TRANSACTIONSOFTHE AMERICANMATHEMATICALSOCIETY Volume368,Number12,December2016,Pages8553–8595 http://dx.doi.org/10.1090/tran6641 ArticleelectronicallypublishedonJanuary27,2016 ARITHMETIC OF ABELIAN VARIETIES IN ARTIN-SCHREIER EXTENSIONS RACHELPRIESANDDOUGLASULMER Abstract. Westudyabelianvarietiesdefinedoverfunctionfieldsofcurvesin positive characteristic p, focusing on their arithmetic in the system of Artin- Schreier extensions. First, we prove that the L-function of such an abelian variety vanishes to high order at the center point of its functional equation under a parity condition on the conductor. Second, we develop an Artin- Schreier variant of a construction of Berger. This yields a new class of Jaco- biansoverfunctionfieldsforwhichtheBirchandSwinnerton-Dyerconjecture holds. Third, we give a formula for the rank of the Mordell-Weil groups of theseJacobiansintermsofthegeometryoftheirfibers ofbadreductionand homomorphisms between Jacobians of auxiliary Artin-Schreier curves. We illustratethesetheoremsbycomputingtherankforexplicitexamplesofJaco- bians of arbitrary dimension g, exhibiting Jacobians with bounded rank and others with unbounded rank in the tower of Artin-Schreier extensions. Fi- nally,wecomputetheMordell-Weillatticesofanisotrivialellipticcurveanda family ofnon-isotrivial elliptic curves. The latter exhibits anexoticphenom- enon whereby the angles between lattice vectors are related to point counts onelliptic curvesover finite fields. Our methodsalsoyieldnew resultsabout supersingularfactorsofJacobiansofArtin-Schreiercurves. 1. Introduction Letkbeafinitefieldofcharacteristicp>0andsupposeF =k(C)isthefunction field of a smooth, projective curve C over k. Given an abelian variety J defined over F, the Birch and Swinnerton-Dyer (BSD) conjecture relates the L-function of J andtheMordell-Weil group J(F). Inparticular, itstatesthatthealgebraicrank of the Mordell-Weil group equals the analytic rank, the order of vanishing of the L-function at s = 1. If the BSD conjecture is true for J over F and if K/F is a finite extension, it is not known in general whether the BSD conjecture is true for J over K. In [Ulm07], the second author studied the behavior of a more general class of L-functions over geometrically abelian extensions K/F. Specifically, for certain self-dual symplectic or orthogonal representations ρ:G →GL (Q ) of weight w, F n (cid:2) thereisafactorizationofL(ρ,K,T),withfactorsindexedbyorbitsofthecharacter groupofGal(K/F)underFrobenius,andacriterionforafactortohaveazeroatthe center point of its functional equation. Under a parity condition on the conductor of ρ, this implies that the order of vanishing of L(ρ,K ,T) at T = |k|−(w+1)/2 is d unbounded among Kummer extensions of the form K = k(t1/d) of F = k(t); see d [Ulm07, Theorem 4.7]. ReceivedbytheeditorsJune10,2013and,inrevisedform,October15,2014. 2010 Mathematics Subject Classification. Primary 11G10, 11G40, 14G05; Secondary 11G05, 11G30,14H25,14J20,14K15. (cid:2)c2016 American Mathematical Society 8553 Thisisafreeoffprintprovidedtotheauthorbythepublisher. Copyrightrestrictionsmayapply. 8554 RACHELPRIESANDDOUGLASULMER The system of rational Kummer extensions of function fields also plays a key role in the papers [Ber08,Ulm13a,Ulm14a]. For example, [Ber08] proves that the BSDconjectureholdsforJacobiansJ /K whenX isintheclassofcurvesdefined X d by equations of the form f(x)−tg(y) over F = k(t) and K is in the Kummer d toweroffieldsK =k(t1/d). Also,[Ulm13a]givesaformulafortherankofJ over d X K whichdependsonhomomorphismsbetweentheJacobiansofauxiliaryKummer d curves. Inthispaper, westudythesephenomenaforthesystemofArtin-Schreierexten- sions of functionfields of positive characteristic. The main