIELD9 isometry φ : Span ( v ) → Span ( w ) given by v w . By Witt’s Theorem, φ extends to anisometry ˜ φ : V → V , which proves the claim. Note that the orthogonal group has 1 orbiton the set of vectors of non-zero norm, and 2 orbits on the set of vectors norm zero. In thelatter case, the two orbits are { } and the set of non-zero vectors of norm zero.We now show that the number of orbits of the commutator subgroup [ O ( V ) , O ( V )] on V is p + 1 for d ≥
5. Again, it suffices to show that the commutator subgroup [ O ( V ) , O ( V )]acts transitively on the set of nonzero vectors of a given norm. Suppose v, w ∈ V are twonon-zero vectors of the same norm. Then by the aforementioned argument, there exists anisomtery ˜ φ ∈ O ( V ) such that ˜ φ ( v ) = w . We want to show that there exists ˜ ψ, ˜ ϕ ∈ O ( V )such that the following equation holds.˜ φ ( v ) = ˜ ψ ˜ ϕ ˜ ψ − ˜ ϕ − ( v )It suffices to consider the case when µ ( v, w ) = 0. If not, then v = w and we can take ˜ φ tobe the identity element in O ( V ). Suppose v, w are orthogonal. Let { v, w, u , u , · · · , u d − } be the orthogonal basis of V . Without loss of generality, we can assume that the norms ofthe basis vectors are the same.Suppose d ≥
5. Let W be the span of { v, u , u , u , w } . Then consider the followingisometries ψ, ϕ : W → W . Here, the matrices are given with respect to the orthogonal basis { v, u , u , u , w } . ψ = , ϕ = Note that ψ − = ψ and ϕ − = ϕ as isometries over W . The matrix form of ψϕψ − ϕ − isgiven as follows, which implies that ψϕψ − ϕ − maps v to w . ψϕψ − ϕ − = By Witt’s theorem, there exist isometries ˜ ψ, ˜ ϕ ∈ O ( V ) such that ˜ ψ ˜ ϕ ˜ ψ − ˜ ϕ − maps v to w . Asimilar argument as for the case of O ( V ) shows that the number of orbits of [ O ( V ) , O ( V )]on V is p + 1. (cid:3) Using the above lemma, we can calculate the average size of Sel p E f for f ∈ F n ( F q ). Theorem 3.13.
Fix a non-isotrivial elliptic curve E : y = x + Ax + B over F q [ t ] suchthat there exists at least one quadratic twist of E whose Néron model admits a multiplicativereduction away from ∞ . Let E be the quadratic twist of E with minimal height among thefamily of quadratic twists of E . Let n be an integer such that n ≥ . Let p be a prime suchthat p ≥ , and coprime to q and all local Tamagawa factors of E . Then the average sizeof Sel p E f for a subfamily of quadratic twists { E f } f ∈ F n ( F q ) is p + 1 when q → ∞ , i.e. lim q →∞ P f ∈ F n ( F q ) | Sel p ( E f ) || F n ( F q ) | = p + 1 Proof.
Denote by { E f } the family of quadratic twists of E . Then note the E must have atleast one place of multiplicative reduction by the proof of Corollary 3.3. Indeed, E satisfiesthe condition for Corollary 3.3. Note that the quadratic twist families { E f } and { ( E ) g } areequal. Hence, we can apply lemma 3.10 to the subfamily of quadratic twists { E f } f ∈ F n ( F q ) .Since F n is an open subscheme of A n +1 , it holds that | F n ( F q ) | = q n +1 + O n ( q ). Then theGrothendieck-Lefschetz trace formula (i.e. Section 2) and lemma 3.10 shows the followingequation. lim q →∞ P f ∈ F n ( F q ) | Sel p ( E f ) || F n ( F q ) | = lim q →∞ P f ∈ F n ( F q ) | H et ( C ¯ F q , E f,p ) Gal(¯ F q / F q ) || F n ( F q ) | = lim q →∞ | τ n,p,E ( F q ) || F n ( F q ) | = lim q →∞ fixed by [ q ]We recall that Γ is the image of π ( F n ) in O (H et ( C ¯ F q , E f,p )), and Γ is the image of π (( F n ) ¯ F q )in O (H et ( C ¯ F q , E f,p )). The class [ q ] is the image of the Frobenius Frob q ∈ Gal(¯ F q / F q ) in Γ / Γ .By theorem 3.11, the geometric monodromy group π (( F n ) ¯ F q ) is isomorphic to a subgroupof O (H et ( C ¯ F q , E f,p )) of index at most 2 and is not SO (H et ( C ¯ F q , E f,p )). Recall from remark3.8 that the Weil pairing on E f [ p ] induces a non-degenerate symmetric bilinear pairing µ onH et ( C ¯ F q , E f,p ). H et ( C ¯ F q , E f,p ) × H et ( C ¯ F q , E f,p ) → F p Hence, the frobenius map Frob q preserves the pairing on H et ( C ¯ F q , E f,p ). We apply lemma3.12 by setting V = H et ( C ¯ F q , E f,p ) and µ to be the non-degenerate symmetric bilinear pairinginduced from the Weil pairing over E f [ p ].Note that the elliptic curve E f has additive reduction at all primes π dividing f . Hence,lemma 3.7 implies that for n ≥
