THE INVERSE GALOIS PROBLEM FOR ORTHOGONAL GROUPS DAVID ZYWINA

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1 THE INVERSE GALOIS PROBLEM FOR ORTHOGONAL GROUPS DAVID ZYWINA ABSTRACT. We rove many new cases of the Inverse Galois Problem for those simle grous arising from orthogonal grous over finite fields. For examle, we show that the finite simle grous Ω 2n+1 () and PΩ + () both occur as the Galois grou of a Galois extension of the rationals for all integers n 2 and all 4n rimes 5. We obtain our reresentations by studying families of twists of ellitic curves and using some known cases of the Birch and Swinnerton-Dyer conjecture along with a big monodromy result of Hall. 1. INTRODUCTION The Inverse Galois Problem asks whether every finite grou G is isomorhic to the Galois grou of some Galois extension of. This roblem is extremely difficult, even in the secial case of nonabelian simle grous which we now restrict our attention to. Many secial cases are known, including alternating grous and all but one of the soradic simle grous. Several families of simle grous of Lie tye are known to occur as Galois grous of an extension of, but usually with congruences imosed on the cardinality of the fields. See [MM99] for background and many examles. Moreover, one can ask whether there is a regular Galois extension K/ (t) with Galois grou isomorhic to G (by regular, we mean that is algebraically closed in K). If such an extension K/ (t) exists, then Hilbert s irreducibility theorem imlies that there are infinitely many Galois extensions of with Galois grou isomorhic to G, and likewise over every number field The grous. We first introduce the simle grous that we are interested in realizing as Galois grous; further background can be found in [Wil09, 3.7] and [CCN + 85, 2.4]. Take any rime l 5. An orthogonal sace over l is a finite dimensional l -vector sace V equied with a non-degenerate and symmetric bilinear airing, : V V l. A homomorhism of orthogonal saces is an l -linear ma which is comatible with the resective airings. The orthogonal grou of V, denoted by O(V ), is the grou of automorhisms of V as an orthogonal sace. The secial orthogonal grou SO(V ) is the index 2 subgrou of O(V ) consisting of those elements with determinant 1. For each v V with v, v 0, we have a reflection r v O(V ) defined by x x 2 x, v / v, v v. The sinor norm of V is the homomorhism s: O(V ) l /( l )2 characterized by the roerty that it satisfies s(r v ) v, v ( l )2 for every v V with v, v 0. The discriminant of V is disc(v ) : s( I); it can also be defined as the coset in l /( l )2 reresented by det( e i, e j ) where {e 1,..., e n } is any basis of V. Now fix an integer n 5, and take any orthogonal sace V over l of dimension n. We define Ω(V ) to be the subgrou of SO(V ) consisting of elements with trivial sinor norm. Using that l /( l )2 has order 2, one can show that Ω(V ) is an index 2 subgrou of SO(V ). The grou Ω(V ) is erfect and its center is either {I} or {±I}. Denote by PΩ(V ) the quotient of Ω(V ) by its center. Suose that n is odd. The grou Ω(V ) has trivial center and is simle. The isomorhism class of O(V ), and hence also Ω(V ), deends only on n and l. Denote by Ω n (l) an abstract grou isomorhic to Ω(V ) (other common notation for this grou is O n (l) and B (n 1)/2 (l)) Mathematics Subject Classification. Primary 12F12; Secondary 14J27, 11G05, 11G40. This material is based uon work suorted by the National Science Foundation under agreement No. DMS

2 Suose that n is even. U to isomorhism, there are two orthogonal saces of dimension n over l and they are distinguishable by their discriminants. We say that V is slit if disc(v ) ( 1) n/2 ( l )2 and non-slit otherwise (note that V is slit if and only if it is an orthogonal sum of hyerbolic lanes). Denote by PΩ + n (l) and PΩ n (l) an abstract grou isomorhic to PΩ(V ) when V is slit or non-slit, resectively. The grous PΩ + n (l) and PΩ n (l) are both simle. (Other common notation for PΩ+ n (l) is O + n (l) and D n/2(l). Other common notation for PΩ n (l) is O n (l) and 2 D n/2 (l 2 ).) 