resultsare analogous to thosedescribedabove: anunboundednessofanalyticranksresult(Corollary2.7.3), aproofoftheBSDconjectureforJacobiansofanewclassofcurvesX overanArtin- Schreiertoweroffields(Corollary3.1.4),andaformulafortherankoftheMordell- WeilgroupofJ overArtin-Schreierextensionswhichdependsonhomomorphisms X between the Jacobians of auxiliary Artin-Schreier curves (Theorem 5.2.1). There are several reasons why the Artin-Schreier variants of these theorems are quite compelling. First, the curves which can be studied using the Artin-Schreier variantincludethosedefinedbyanequationoftheformf(x)−g(y)−toverF =k(t). Thegeometryofthesecurvesiscomparativelyeasytoanalyze,allowingustoapply themainresultsinbroadgenerality. Forexample,Proposition4.4.1illustratesthat the hyperelliptic curve x2 = g(y)+t with g(y) ∈ k[y] of degree N satisfies the BSD conjecture, with unbounded rank in the tower of Artin-Schreier extensions of k(t), under the very mild conditions that p (cid:2) N and the finite critical values of g(y) are distinct. Second, the structure of endomorphism rings of Jacobians of Artin-Schreier curves is sometimes well understood. This allows us to compute the exact value of the rank of the Mordell-Weil group in several natural cases. Finally, some apparently unusual latticesappear asMordell-Weil latticesof elliptic curves covered by our analysis. We illustrate this for the family of elliptic curves Y2 =X(X+16b2)(X+t2)(wherebisaparameterinafinitefield)inSubsection7.3. Here is an outline of the paper. In Section 2, we consider certain elementary abelian extensions K of F =k(C) with deg(K/F)=q a power of p, and we study the L-functions L(ρ,K,T) of certain self-dual representations ρ : G → GL (Q ). F n (cid:2) Using results about Artin conductors of twists of ρ by characters of Gal(K/F), we prove a lower bound for the order of vanishing of L(ρ,K,T) at the center point of thefunctionalequation. InthecaseofanabelianvarietyJ overF whoseconductor satisfies a parity condition, this yields a lower bound for the order of vanishing of L(J/K,s) at s=1, Corollary 2.7.3. In Section 3, we prove that a new class of surfaces has the DPCT property introduced by Berger. More precisely, we prove that a surface associated to the curveX givenbyanequationoftheformf(x)−g(y)−toverF =k(t)isdominated by a product of curves and, furthermore, this DPC property is preserved under pullback to the field K := F(u)/(uq −u −t) for all powers q of p. It follows q that the BSD conjecture holds for the Jacobians of this class of curves X over this Artin-Schreier tower of fields, Corollary 3.1.4. In Section 4, we combine the results from Sections 2 and 3 to give a broad array of examples of Jacobians over rational function fields k(u) which satisfy the BSD conjecture and have large Mordell-Weil rank; see, e.g., Proposition 4.4.1. Section5containsaformulafortherankofJ overK intermsofthegeometry X q of the fibers of bad reduction of X and the rank of the group of homomorphisms Thisisafreeoffprintprovidedtotheauthorbythepublisher. Copyrightrestrictionsmayapply. ABELIAN VARIETIES IN ARTIN-SCHREIER EXTENSIONS 8555 Section 3 Section 2 Surfaces Analytic dominated ranks by a product Section 4 Examples— Lower bounds Section 5 A rank formula Section 7 Section 6 Examples— Examples— Explicit Exact ranks points Figure 1. Leitfaden between the Jacobians of auxiliary curves. (The auxiliary curves are C and D , q q defined by equations zq−z =f(x) and wq−w =g(y), and we consider homomor- phismswhichcommutewiththeF -GaloisactionsonC andD ;seeTheorem5.2.1.) q q q Section6containsthreeapplicationsoftherankformula: first,byconsideringcases where C is ordinary and D has p-rank 0, we construct examples of curves X over q q F