5, the dimension of the vector space V is greater than 5.Hence for n ≥
5, the orbits of π (( F n ) ¯ F q ) are the sets of non-zero vectors of a fixed normand the set { } . Therefore, Frob q preserves the orbits of π (( F n ) ¯ F q ). Hence, the number oforbits of π (( F n ) ¯ F q ) fixed by Frob q is p + 1 for all q .Hence, we have the following equation, which proves the theorem.lim q →∞ P f ∈ F n ( F q ) | Sel p ( E f ) || F n ( F q ) | = p + 1 (cid:3) Main Theorem
In this section, we prove the main theorem by using theorem 3.13. Fix a non-isotrivialelliptic curve E : y = x + Ax + B over F q [ t ] such that there exists at least one quadratictwist of E whose Néron model admits a multiplicative reduction away from ∞ . We alsoassume that char( F q ) ≥
5. Denote by ∆ E the discriminant of the elliptic curve E . We thenorder the family of quadratic twists { E f } with square-free polynomials f over F q based onthe canonical height of E f .The idea of the proof is as follows. Let E be the quadratic twist of E with minimal heightamong the family of quadratic twists of E . Then by corollary 3.3, the Néron model of E admits a multiplicative reduction away from ∞ . Note the quadratic twist families { E f } and N THE AVERAGE OF p -SELMER RANK IN QUADRATIC TWIST FAMILIES OF ELLIPTIC CURVES OVER FUNCTION FIELD11 { ( E ) f } are the same. Hence, we can reorder the family { E f } by { ( E ) f } . Such reorderingof family of quadratic twists will allow us to ensure that the subfamily of quadratic twistswhose p -Selmer rank is undetermined does not affect the average size of p -Selmer group of { E f } as q → ∞ . Remark 4.1.
One may ask whether it is possible to compute the average size of p -Selmergroups by ordering the family of quadratic twists { E f } by the degree of the twisting poly-nomial f . The problem with this approach is that Chris Hall’s construction of étale F p -lissesheaf only works for elliptic curves E with at least one multiplicative reduction. Suppose E has at least one place of multiplicative reduction. Then the quadratic twist family { E f } canbe decomposed into the following two disjoint sets. { E f } = { E f } { f | ( f, ∆ E )=1 } ⊔ { E f } { f | ( f, ∆ E ) =1 } The subfamily { E f } { f | ( f, ∆ E )=1 } , which dominates the family { E f } when deg f → ∞ , consistsof quadratic twists of E having at least one place of multiplicative reduction, the subfamilyon which the average p -Selmer rank is known to be p + 1.However, the elliptic curve E ′ := E ∆ E has no multiplicative reduction. Note that ∆ E ′ =∆ E . Contrary to { E f } , the family of quadratic twists of { E ′ f } is given as follows. { E ′ f } = { E ′ f } { f | ( f, ∆ E ′ )=( f, ∆ E )=1 } ⊔ { E f } { f | ( f, ∆ E ′ )=( f, ∆ E ) =1 } Here, the subfamily { E ′ f } { f | ( f, ∆ E )=1 } , which dominates the family { E ′ f } when deg f → ∞ ,consists of quadratic twists of E ′ having no multiplicative reductions, the subfamily on whichthe average size of p -Selmer group is unknown.But notice that { E f } and { E ′ f } are the same family of quadratic twists. Hence, we cannotdetermine the average size of p -Selmer group by ordering the family of quadratic twists basedon the degree of the twisting polynomials.Before we present the proof of the main theorem, we state the following definitions andnotations. Definition 4.2.
Denote by F ( n ) the following scheme defined over ¯ F q . F ( n ) := { f ∈ ¯ F q [ t ] | f ∈ F d for d ≤ n } = n G d =1 F d As mentioned before F ( n ) is an open subscheme of A n +1 . Hence, the following equationholds | F ( n )( F q ) | = q n +1 − O( q n ) Definition 4.3.