1.2. Main results. Theorem 1.1. Take any integer n 5 and rime l 5. (i) If n is odd, then the simle grou Ω n (l) occurs as the Galois grou of a regular extension of (t). (ii) If n 0 (mod 4) or l 1 (mod 4), then the simle grou PΩ + n (l) occurs as the Galois grou of a regular extension of (t). (iii) If n 2 (mod 4) and l 3 (mod 4), then the simle grou PΩ n (l) occurs as the Galois grou of a regular extension of (t). (iv) If n is even and 2, 3, 5 or 7 is not a square modulo l, then the grous PΩ + n (l) and PΩ n (l) both occur as the Galois grou of a regular extension of (t). The following is a restatement of Theorem 1.1(ii) and (iii). Corollary 1.2. Take any even integer n 6 and rime l 5. If V is an orthogonal sace of dimension n over l with disc(v ) ( l )2, then the simle grou PΩ(V ) occurs as the Galois grou of a regular extension of (t). The following is a consequence of Theorem 1.1 and the excetional isomorhisms Ω 5 (l) PS 4 ( l ), PΩ + 6 (l) PSL 4 ( l ) and PΩ 6 (l) PSU 4 ( l ). Corollary 1.3. Take any rime l 5. (i) The simle grou PS 4 ( l ) occurs as the Galois grou of a regular extension of (t). (ii) If l is not congruent to 311, 479, 551, 671, 719 and 839 modulo 840, then the simle grou PSL 4 ( l ) occurs as the Galois grou of a regular extension of (t). (iii) If l is not congruent to 1, 121, 169, 289, 361 and 529 modulo 840, then the simle grou PSU 4 ( l ) occurs as the Galois grou of a regular extension of (t) Some revious work and related cases. Reiter [Rei99] roved Theorem 1.1(i) in the secial case where 2 or 3 is not a square modulo l; in articular, it covers the case l 3 which we excluded. Additional secial cases of Theorem 1.1(i) for n 5 and 7 were roved by Häfner [Häf92]. Theorem 1.1(iv) covers the various cases of the regular inverse Galois roblem for PΩ + n (l) and PΩ n (l) with l 5 that are due to Reiter [Rei99] and Malle-Matzat [MM99, 10.2]. The cases of the regular inverse Galois roblem in Theorem 1.1(ii) and (iii) aear to be new. We now briefly discuss the excluded cases n 3 and 4; we do not obtain non-abelian simle grous when n 2. These cases are esecially interesting because of the excetional isomorhisms Ω 3 (l) PSL 2 ( l ), PΩ + 4 (l) PSL 2 ( l ) PSL 2 ( l ) and PΩ 4 (l) PSL 2 ( l 2). The simle grou PSL 2 ( l ) is known to occur as the Galois grou of a regular extension of (t) if 2, 3, 5 or 7 is not a square modulo l; the cases 2, 3 and 7 are due to Shih [Shi74] and 5 is then due to Malle [Mal93]. The conclusion of Theorem 1.1(i) and Theorem 1.1(ii) with n 3 and n 4, resectively, remains oen. In [Zyw13], the author showed that PSL 2 ( l ) occurs as a Galois grou of an extension of for all rimes l. The construction of such extensions in [Zyw13] is similar to those of this aer; however, regular extensions of (t) are not obtained. 2

3 The grou PSL 2 ( l 2) is already known to occur as the Galois grou of a regular extension of (t) if 2, 3, 5 or 7 is not a square modulo l; see [Shi04] and [Shi03] for 2 and 3, [Mes88] for 5, and [DW06] for 7. For many other l, the grou PSL 2 ( l 2) is known to occur as the Galois grou of an extension of ; for examles, see [Rib75, 7], [RV95] and [DV00]. The simle grou G 2 (l) occurs as the Galois grou of a regular extension of (t) for all rimes l 5 (cf. [FF85] for l > 5 and [Tho85] for l 5). The simle grou E 8 (l) occurs as the Galois grou of a regular extension of (t) for all rimes l 7, cf. [GM12]; this was first shown to be true by Yun for all l sufficiently large [Yun14]. Theorem 1.1(i) and Theorem 1.1(ii) with n 0 (mod 4) are the first cases where one has analogous results for finite simle grous of a fixed classical Lie tye A secial case. We now give an overview of the ideas behind the roof of Theorem 1.1 in the secial case n 5. In articular, for a fixed l 5, we will describe a regular extension of (t) with Galois grou isomorhic to Ω 5 (l). This section can be safely skied and will not be referred to later on. Define S : {2, 3, l} and the ring R : [S 1 ]. Define the R-scheme M Sec R[u, u 1, (u 1) 1, (u + 1) 1 ]; it is an oen subscheme of 1 R Sec R[u]. Let k be any finite field that is an R-algebra, i.e., a finite field whose characteristic is not 2, 3 or l. Denote the cardinality of k by q. Take any m M(k), i.e., any m k {0, 1, 1}. Let E m be the ellitic curve over the function field k(t) defined by the Weierstrass equation (t m) y 2 x 3 + 3(t 