witharbitrarygenusforwhichtherankofJ overK isboundedindependently X q of q; second, looking at the case when f = g, we construct examples of curves X over F with arbitrary genus for which the rank of X over K goes to infinity with q q; third, wecombinethelowerboundfortheanalyticrankandtherankformulato deduce the existence of supersingular factors of Jacobians of Artin-Schreier curves. Finally, in Section 7, we construct explicit points and compute heights for two examples. When q ≡2mod3, the isotrivial elliptic curve E defined by Y2+tY = X3 has rank 2(q−1) over K =F (u)where uq−u=t. Weconstruct a subgroup q q2 of finite index in the Mordell-Weil group, and we conjecture that the index is X| (E/K )|1/2 (which is known to be finite in this case). For b (cid:5)∈ {0,1,−1}, the q non-isotrivial curve Y2 =X(X+16b2)(X+t2) has rank q−1 over K , and again q we construct a subgroup of finite index in the Mordell-Weil group. In this case, the lattice generated by q−1 explicit points is in a certain sense a perturbation of the lattice A∗ where the fluctuations are determined by point counts on another q−1 family of elliptic curves. This rather exotic situation has, to our knowledge, not appeared in print before. Anappendix,Section8,collectsalltheresultsweneedaboutramification,New- ton polygons, and endomorphism algebras of Artin-Schreier curves. Thisisafreeoffprintprovidedtotheauthorbythepublisher. Copyrightrestrictionsmayapply. 8556 RACHELPRIESANDDOUGLASULMER Figure 1 shows dependencies between the sections. A dashed line indicates a verymild dependencywhich canbeignoredtofirst approximation, whereasasolid line indicates a more significant dependency. We have omitted dependencies in the appendix; these exist in Sections 2, 6, and 7, and at one place in Section 3. 2. Analytic ranks Inthissection,weuseresultsfrom[Ulm07]toshowthatanalyticranksareoften large in Artin-Schreier extensions. The main result is Corollary 2.7.3. 2.1. Notation. Let p be a prime number, let F be the field of p elements, and let p k be a finite field of characteristic p. We write r =|k| for the cardinality of k. Let F =k(C) be the function field of a smooth, projective, irreducible curve C over k. Let Fsep be a separable closure of F. We write F for the algebraic closure of F p p in Fsep. Let G =Gal(Fsep/F) be the Galois group of F. F Let (cid:3) (cid:5)= p be a prime number and let Q be an algebraic closure of the (cid:3)-adic (cid:2) numbers. Fix a representation ρ : G → GL (Q ) satisfying the hypotheses of F n (cid:2) [Ulm07,§4.2]. Inparticular,ρisassumedtobeself-dual ofsomeweightw andsign −(cid:4). When (cid:4)=1 we say ρ is symplectic, and when (cid:4)=−1 we say ρ is orthogonal. The representation ρ gives rise to an L-function L(ρ,F,T) given by an Euler product as in [Ulm07, §4.3]. We write L(ρ,K,T) for L(ρ| ,K,T) for any finite GK extension K of F contained in Fsep. In [Ulm07, §4], we studied the order of vanishing of L(ρ,K,T)/L(ρ,F,T)at the center point T = r−(w+1)/2 when K/F is a Kummer extension. Here we want to study the analogous order when K/F is an Artin-Schreier extension. 2.2. Extensions. Letq be apower ofpand write℘ (z) forthe polynomial zq−z. q We will consider field extensions K of F of the form (2.2.1) K =K =F[z]/(℘ (z)−f) ℘q,f q for f ∈ F \ k. We assume throughout that F K is a field, a condition which p is guaranteed when f has a pole of order prime to p at some place of F. As describedinLemma8.1.1,underthisassumption, thedegreeq fieldextensionK/F is“geometricallyabelian”inthesense that F K/F F isGaloiswithabelianGalois p p group. In fact, setting H = Gal(F K/F F), we have a canonical isomorphism p p H ∼= F , where F is the subfield of Fsep of cardinality q. The element α ∈ F q q q corresponds