Denote by ˜ τ ( n, p, E ) the sheaf over F ( n ) obtained by gluing étale F p -lissesheaves { τ d,p,E → F d } for all d ≤ n .The following theorem states an analogue of Theorem 3.12 for the subfamily of quadratictwists { E f } f ∈ F ( n )( F q ) such that E satisfies the aforementioned two conditions. Theorem 4.4.
Fix a non-isotrivial elliptic curve E : y = x + Ax + B over F q [ t ] suchthat there exists at least one quadratic twist of E whose Néron model admits a multiplicativereduction away from ∞ . Let E be the quadratic twist of E with minimal height among thefamily of quadratic twists of E . Let n be an integer such that n ≥ . Let p be a prime such that p ≥ , and coprime to q and all local Tamagawa factors of E . Then the average sizeof Sel p E f for the subfamily of quadratic twists { E f } f ∈ F ( n )( F q ) is p + 1 as q → ∞ , i.e. lim q →∞ P f ∈ F ( n )( F q ) | Sel p ( E f ) || F ( n )( F q ) | = p + 1 Proof.
As stated before, | F ( n )( F q ) | = q n +1 + O ( q n ). As in the proof of theorem 3.13, theGrothendieck-Lefschetz trace formula and lemma 3.10 shows the following equation, whereFrob q ∈ Gal(¯ F q / F q ).lim q →∞ P f ∈ F ( n )( F q ) | Sel p E f || F ( n )( F q ) | = lim q →∞ P nd =1 P f ∈ F d ( F q ) | Sel p E f | P nd =1 | F d ( F q ) | = lim q →∞ P nd =1 P f ∈ F d ( F q ) | Sel p E f | P nd =1 | F d ( F q ) | = lim q →∞ P nd =1 P f ∈ F d ( F q ) | H et ( C ¯ F q , E f,p ) Gal(¯ F q / F q ) | P nd =1 | F d ( F q ) | = lim q →∞ P nd =1 | τ d,p,E ( F q ) | P nd =1 | F d ( F q ) | = lim q →∞ | ˜ τ ( n, p, E )( F q ) || F ( n )( F q ) | ! = lim q →∞ n X d =1 | τ d,p,E ( F q ) || F d ( F q ) | | F d ( F q ) || F ( n )( F q ) | ! By theorem 3.13, the following equation holds.lim q →∞ P f ∈ F ( n )( F q ) | Sel p E f || F ( n )( F q ) | = lim q →∞ n X d =1 ( p + 1) | F d ( F q ) || F ( n )( F q ) | ! = lim q →∞ ( p + 1) | F n ( F q ) || F ( n )( F q ) | = p + 1 (cid:3) We now order the family of quadratic twist of elliptic curves based on the canonical heightof the elliptic curve. Recall that the canonical height of the elliptic curve is given as follows,where E ′ : y = x + C ( t ) x + D ( t ) is an elliptic curve isomorphic to E . h ( E ) := inf E ′ ∼ = E (max { C, D } ) Remark 4.5.
There is a unique equation for E of the form y = x + Ax + B satisfy-ing that for any prime p ∈ F q [ t ], p | A implies p ∤ B . In such case, h ( E ) is equal tomax { A, B } .In particular E has the least height among all quadratic twists if and only if for anyprime p ∈ F q [ t ], p | A implies p ∤ B . Otherwise, there exists a quadratic twist E p : py = x + Ax + B ≃ y = x + Ap x + Bp , which has smaller height than E . Remark 4.6.
In order to apply the construction of the étale F p -lisse sheaf τ n,p,E → F n from[Hal06], we need the assumption that the quadratic twist family we start with has memberswhose Néron model admits at least one multiplicative reduction away from ∞ .This is essentially necessary because one can find families of quadratic twists of somegiven elliptic curves, such that the whole family has no elliptic curve whose Néron model N THE AVERAGE OF p -SELMER RANK IN QUADRATIC TWIST FAMILIES OF ELLIPTIC CURVES OVER FUNCTION FIELD13 has multiplicative reductions. For instance, suppose 4 and 27 are invertible over the field F q . Then there are π , π ∈ F q [ t ] such that 4 π t + 27 π ( t + 1) = 1 with deg ( π ) = 1 and deg ( π ) = 2. One can readily check that the elliptic curve E : y = x + π π t + π π ( t + 1)has discriminant ∆ E = π π (4 π t + 27 π ( t + 1) ) = π π . Therefore, the Néron model of E has no multiplicative reduction by Tate’s algorithm. This elliptic curve E has the leastheight in the quadratic twist family by remark 4.5. So, the whole quadratic twist family hasno member whose Néron model admits multiplicative reductions away from ∞ by remark3.4.Now we prove the main theorem. Proof of Theorem 1.1.