2 1) 3 x 2(t 2 1) 5. Denote by L(T, E m ) the L-function of the ellitic curve E m /k(t), see 2.3 for details. One can show that L(T, E m ) is a olynomial in [T] of degree 5. For examle, with k 5 and m 1, one can comute that L(T, E m ) 1 2T + T 2 5T T T 5. Using the cohomological descrition of L-functions, we will construct an orthogonal sace V l over l of dimension 5 and a continuous reresentation θ l : π 1 (M) O(V l ) such that for any k and m M(k) as above, we have (1.1) det(i θ l (Frob m )T) L(T/q, E m ) (mod l). Here π 1 (M) is the étale fundamental grou of M (with suressed base oint) and Frob m is the geometric Frobenius conjugacy class of m in π 1 (M). The reresentation θ l has big monodromy, i.e., θ l (π 1 (M )) Ω(V l ). This will be shown following the aroach of Hall in [Hal08] (we will directly use Hall s results for the cases with n > 5). The key ste is to show that the grou generated by the reflections in θ l (π 1 (M )) acts irreducibly on V l. The classification of finite irreducible linear grous generated by reflections then gives a finite number of small ossibilities for that need to be ruled out to ensure that Ω(V l ). The image of θ l can sometimes be the full orthogonal grou O(V l ) (in fact, this haens if l ±3 (mod 8)). Let W be the oen subscheme Sec R[u, u 1, (u 2 3) 1, (u 2 + 3) 1 ] of 1 R. The morhism h: W M given by w ( w 2 + 3)/(w 2 + 3) is étale of degree 2, so we have a reresentation We claim that there are inclusions ϑ l : π 1 (W ) h π 1 (M) θ l O(V l ). (1.2) Ω(V l ) ϑ l (π 1 (W )) ϑ l (π 1 (W )) ±Ω(V l ), 3

4 where ±Ω(V l ) is the grou generated by Ω(V l ) and I. The natural ma Ω(V l ) (±Ω(V l ))/{±I} is an isomorhism, so the claimed inclusions give a surjective homomorhism β : π 1 (W ) ϑ l ±Ω(V l ) (±Ω(V l ))/{±I} Ω(V l ) that satisfies β(π 1 (W )) Ω(V l ). Therefore, β gives rise to a regular extension of (u), i.e., the function field of W, that is Galois with Galois grou isomorhic to Ω(V l ) Ω 5 (l); this gives the desired extension for the n 5 case of Theorem 1.1(i). We now exlain why the inclusions of (1.2) hold. We have the inclusion ϑ l (π 1 (W )) Ω(V l ) of (1.2) since the simle non-abelian grou Ω(V l ) is a normal subgrou of θ l (M ) and the cover h is abelian. Now let κ be any coset of Ω(V l ) in θ l (π 1 (M)) with det(κ) { 1}. Since det(κ) { 1}, one can show that there is an element A κ such that det(i A) 0. From a formula of Zassenhaus, we find that s( A) 2 det(i A) ( l )2. Using equidistribution, there is a rime 6l and an m M( ) such that A is conjugate to θ l (Frob m ) in O(V l ). We have L(T/, E m ) det(i AT) (mod l) and hence L(1/, E m ) det(i A) (mod l). Therefore, s( A) 2 L(1/, E m ) ( l )2. The secial value L(1/, E m ) is linked to the arithmetic of the curve E m / (t). We have L(1/, E m ) 0 (since it is non-zero modulo l), so the Birch and Swinnerton-Dyer (BSD) conjecture redicts that the Mordell-Weil grou E m ( (t)) is finite. In fact, this is known unconditionally by work of Artin and Tate. Moreover, from Artin, Tate and Milne, the following refined version of BSD is known to hold: we have X Em c Em L(1/, E m ) E m ( (t)) 2 1+χ, Em where c Em is the roduct of Tamagawa numbers of E m over the laces of (t), 12χ Em is the degree of the minimal discriminant of E m, and X Em is the (finite!) Tate-Shafarevich grou of E m. Since X Em is finite, a airing of Cassels on X Em shows that X Em is a square. An alication of Tate s algorithm shows that χ Em 3 and that c Em is a ower of 2. So 2L(1/, E m ) 2c Em ( ) 2 and s( A) 2c Em ( l )2. Using Tate s algorithm, one can show that 2c Em {16, 64} if 3(m 2 1) is a square and 2c Em 32 otherwise. Therefore, ( s( A) l )2 if 3(m 2 1) is a square, 2( l )2 if 3(m 2 1) is not a square. Now suose that κ is actually a coset of Ω(V l ) in ϑ l (π 1 (W )). Then m h(w) for some w W ( ). We have 3(m 2 1) 6 2 w 2 /(w 2 + 3) 2 which is clearly a square (our degree 2 cover h: W M was chosen to ensure this held), so s( A) ( l )2. We have A Ω(V l ) since s( A) ( l )2 and det( A) ( 1) 5 det(a) 1. Therefore, κ AΩ(V l ) Ω(V l ). We now know that ϑ l (π 1 (W )) contains Ω(V l ) and the only ossibly Ω(V l )-coset with determinant 1 is Ω(V l ). Therefore, ϑ l (π 1 (W )) is either ±Ω(V l ) or is a subgrou of SO(V l ). So to exlain the last inclusion of (1.2), we need only show that ϑ l (π 1 (W )) contains an element with determinant 1. For any / S and m M( ), there is a functional equation T 5 L(T 1 /, E m ) ɛ Em L(T/, E m ), where ɛ Em {±1} is the root number of E m. Since A : θ l (Frob m ) belongs to O(V l ), we have T 5 det(i T 1 A) det( A) det(i TA). From (1.1), we deduce that det( A) ɛ Em and hence det(θ l (Frob m )) 4