to the automorphism of F K which sends the class of z in (2.2.1) to p z+α. It will be convenient to consider a more general class of geometrically abelian extensions whose Galois groups are elementary abelian p-groups. Suppose that A isamonic, separable, additivepolynomial, inotherwordsapolynomialoftheform ν(cid:2)−1 A(z)=zpν + a zpi i i=0 with a ∈ F and a (cid:5)= 0. We will see in Subsection 8.2 that there is a bijection i p 0 between such polynomials A and subgroups of F which associates to A the group p H of its roots. The field generated by the coefficients of A is the field of pμ A elements, where pμ is the smallest power of p such that H is stable under the A pμ-power Frobenius. Thisisafreeoffprintprovidedtotheauthorbythepublisher. Copyrightrestrictionsmayapply. ABELIAN VARIETIES IN ARTIN-SCHREIER EXTENSIONS 8557 Suppose f ∈F has apoleoforder prime topatsome placeofF andthatAhas coefficients in k. Then we have a field extension K =K =F[x]/(A(z)−f). A,f It is geometrically Galois over F, with Gal(F K/F F) canonically isomorphic to p p H . A ByLemma8.2.2,ifAhasrootsinF ,thenthereexistsanothermonic,separable, q additivepolynomialBsuchthatthecompositionA◦Bequals℘ . Furthermore,this q implies that K is a subfield of K and that Gal(F K /F F) is a quotient A,f ℘q,f p A,f p of F , namely B(F ). In particular, for many questions, we may reduce to the case q q where K is the Artin-Schreier extension K . A,f ℘q,f 2.3. Characters. Let K = K be an Artin-Schreier extension of F as in Sub- section 2.2, and let H = Gal(F℘qK,f/F F) ∼= F . Fix once and for all a non-trivial p p q additive character ψ : F → Q ×. Let Hˆ = Hom(H,Q ×) be the group of Q - 0 p (cid:2) (cid:2) (cid:2) valuedcharactersofH. ThenwehaveanidentificationHˆ ∼=F underwhichβ ∈F q q × corresponds to the character χβ :H →Q(cid:2) , α(cid:8)→ψ0(TrFq/Fp(αβ)). Next we consider actions of G = Gal(k/k) on H and Hˆ. To define them, k consider the natural projection G →G , and let Φ be any lift of the (arithmetic) F k generator of G , namely the r-power Frobenius. Using this lift, G acts on H = k k Gal(F K/F F) on the left by conjugation, and it is easy to see that under the p p identification H ∼=F , Φ acts on F via the r-th power Frobenius. q q We also have an action of G on Hˆ on the right by precomposition: (χ )Φ(α)= k β χ (Φ(α))=χ (αr). Since β β TrFq/Fp(αrβ)=TrFq/Fp(αβr−1) we see that (χβ)Φ =χβr−1. If A is a monic, separable, additive polynomial with coefficients in k and group of roots H , then the character group of H is naturally a subgroup of Hˆ, and it A A is stable under the r-power Frobenius. More precisely, by Lemma 8.2.2(2), H is A the quotient B(F ) of F , and so its character group is identified with (kerB)⊥ = q q (ImA)⊥ where the orthogonal complements are taken with respect to the trace pairing (x,y)(cid:8)→TrFq/Fp(xy). As seen in Example 8.2.3, if r is a power of an odd prime p and A(z)=zrν +z, thenthegroupH ofrootsofAgeneratesF whereq =r2ν. Inthiscase,A◦B =℘ A q q whenB =℘ . Iff ∈F hasapoleoforderprimetopatsomeplaceofF,thenthe rν fieldextensionK isasubextensionofK anditscharactergroupisidentified A,f ℘q,f with (kerB)⊥ =H . A 2.4. Ramification and conductor. We fix a place v of F and consider a decom- position subgroup G of G=G at the place v and its inertia subgroup I . v F v Recall from [Ser79, Chap. IV] that the upper numbering of ramification groups is compatible with passing to a quotient, and so defines a filtration on the inertia group I , which we denote by It for real numbers t≥0. By the usual convention, v v we set It =I for −1<t≤0. v v Let ρ:G →GL (Q ) be a Galois representation as above, acting on V =Q n. F n (cid:2) (cid:2) We denote the local exponent at a place v of the conductor of ρ by f (ρ). We refer v to [Ser70] for the definition. Thisisafreeoffprintprovidedtotheauthorbythepublisher. Copyrightrestrictionsmayapply. 