Let E : y = x + A x + B be any non-isotrivial elliptic curve over F q [ t ]. We can always replace E by E ∈ { E f } that has minimal canonical height amongall quadratic twists. Since we assumed that there exists at least one quadratic twist of E whose Néron model admits a multiplicative reduction away from ∞ , E admits at least onemultiplicative reduction. Setup
Choose large enough q such that the discriminant of E/ F q [ t ], denoted by ∆ E , splits com-pletely into linear factors as follows.∆ E = π r π r · · · π r m m Without loss of generality, assume E has multiplicative reduction at π . Note we can guaran-tee that the primes π i ’s are all linear. For those large enough q , we will explicitly determinethe collection of all possible quadratic twists of E/ F q [ t ] whose height is bounded by n .We order the family of quadratic twists of E by canonical height. For any elliptic curve E/ F q [ t ], there exists a unique way to write E as y = x + Ax + B such that for any irreduciblepolynomial p ∈ F q [ t ], if p | A , then p ∤ B . Then the canonical height of E is given as follows. h ( E ) = max { A, B } In particular, we chose E to have the least height in the family of quadratic twists of E .Hence, the coefficients A, B of E satisfy the aforementioned condition.Any quadratic twist of E/ F q [ t ] can be uniquely written as E f : f y = x + Ax + B suchthat f ∈ F q [ t ] is square-free. Assume f = π a π i π i · · · π i s g where π i j ’s are distinct primesbelong to the set { π , · · · , π m } , a = 0 or 1, and g is a square-free polynomial such that( g, ∆ E ) = 1 in ¯ F q [ t ]. Denote by J the subset { π i , π i , · · · , π i s } of { π , π , · · · , π m } .Fix a positive integer n . We now consider the quadratic twists { E f } whose height h ( E f ) ≤ n . Case 1
Suppose that a = 0, i.e. π is not a prime factor of f . Remark 4.5 implies that the twist E f is isomorphic to the following elliptic curve, which is a minimal model. y = x + Y π ij ∈ J π i j g Ax + Y π ij ∈ J π i j g B (*)We will use use ( ∗ ) to explicitly compute the height of E f . We also note that for anyprime p | g , v p ( A ) = 0 or v p ( B ) = 0. Otherwise, g is not coprime to ∆ E . Therefore, we have the following equivalent relation. h ( E f ) ≤ n ⇐⇒ max deg Y π ij ∈ J π i j g A , deg Y i j ∈ J π i j g B ≤ n We set M := max { A, B } and M J := M + 6 deg (cid:16)Q i j ∈ I π i j (cid:17) = M + 6 | J | . Hencethe following equivalent relation holds. h ( E f ) ≤ n ⇐⇒ deg g ≤ n − M J E f still has at least one place of multiplicative reductionat π , which can be checked using Tate’s algorithm.Denote by E J the following elliptic curve, which is minimal by remark 4.5. E J : y = x + Y π ij ∈ J π i j Ax + Y π ij ∈ J π i j B Then the following equation holds. X f = π i π i ··· π is g ( g, ∆ E )=1 h ( E f ) ≤ n | Sel p E f | = X g ∈ F (cid:16) n − MJ (cid:17) ( F q ) | Sel p ( E J ) g | = (cid:12)(cid:12)(cid:12)(cid:12) ˜ τ (cid:18) n − M J , p, E J (cid:19) ( F q ) (cid:12)(cid:12)(cid:12)(cid:12) = ( p + 1) (cid:18) q n − MJ +1 + O n,p ( q n − MJ ) (cid:19) Case 2