5 ɛ Em. One can exress ɛ Em as a roduct of local root numbers and a comutation shows that ɛ Em is 1 if 3m is a square and 1 otherwise. So if ϑ l (π 1 (W )) is a subgrou of SO(V l ), then 3h(w) 3( w 2 + 3)/(w 2 + 3) is a square in for all rimes / S and w W ( ). This is easily seen to be false, so we deduce that ϑ l (π 1 (W )) is not a subgrou of SO(V l ) Overview. In 2, we give background on ellitic curves defined over global function fields. We will mainly be interested in non-isotrivial ellitic curves E defined over a function field k(t), where k is a finite field of order q with (q, 6) 1. We will recall the definition of the L-function L(T, E) of E; it is a olynomial with integer coefficients. For almost all rimes l, we will construct an orthogonal sace V E,l over l and a reresentation θ E,l : Gal(k/k) O(V E,l ) such that det(i θ E,l (Frob q )T) L(T/q, E) (mod l), where Frob q is the geometric Frobenius (i.e., the inverse of x x q ). The determinant of θ E,l (Frob q ) is related to the root number of E. In many case, we can comute the sinor norm of θ E,l (Frob q ) by using the secial value of L(T, E) arising in the Birch and Swinnerton-Dyer conjecture. We will discuss the Birch and Swinnerton-Dyer conjecture in 2.4. We shall use known cases to give an simle descrition of the square class L(1/q, E) ( ) 2 when L(1/q, E) is non-zero. In 3, we follow Hall and consider families of quadratic twists. We shall constructing a reresentation that encodes the reresentations θ E,l as E varies in our family. In 3.3, we state an exlicit version of a big monodromy result of Hall. In 4, we state a criterion to ensure that the reresentation of 3 will roduce a grou Ω(V ) as the Galois grou of a regular extension of (t). For any given examle, the conditions are straightforward to verify using some basic algebra and Tate s algorithm. In sections 5, 6 and 7, we give many examles and use our criterion to rove Theorem 1.1 for all n > 5. We shall not exlain how the equations in these sections were found; they were discovered through many numerical exeriments (though the aer [Her91] served as a useful starting oint since it gives many ellitic surfaces with only four singular fibers). Finally, in 8 we comlete the roof of Theorem 1.1 for n 5. The big monodromy criterion of Hall does not aly for our examle, so we need to rove it directly. Notation. Throughout the aer, we will feely use Tate s algorithm, see [Sil94, IV 9] or [Tat75]. All fundamental grous, cohomology and sheaves in the aer are with resect to the étale toology. We will often suress base oints for our fundamental grous, so many grous and reresentations will only be determined u to conjugacy. We will indicate base change of schemes by subscrits, for examle given a scheme X over, we denote by X the corresonding scheme base changed to. Acknowledgements. Numerical exeriments and comutations were done using Magma [BCP97]. Thanks to Brian Conrad for comments and corrections. 2. L-FUNCTIONS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS In this section, we give some background on the arithmetic of ellitic curves defined over global function fields. Fix a finite field k whose cardinality q is relatively rime to 2 and 3. Let C be a smooth, roer and geometrically irreducible curve of genus g over k and denote its function field by K. Let C be the set of closed oints of C. For each x C, let x be the residue field at x and let deg x be the degree of x over k. Each closed oint x C gives a discrete valuation v x : K { } and we denote by K x the corresonding local field. 5