8558 RACHELPRIESANDDOUGLASULMER × Now let χ : G → Q be a finite order character. We say “χ is more deeply F (cid:2) ramifiedthanρatv”ifthereexistsanon-negative realnumber tsuchthatρ(It)= v {id}andχ(It)(cid:5)={id}. Inotherwords,χisnon-trivialfurtherintotheramification v filtration than ρ is. Let t0 be the largest number such that χ is non-trivial on Ivt0 and recall that f (χ)=1+t [Ser79, VI, §2, Proposition 5]. v 0 Lemma 2.4.1. If χ is more deeply ramified than ρ at v, then f (ρ⊗χ)=deg(ρ)f (χ). v v Proof. Thisisaneasyexerciseandpresumablywellknowntoexperts. Itisasserted in [DD13, Lemma 9.2(3)], and a detailed argument is given in [Ulm13b]. (cid:2) A particularly useful case of the lemma occurs when ρ is tamely ramified and χ is wildly ramified, e.g., when χ is an Artin-Schreier character. 2.5. Factoring L(ρ,K,T). Fix a monic, separable, additive polynomial A with coefficients in k and a function f ∈F such that f has a pole of order prime to p at some place of F. Let K = K be the corresponding extension whose geometric A,f Galois group Gal(F K/F F) is canonically identified with the group H = H of p p A roots of A. Let F be the subfield of Fsep generated by H . Recall the Galois q A representation ρ fixed above. In this section, we record a factorization of the L- function L(ρ,K,T). InSubsection2.3above, we identifiedthecharacter groupofH withasubgroup of F which is stable under the r-power Frobenius. As in [Ulm07, §3], we write q o ⊂ Hˆ ⊂ F for an orbit of the action of Fr . Note that the cardinality of the q r orbit o through β ∈ F is equal to the degree of the field extension k(β)/k and is q therefore at most 2ν. As in [Ulm07, §4.4], we have a factorization (cid:3) L(ρ,K,T)= L(ρ⊗σ ,F,T) o o⊂Hˆ and a criterion for the factor L(ρ⊗σ ,F,T) to have a zero at T = (cid:4)r−(w+1)/2 (or o more generally to be divisible by a certain polynomial). To unwind that criterion, we need to consider self-dual orbits. More precisely, notethattheinverseofχβ is(χβ)−1 =χ−β. Thusanorbitoisself-dualinthesense of [Ulm07, §3.4] if and only if there exists a positive integer ν such that βrν =−β forallβ ∈o. Thetrivialorbito={0}isofcourseself-dualinthissense. Toensure that that there are many other self-dual orbits, we may assume r is odd and take A(x)=xrν +x for some positive integer ν. Then if β is a non-zero root of A, the orbit through β is self-dual. Since the size of this orbit is at most 2ν, we see that there are at least (rν −1)/(2ν) non-trivial self-dual orbits in this case. We also note that if β (cid:5)=0, then the order of the character χ is p, and since we β are assuming r, and thus p, is odd, we have that χ has order >2. Summarizing, β we have the following. Lemma 2.5.1. Let k be a finite field of cardinality r and characteristic p > 2. Suppose A(z) = zrν +z. Suppose f ∈ F has a pole of order prime to p, and let K =K . Let ρ be a representation of G as in Subsection 2.1. Then we have a A,f F factorization (cid:3) L(ρ,K,T)= L(ρ⊗σ ,F,T) o o⊂Hˆ Thisisafreeoffprintprovidedtotheauthorbythepublisher. Copyrightrestrictionsmayapply. ABELIAN VARIETIES IN ARTIN-SCHREIER EXTENSIONS 8559 where the product is over the orbits of the r-power Frobenius on the roots of A. Aside from the orbit o={0}, there are at least (rν−1)/2ν orbits, each of which is self-dual, has cardinality at most 2ν, and consists of characters of order p>2. 