Now assume a = 1, i.e. f = π π i π i · · · π i s g such that ( g, ∆ E ) = 1 and J = { π i , π i , · · · , π i s } is a subset of { π , π , · · · , π m } . Define M and M J analogously to Case 1. Then, by the afor-mentioned argument in Case 1, we have that E f is isomorphic to the following minimalmodel. y = x + Y π ij ∈ J π i j π g Ax + Y π ij ∈ J π i j π g B (*)Thus the height of E f can be written as follows. h ( E f ) ≤ n ⇐⇒ deg g ≤ n − M J − π n − M J − h ( E f ) to estimate the summation of the size of the p -Selmer groupfor quadratic twists E f by using lemma 3.7. Corollary 3.3 and lemma 3.7 implies that themaximal size of p -Selmer group of E f is the following. | Sel p E f | ≤ p Ef +4 ≤ p h ( E f )+4 = p n +4 N THE AVERAGE OF p -SELMER RANK IN QUADRATIC TWIST FAMILIES OF ELLIPTIC CURVES OVER FUNCTION FIELD15 Therefore, the following equation gives the approximation on the size of the p -Selmer groupover { E f } for the desired collection of f . X f = π π i π i ··· π is g ( g, ∆ E )=1 h ( E f ) ≤ n | Sel p E f | = X g ∈ F (cid:16) n − MJ − (cid:17) ( F q ) Sel p ( E J ) g = M (cid:18) q n − MJ + O n,p ( q n − MJ − ) (cid:19) Here, M is a positive integer such that 1 ≤ M ≤ p n +4 . Average Size
Using both aforementioned cases, we can now calculate the average size of p -Selmer groupas k → ∞ . Recall that E n,k,p is the average value of p -Selmer groups over families of quadratictwists of canonical height at most n . Then the following equation holds for any fixed n ≥ n ≥
30 because of the conditions on the degree of twistingpolynomials from lemma 3.12 and theorem 3.13.lim q →∞ E n,p = lim q →∞ P J ⊂{ π , ··· ,π m } ( p + 1) (cid:18) q n − MJ +1 + O n,p ( q n − MJ ) (cid:19) + M (cid:18) q n − MJ + O n,p ( q n − MJ − ) (cid:19)P J ⊂{ π , ··· ,π m } | F ( n − M J )( F q ) | + | F ( n − M J − F q ) | = lim q →∞ X J ⊂{ π , ··· ,π m } ( p + 1) (cid:18) q n − MJ +1 + O n,p ( q n − MJ ) (cid:19) + M (cid:18) q n − MJ + O n,p ( q n − MJ − ) (cid:19) | F ( n − M J )( F q ) | + | F ( n − M J − F q ) |× lim q →∞ | F ( n − M J )( F q ) | + | F ( n − M J − F q ) | P J ⊂{ π , ··· ,π m } | F ( n − M J )( F q ) | + | F ( n − M J − F q ) | = lim q →∞ X J ⊂{ π , ··· ,π m } ( p + 1) (cid:18) q n − MJ +1 + O n,p ( q n − MJ ) (cid:19) + M (cid:18) q n − MJ + O n,p ( q n − MJ − ) (cid:19) q n − MJ +1 + O n,p ( q n − MJ ) + q n − MJ + O n,p ( q n − MJ − ) × lim q →∞ | F ( n − M J )( F q ) | + | F ( n − M J − F q ) | P J ⊂{ π , ··· ,π m } | F ( n − M J )( F q ) | + | F ( n − M J − F q ) | = lim q →∞ X J ⊂{ π , ··· ,π m } ( p + 1) | F ( n − M J )( F q ) | + | F ( n − M J − F q ) | P J ⊂{ π , ··· ,π m } | F ( n − M J )( F q ) | + | F ( n − M J − F q ) | = ( p + 1) lim q →∞ X J ⊂{ π , ··· ,π m } | F ( n − M J )( F q ) | + | F ( n − M J − F q ) | P J ⊂{ π , ··· ,π m } | F ( n − M J )( F q ) | + | F ( n − M J − F q ) | = p + 1Therefore, the average size of p -Selmer groups of family of quadratic twists of any ellipticcurve E over F q [ t ] is given by p + 1:lim n →∞ lim q →∞ E n,p = p + 1 (cid:3) References [Ces13] Kestutis Cesnavicius. Selmer groups as flat cohomology groups, 2013.[CH05] Alina Carmen Cojocaru and Chris Hall. Uniform results for serre’s theorem for elliptic curves, 2005.[Ell14] Jordan S Ellenberg. Arizona winter school 2014 course notes: Geometric analytic number theory,2014.[Hal06] Chris Hall. Big symplectic or orthogonal monodromy modulo l , 2006.[Kat98] Nicholas M Katz. Twisted L-Functions and Monodromy
JAMS , 25(1):245–269, 2012.[Sil91] Joseph H Silverman.
Advenced Topics in the Arithmetic of Elliptic Curves . Springer-Verlag, 1991.
Department of Mathematics, University of Wisconsin – Madison, 480 Lincoln Dr., Madi-son, WI 53706, USA
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