6 Fix an ellitic curve E defined over K whose j-invariant j E is non-constant (i.e., j E K k). Let π: C be the Néron model of E/K, cf. [BLR90, 1.4]. Let U C be the dense oen subset comlementary to the finite number of oints of bad reduction for E. By abuse of notation, we let E U be the (relative) ellitic curve π 1 (U) π U; the fiber over the generic oint of U is E/K Kodaira symbols. For each closed oint x of C, we can assign a Kodaira symbol to the ellitic curve E after base extending by the local field K x ; it can be quickly comuted using Tate s algorithm, cf. [Sil94, IV 9] or [Tat75]. The ossible Kodaira symbols are the following: I n (n 0), I n (n 0), II, III, IV, IV, III, II. Let Kod(E) be the multiset consisting of the Kodaira symbols of E at the oints x C for which E has bad reduction; we count a Kodaira symbol at x with multilicity deg x. Note that the multiset Kod(E) does not change if we relace E by its base extension to the function field of C k for any finite extension k /k Some invariants. We now define some numerical invariants of the curve E. For each x C, we define integers f x (E), e x (E), γ x (E), λ x (E), r x (E) and b x (E) as in the following table: Kodaira symbol at x I 0 I 0 I n (n 1) II III IV I n (n 1) IV III II f x e x 0 6 n n γ x 1 1 n/ gcd(2, n) / gcd(2, n) λ x 1 1 n n r x b x Define N E 4 + 4g + f x (E) deg(x), χ E 1 e x (E) deg x, and γ E γ x (E) deg x. 12 x C Define E to be the roduct of the rimes l 5 that divide λ x (E) for some x C ; it is also the roduct of rimes l 5 that divide max{1, v x (j E )} for some x C. Remark 2.1. x C (i) The integers N E, χ E, γ E and E can all be determined directly from the multiset Kod(E). (ii) One way to rove that χ E is an integer is to show that it agrees with the Euler characteristic of the sheaf X, where X C is a relatively minimal ellitic surface that extends E U with X smooth and rojective. Let x / x be the fiber of the Néron model C of E/K at x. We define c x (E) to be the order of the grou x ( x )/ x ( x), where x is the identity comonent of the grou scheme x. Define the integer c E c x (E); x C it is well-defined since c x (E) 1 whenever E has good reduction at x. The integer c E serves as a fudge factor in the Birch and Swinnerton-Dyer conjecture, cf. 2.4, and can be quickly comuted using Tate s algorithm. Let A E be the set of closed oints x of C for which E has bad reduction of additive tye. For each x A E, let χ x : x {±1} be the non-trivial quadratic character (recall that q is odd). Let m + be the number of closed oints x of C for which E has slit multilicative reduction. Define ɛ E : ( 1) m + χ x A x ( r x (E)) {±1}; E it is the root number of E, cf. Theorem x C

7 2.3. L-functions. Take any closed oint x of C. If E has good reduction at x, define the olynomial L x (T) : 1 a x (E)T deg x + q deg x T 2 deg x, where E x / x is the fiber of E over x and a x (E) : q deg x + 1 E x ( x ). If E has bad reduction at x, define a x (E) to be 1, 1 or 0 when E has slit multilicative, non-slit multilicative or additive reduction, resectively, at x; define the olynomial L x (T) 1 a x (E)T deg x. The L-function of E is the formal ower series L(T, E) : L x C x(t) 1 [[T]]. The following gives some fundamental roerties of L(T, E); we will give a sketch in 2.7. Theorem 2.2. The L-function L(T, E) is a olynomial of degree N E with integer coefficients and satisfies the functional equation T N E L(T 1 /q, E) ɛ E L(T/q, E). If q is a ower of 2 or 3, then Theorem 2.2 will hold excet the numerical recies for N E and ɛ E need to be refined. One can also consider the case where E has constant j-invariant; L(T, E) is still a rational function but is no longer a olynomial. Another imortant roerty of L(T, E), which we will not require, is that all its comlex roots have absolute value q 1. This will follow from the cohomological interretation and the work of Deligne The Birch and Swinnerton-Dyer conjecture. The Mordell-Weil theorem for E says that the abelian grou E(K) is finitely generated. It is straightforward to comute the torsion subgrou of E(K) but comuting its rank is more difficult. Before stating the conjecture, we mention several invariants of E: Let X E be the Tate-Shafarevich grou of E/K. Let E be the minimal discriminant of E/K; we may view it as a divisor of C. Using Tate s algorithm (and our ongoing assumtion that q is not a ower of 2 or 3), we find that the degree of E is 12χ E. The integer c E from 2.2. Let