2.6. Parity conditions. Accordingto[Ulm07,Thm.4.5],L(ρ⊗σ ,F,T)vanishes o atT =r−(w+1)/2 ifρissymplecticofweightw,oisaself-dualorbit,andthedegree of Cond(ρ⊗χ ) is odd for one, and therefore for all, β ∈o. Thus to obtain a large β orderofvanishing, weshouldarrangematterssothatρ⊗χ satisfiestheconductor β parity condition for many orbits o. This is not hard to do using Lemma 2.4.1. Indeed, let S be the set of places where χβ is ramified, and s(cid:4)uppose that χβ is more deeply ramified than ρ at each v ∈ S. Suppose also that f (ρ)deg(v) v(cid:5)∈S v is odd. Then using Lemma 2.5.1 we have (cid:2) (cid:2) degCond(ρ⊗χ )= deg(ρ)f (χ )deg(v)+ f (ρ)deg(v). β v β v v∈S v(cid:5)∈S Since ρ is symplectic, it has even degree, and so our assumptions imply that degCond(ρ⊗χ ) is odd. β 2.7. High ranks. Putting everything together, we get results guaranteeing large analytic ranks in Artin-Schreier extensions: Theorem 2.7.1. Let k be a finite field of cardinality r and characteristic p > 2. Let ν ∈ N and let k(cid:7) be the field of q = r2ν elements. Let F = k(C) and ρ : G → GL (Q ) be as in Subsection 2.1. Assume that ρ is symplectically self-dual F n (cid:2) of weight w. Choose f ∈F with at least one pole of order prime to p. Suppose that either (1) K =K where A(z)=zrν+z or (2) K =K where ℘ (z)=zq−z A,f ℘q,f q as in Subsection 2.2. Let S be a set of places of F where K/F is ramified and (cid:4)suppose that ρ is at worst tamely ramified at each place v ∈ S. Suppose also that f (ρ)deg(v) is odd. Then v(cid:5)∈S v L(ρ,K,s) ord ≥(rν −1)/(2ν) s=(w+1)/2 L(ρ,F,s) and L(ρ,k(cid:7)K,s) ord ≥(rν −1). s=(w+1)/2 L(ρ,k(cid:7)F,s) Proof. For case (1), the first inequality is an easy consequence of the preceding subsectionsand[Ulm07,Thm.4.5]. Indeed,byLemma2.5.1,wehaveafactorization (cid:3) L(ρ,K,T)= L(ρ⊗σ ,F,T) o o⊂Hˆ where the product is over the orbits of the r-power Frobenius on the roots of A. The factor on the right corresponding to the orbit o = {0} is just L(ρ,F,T), and by the lemma, all the other orbits are self-dual and consist of characters of order > 2. The hypotheses on the ramification of ρ allow us to apply Lemma 2.4.1 to conclude that the parity of degCond(ρ⊗χ ) is odd for all roots β (cid:5)=0 of A. Thus β [Ulm07, Thm. 4.5] implies that each of the factors L(ρ⊗σ ,F,T) is divisible by o 1−(r(w+1)/2T)|o| and, in particular, has a zero at T =r−(w+1)/2. Since there are (rν −1)/2ν non-trivial orbits, we obtain the desired lower bound. Thisisafreeoffprintprovidedtotheauthorbythepublisher. Copyrightrestrictionsmayapply. 8560 RACHELPRIESANDDOUGLASULMER Over any extension k(cid:7) of k of degree divisible by 2ν, we have a further factoriza- tion (cid:3) L(ρ⊗σ ,k(cid:7)F,T)= L(ρ⊗χ ,k(cid:7)F,T), o β β∈o andeachfactorL(ρ⊗χ ,k(cid:7)F,T)isdivisibleby(1−|k(cid:7)|(w+1)/2T)andthusvanishes β at s=(w+1)/2. This establishes the second lower bound in case (1). The lower bounds for case (2) are an immediate consequence of those for case (1) since K is a subextension of K by Example 8.2.3. (cid:2) A,f ℘q,f Remark 2.7.2. If F =F (t) and f =t, then the Artin-Schreier extension given by p uq −u = t is again a rational function field. Thus starting with a suitable ρ and takingalargedegreeArtin-Schreierextension,orbytakingmultipleextensions,we obtainanotherproofofunboundedanalyticranksoverthefixedgroundfieldF (u). p As an illustration, we specialize Theorem 2.7.1 to the case where ρ is given by the action of G on the Tate module of an abelian variety over F. F Corollary 2.7.3. Let k be a finite field of cardinality r and characteristic p > 2. Letν ∈Nandletk(cid:7) bethefieldofq =r2ν elements. SupposeJ isanabelianvariety over a function field F =k(C) as in Subsection 2.1. Choose f ∈F with at least one pole of order prime to p. Suppose that either (1) K =K where A(z)=zrν +z A,f or (2) K = K where ℘ (z) = zq −z as in Subsection 2.2. Let S be the set of ℘q,f q places of F where K/F is ramified. Suppose that J is at worst tamely ramified at all places in S and that the degree of the part of the conductor of J away from S is odd. Then ord L(J/K,s)≥(rν −1)/(2ν) s=1 and ord L(J/k(cid:7)K,s)≥(rν −1). s=1 2.8. Orthogonalρandsupersingularity. Considertheset-upofTheorem2.7.1, exceptthatweassumethatρisorthogonallyself-dualinsteadofsymplecticallyself- dual, and we replace the parity condition there with the assumption that (cid:2) (cid:2) deg(ρ) (−ord (f)+1)deg(v)+ f (ρ)deg(v) v v v∈S v(cid:5)∈S is odd. Then [Ulm07, Thm. 4.5] implies that if o is an orbit with o (cid:5)= {0}, then L(ρ⊗σ ,F,T) is divisible by 1+(r(w+1)/2T)|o|. In particular, over a large enough o finite extension k(cid:7) of k, at least rν − 1 of the inverse roots of the L-function L(ρ,K,T)/L(ρ,F,T) are equal to |k(cid:7)|(w+1)/2. We apply this result to the case when ρ is the trivial representation to conclude thattheJacobians ofcertainArtin-Schreier curveshave many copies of asupersin- gular elliptic curve as isogeny factors. This implies that the slope 1/2 occurs with high multiplicity in their Newton polygons as defined in Subsection 8.3. However, as explained in Subsection 8.4, the occurrence of slope 1/2 in the Newton poly- gon of an abelian variety usually does not give any information about whether the abelian variety has a supersingular elliptic curve as an isogeny factor. This gives the motivation for this result. More precisely: Proposition 2.8.1. With the notation of Corollary 2.7.3, write (cid:2)m div∞(f)= aiPi i=1 Thisisafreeoffprintprovidedtotheauthorbythepublisher. Copyrightrestrictionsmayapply. ABELIAN VARIETIES IN ARTIN-SCHREIER EXTENSIONS 8561 w(cid:4)here the Pi are distinct k-valued points of C. Assume that p(cid:2)ai for all i and that m (a +1) is odd. Let J (resp. J , J ) be the Jacobian ofC (resp. the cover i=1 i A,f ℘q,f C of C defined by A(z)=f, the cover C of C defined by ℘ (z)=f). Then up A,f ℘q,f q to isogeny over k, the abelian varieties J /J and J /J each contain at least A,f ℘q,f (rν −1)/2 copies of a supersingular elliptic curve. Proof. Wegiveonlyabriefsketch, sincethisresultplaysaminorroleintherestof the paper. An argument parallel to that in the proof of Theorem 2.7.1 shows that the numerator of the zeta function of C divided by that of C is divisible by A,f (cid:5) (cid:6) 1+rνT2ν (rν−1)/(2ν). Thus over a large extension k(cid:7) of k, at least rν −1 of the inverse roots of the zeta function are equal to |k(cid:7)|1/2. Honda-Tate theory then shows that the Jacobian has a supersingular elliptic curve as an isogeny factor with multiplicity at least (rν −1)/2. (cid:2) We will see in Section 8 that the lower bound of Proposition 2.8.1 is not always sharp. 2.9. The case p = 2. The discussion of the preceding subsections does not apply when p = 2 since in that case all characters of H have order 2. To get high ranks when p = 2, we can use the variant of [Ulm07, Thm. 4.5] suggested in [Ulm07, 4.6]. In this variant, instead of symmetric or skew-symmetric matrices, we have orthogonal matrices, and zeroes are forced because 1 is always an eigenvalue ofanorthogonalmatrixofoddsize,and±1arealwayseigenvaluesofanorthogonal matrix of even size and determinant −1. The details are somewhat involved and tangential to the main concerns of this paper, so we will not include them here. 2.10. Artin-Schreier-Wittextensions. TheargumentleadingtoTheorem2.7.1 generalizes easily to the situation where we replace Artin-Schreier extensions with Artin-Schreier-Witt extensions. This generalization is relevant even if p = 2. We sketch very roughly the main points. Let W (F) be the ring of Witt vectors of length n with coefficients in F. We n choose f ∈ W (F) and we always assume that its first component f is such that n 1 xq −x−f