E(K) tors be the torsion subgrou of E(K). The regulator of E is the real number R E : det( P i, P j ), where, : E(K) E(K) is the canonical height airing and P 1,..., P r E(K) are oints that give a basis for the free abelian grou E(K)/E(K) tors. The following is a conjectural relation between the rank of the Mordell-Weil grou E(K) and its L- function. Conjecture 2.3 (Birch and Swinnerton-Dyer). Let r be the rank of E(K). (a) The rank r agrees with multilicity of 1/q as a root of L(T, E). (b) For some rime l, the l-rimary comonent of X E is finite. (c) The grou X E is finite and as s 1. L(q s X, E) E R E c E E(K) tors 2 q g 1+χ E (s 1) r A nice exosition of the conjecture, with the exlicit constant α, is given by Gross in [Gro11]; he also gives similar details for the more familiar number field analogue. Note that the regulator in [Gro11] is equal to R E / E(K) tors 2. Conjecture 2.3(c) clearly imlies the other two arts; they are in fact equivalent. Theorem 2.4 (Artin-Tate, Milne). Statements (a), (b) and (c) of Conjecture 2.3 are equivalent. 7

8 Proof. This follows from Theorem 8.1 of [Mil75]; it builds on the work of Artin and Tate resented in Tate s 1966 Bourbaki seminar [Tat66]. It should be noted that the L-functions in [Tat66] do not include the now familiar factors at the bad laces; those were later worked out by Tate in [Tat75, 5]. The following gives an a riori inequality between the two quantities in Conjecture 2.3(a); it follows from the injectivity of the homomorhism h of Theorem 5.2 of [Tat66]. Proosition 2.5. The rank of E(K) is always less than or equal to the multilicity of 1/q as a root of L(T, E). In this aer, we will use only the following secial consequence of the above results. Corollary 2.6. Suose that L(1/q, E) 0. Then L(1/q, E) q g 1+χ E c E ( ) 2. Proof. Let r be the rank of E(K). Proosition 2.5 and our assumtion L(1/q, E) 0 imlies that r 0. Therefore, r 0, i.e., E(K) is finite. Since Conjecture 2.3(a) holds, Theorem 2.4 imlies that the grou X E is finite and that X L(1/q, E) E c E E(K) tors 2 q g 1+χ E (we have R E 1 since r 0). Therefore, L(1/q, E) X E q g 1+χE c E ( ) 2. Cassels constructed an alternating and non-degenerate airing X E X E /, cf. [Mil06, Theorem 6.13], from which one can deduce that X E has square cardinality (we are using of course that X E is finite in our case). The result is then immediate L-functions modulo l. Take any rime l 6q E. Let E[l] be the l-torsion subscheme of E; it is a sheaf of l -modules on U that is free of rank 2. The lisse sheaf E[l] corresonds to a reresentation ρ E,l : π 1 (U, η) Aut l (E[l] η ) GL 2 ( l ), where η is a geometric generic oint of U. The Weil airing E[l] E[l] l (1) is non-degenerate and alternating, so ρ E,l (π 1 (U k )) SL 2 ( l ). The following big monodromy result will be roved in 2.6. Proosition 2.7. We have ρ E,l (π 1 (U k )) SL 2 ( l ). Pushing forward, we obtain a sheaf j (E[l]) of l -modules on C, where j : U C is the inclusion morhism. Define the l -vector sace V E,l : H 1 (C k, j (E[l])). The Weil airing E[l] E[l] l (1) is non-degenerate and alternating, and gives rise to an isomorhism E[l] (1) E[l] of sheaves. Using this isomorhism and Poincaré duality (for examle, as in [Mil80, V Proosition 2.2(b)]), we obtain a non-degenerate and symmetric airing, : V E,l V E,l H 2 (C k, l (1)) l. Therefore, V E,l with the airing, is an orthogonal sace over l. There is a natural action of Gal(k/k) on the vector sace V E,l. The Galois action on V E,l resects the airing and hence gives rise to a reresentation θ E,l : Gal(k/k) O(V E,l ). Let Frob q Gal(k/k) be the geometric Frobenius (i.e., the inverse of x x q ). The following says that we can recover L(T, E) modulo l from the characteristic olynomial of θ E,l (Frob q ). 8