is irreducible in F[x] and so defines an extension of F of degree q. 1 Then adjoining to F the solutions (in W (Fsep)) of the equation Fr (x)−x = f n q yieldsafieldextensionofF which isgeometricallyGalois withgroupW (F ). The n q character group of this Galois group can be identified with W (F ), and we have n q anactionofG (i.e.,ther-powerFrobeniuswherer =|k|)onthecharactersofthis k group. Choose a positive integer ν and consider the situation above where q =r2ν. We claim that there are rnν solutions in W (F ) to the equation Fr (x)+x=0. For n q rν p > 2, this is clear—just take Witt vectors whose entries satisfy xrν +x = 0. For p = 2, the entries of −x are messy functions of those of x, so we give a different argument. Namely, let us proceed by induction on n. For n = 1, x = (1) is a 1 solution. Suppose that xn−1 = (a1,...,an−1) satisfies Frrν(x)+x = 0. Then we have Frrν(a1,...,an−1,0)+(a1,...,an−1,0)=(0,...,0,bn), and it is easy to see that b lies in the field of rν elements. We can thus solve the n equationarν+a =b , andthenx =(a ,...,a )solvesFr (x )+x =0. With n n n n 1 n rν n n Thisisafreeoffprintprovidedtotheauthorbythepublisher. Copyrightrestrictionsmayapply. 8562 RACHELPRIESANDDOUGLASULMER onesolutionwhichisaunitinW (F )inhand, weremarkthatanymultiple ofour n q solution by an element of W (F ) is another solution, so we have rnν solutions in n rν all. Nextwenotethattheself-dualorbitso⊂W (F )(i.e.,thoseorbitsstableunder n q x (cid:8)→ −x) are exactly the orbits whose elements satisfy Fr (x)+x = 0. These rν orbitsareofsizeatmost2ν. Ifp>2,allbuttheorbito={0}consistofcharacters of order >2, whereas if p=2, all but pν of the orbits consist of characters of order > 2. Thus taking p > 2 or p = 2 and n > 1, we have a plentiful supply of orbits which are self-dual and consist of characters of order >2. ThelastingredientneededtoensureahighorderofvanishingfortheL-function is a conductor parity condition. This can be handled in a manner quite parallel to the cases considered in Subsection 2.6. Namely, we choose f ∈ W (F) so that at n places where ρ and characters χ are ramified, χ should be so more deeply, and the remaining part of the conductor of ρ should have odd degree. Then ρ⊗χ will have conductor of odd degree. 3. Surfaces dominated by a product of curves in Artin-Schreier towers In this section, we extend a construction of Berger to another class of surfaces, following [Ulm13a, §§4-6]. 3.1. Construction of the surfaces. Let k be a field with Char(k) = p and let K = k(t). Suppose C and D are smooth projective irreducible curves over k. Suppose f :C →P1 and g :D →P1 are non-constant separable rational functions. Write the polar divisors of f and g as (cid:2)m (cid:2)n div∞(f)= aiPi and div∞(g)= bjQj i=1 j=1 where the P and the Q are distinct k-valued points of C and D. Let i j (cid:2)m (cid:2)n M = a and N = b . i j i=1 j=1 We make the following standing assumption: (3.1.1) p(cid:2)a for 1≤i≤m and p(cid:2)b for 1≤j ≤n. i j We use the notation P1 to denote the projective line over k with a chosen k,t parametert. Definearationalmapψ :C× D(cid:3)(cid:3)(cid:4)P1 bytheformulat=f(x)−g(y) 1 k k,t or more precisely ⎧ ⎪⎨[f(x)−g(y):1] if x(cid:5)∈{P } and y (cid:5)∈{Q }, i j ψ (x,y)= [1:0] if x∈{P } and y (cid:5)∈{Q }, 1 ⎪⎩ i j [1:0] if x(cid:5)∈{P } and y ∈{Q }. i j The map ψ is undefined at each of the points in the set 1 B={(P ,Q )|1≤i≤m,1≤j ≤n}. i j Let U =C× D−B and note that the restriction ψ | :U →P1 is a morphism. k 1 U k,t WecallthepointsinB“basepoints”becausetheyarethebasepointsofthepencil of divisors on C × D defined by ψ . Namely, for each closed point v ∈ P1 , let k 1 k,t Thisisafreeoffprintprovidedtotheauthorbythepublisher. Copyrightrestrictionsmayapply.