9 Proosition 2.8. The vector sace V E,l has dimension N E over l and det(i θ E,l (Frob q )T) L(T/q, E) (mod l). In many cases, the following roosition gives a way to comute the determinant and the sinor norm of θ E,l (Frob q ) in terms of some of the invariants of E. Our assumtion l 6q E ensures that l does not divide the integer 2 N E q g 1+χ E c E γ E (if c x (E) is divisible by a rime 5 for some x C, then Tate s algorithm shows that E must have Kodaira symbol I n at x where n c x (E), and hence divides E ). Proosition 2.9. (i) We have det(θ E,l (Frob q )) ( 1) N E ɛ E. (ii) If det(i θ E,l (Frob q )) 0, then s( θ E,l (Frob q )) 2 N E q g 1+χ E c E ( l )2. (iii) If det(i + θ E,l (Frob q )) 0, then s(θ E,l (Frob q )) 2 N E q g 1+χ E c E γ E ( l )2. (iv) If det(i ± θ E,l (Frob q )) 0, then disc(v E,l ) γ E ( l )2. The roofs of Proositions 2.8 and 2.9 will be given in 2.7 and 2.8, resectively. Remark Let [l] be the l-torsion subscheme of the Néron model C; it is a sheaf of l -modules on C. For any non-emty oen subvariety U of C, one can show that [l] is canonically isomorhic to j j ( [l]), where j : U C is the inclusion morhism. In articular with U U, we find that V E,l H 1 (C k, [l]). If U U, then we have V E,l H 1 (C k, j (E[l] U )). Remark Let X C be a relatively minimal morhism extending E U, where X is a smooth and rojective surface over k. One can give a filtration of H 2 (X, l (1)) as an l [Gal(k/k)]-module such that one of the quotients is V E,l (and the cu airing on H 2 induces our airing on V E,l ). Similar remarks hold for the more general constructions of Proof of Proosition 2.7. Take any roer subgrou H of SL 2 ( /l ). For a fixed algebraically closed field F whose characteristic is not l, let X (l) be the modular curve over F arametrizing ellitic curves with level l-structure; it is smooth and rojective. There is a natural action of SL 2 ( /l ) on X (l). Define the curve X H X (l)/h and let π H : X H X (l)/ SL 2 ( /l ) 1 F be the morhism down to the j-line. Let m be the least common multile of the order of the oles of π H. We claim that m l. There is a model of the modular curve X H over Sec [1/l] such that the the divisor consisting of cuss is étale over Sec [1/l], for background see 3 in art IV of [DR73]. The integer m is thus indeendent of F, so we may take F. Let Γ be the congruence subgrou consisting of matrices A SL 2 ( /l ) such that A modulo l belongs to H. The ma π H : X H ( ) 1 ( ) of comact Riemann surfaces comes from comactifying the natural quotient ma h/γ h/ SL 2 ( ), where SL 2 ( ) acts on the uer-half lane h via linear fractional transformations. Therefore, m is equal to the least common multile of the width of the cuss of Γ. Since Γ has level l, we have m l (in [Woh64], the quantity m is called the general level of Γ and it is shown to agree with the usual level). We now focus on the case F k. Suose that H ρ E,l (π 1 (U k )) is a roer subgrou of SL 2 ( /l ). Let J : U k 1 be the morhism given by the j-invariant of E; it is dominant since E is non-isotrivial. k Since ρ E,l (π 1 (U k )) H, the morhism J factors as π H U k X H 1. k Let m be the least common multile of the order of the oles of the morhism C k 1 extending J. k The integer m is divisible by m; the least common multile of the order of the oles of π H. By our claim, m is divisible by l. However, l dividing m imlies that there is a closed oint x of C such that v x (j E ) is negative and divisible by our rime l 5; this in turn imlies that l divides E. This contradicts our ongoing assumtion that l E, so H SL 2 ( /l ) as desired. 9

10 2.7. Proof of Proosition 2.8 and Theorem 2.2. We first recall a cohomological descrition of L(T, E). For each integer n 1, let E[l n ] be the l n -torsion subscheme of E; it is a lisse sheaf of /l n -modules on U that is free of rank 2. The sheaves {E[l n ]} n 1 with the multilication by l morhism E[l n+1 ] E[l n ] form a lisse sheaf of l -modules on U which we denote by T l (E). Define the l -sheaf : j (T l (E)) l l, where j : U C is the inclusion morhism, and let be its dual. Take any closed oint x of C and let x be a geometric oint of C maing to x arising from a choice of algebraic closure x of x. The geometric Frobenius Frob x Gal( x / x ) acts on the fibers x and. One can show that x det(i Frob x T deg x x ) L x(t). The Weil airings give an isomorhism ( 1), and hence det(i Frob x T deg x x ) L x (T/q). Therefore, L(T/q, E) det(i Frob x T deg x x ) 1. x C By the Grothendieck-Lefschetz trace formula, we have L(T/q, E) det I Frob i q T H i (C k, ) ( 1)i+1. Lemma 2.12(ii) below then shows that L(T/q, E) is equal to the olynomial det(i Frob q T M l l ), where M is the l -module H 1 (C k, j (T l (E))). That the olynomial L(T, E) has integer coefficients is clear from its ower series definition. Lemma Take any integer i 1. (i) We have H i (C k, j (E[l n ])) 0 for all n 1. (ii) We have H i (C k, j (T l (E))) 0 and H i (C k, ) 0. Proof. The lisse sheaf E[l n ] corresonds to a reresentation ρ E,l n : π 1 (U, η) Aut /l n (E[l n ] η ) GL 2 ( /l n ), where η is a geometric generic oint of U. The Weil airing on E[l n ] is non-degenerate and alternating, so H : ρ E,l n(π 1 (U k )) SL 2 ( /l n ). Proosition 2.7 imlies that the image of H modulo l is SL 2 ( /l ). We thus have H SL 2 ( /l n ) since SL 2 ( /l n ) has no roer subgrous whose image modulo l is SL 2 ( /l ), cf. Lemma 2 on age IV-23 of [Ser68]. We now rove (i). Since C has dimension 1, we need only consider i {0, 2}. The Weil airing on E[l n ] gives rise to an isomorhism E[l n ] (1) E[l n ] of sheaves on U. Using this isomorhism and Poincaré duality (for examle, as in [Mil80, V Proosition 2.2(b)]), we obtain a non-degenerate airing H 0 (C k, j (E[l n ])) H 2 (C k, j (E[l n ]))) /l n. So we may assume that i 0. We have H 0 (C k, j (E[l n ])) H 0 (U q, E[l n ]) (E[l n ] η ) π 1(U k,η) 0, where the last equality uses that ρ E,l n(π 1 (U k )) SL 2 ( /l n ). We have a short exact sequence 0 T l (E) l T l (E) E[l] 0 of sheaves on U. Pushing forward, we have a short exact sequence of sheaves on C which gives an exact sequence 0 j (T l (E)) l j (T l (E)) j (E[l]) 0 (2.1) 0 H 0 (C k, j (E[l])) M l M V E,l H 2 (C k, j (T l (E))) 0, 10

11 where we have used Lemma 2.12 for the H 0 and H 2 terms. From (2.1), the finitely generated l -module M has trivial l-torsion and is thus a free l -module of finite rank. From (2.1), we have an isomorhism of M/lM and V E,l that resects the action of Frob q. Therefore, L(T/q, E) det(i Frob q T M l l ) is congruent modulo l to det(i Frob q T V E,l ). To comlete the roof of Proosition 2.8, it remains to show that V E,l has dimension N E over l. As a consequence, we will deduce that L(T, E) has degree N E. Define χ l i ( 1)i dim l H i (C k, j (E[l])). By [Mil80, V Theorem 2.12], we have χ l (2 2g) dim l j (E[l]) η f x x 4 4g f x x, where the sums are over the closed oints of C k and f x is the (exonent of the) conductor of the sheaf j (E[l]) at x. Since the sheaf j (E[l]) is tamely ramified (q is not a ower of 2 or 3), we have f x dim l j (E[l]) η dim l j (E[l]) x 2 dim j (E[l]) x. In articular, f x is 0, 1, or 2 if E has good, multilicative or additive reduction, resectively, at x. The sum of the f x over the closed oints x of C k is equal to x C f x(e) deg x. Therefore, χ l equals N E. Using Lemma 2.12(i), we deduce that V E,l H 1 (C k, j (E[l])) has dimension χ l N E over l. Let us rove the functional equation for L(T, E) using what we have already roved. Take any rime l 6q E. We have shown that L(T/q, E) det(i AT) (mod l) for some A O(V E,l ). We have T N E det(i AT 1 ) ± det(i AT) for every A O(V E,l ), so T N E L(T 1 /q, E) ±L(T, E) (mod l). Since this holds for all but finitely many rimes l, we must have T N E L(T 1 /q, E) ɛl(t/q, E) for a unique ɛ {±1}. We can exress ɛ as a roduct of local root numbers ɛ x (E) over the closed oints x of C; note that ɛ x (E) 1 if E has good reduction at x. Fix a closed oint x of C for which E has bad reduction and let κ be the Kodaira symbol of E at x. If κ is not of the form I n or I n with n > 0, then ɛ x(e) χ x ( r x (E)) by Theorem 3.1 of [CCH05]; this uses the ongoing assumtion that gcd(q, 6) 1 and also that the e of loc. cit. is 12/ gcd(e x (E), 12). If κ is of the form I n for some n 0, then ɛ x(e) χ x ( r x (E)) by Theorem 3.1(2) of [CCH05]. If κ is of the form I n for some n > 0, then ɛ x (E) is 1 or 1 when E has slit or non-slit multilicative reduction, resectively, at x; cf. Theorem 3.1(2) and Lemma 2.2 of [CCH05]. This shows that ɛ agrees with our value ɛ E. This comletes the roof of Theorem Proof of Proosition 2.9. To comute sinor norms, we will use the following result of Zassenhaus. Lemma Let V be an orthogonal sace of dimension N defined over a finite field of odd characteristic. If B O(V ) satisfies det(i + B) 0, then s(b) 2 N det(i + B) ( ) 2. Proof. This is a secial case of Zassenhaus formula for the sinor norm in 2 of [Zas62]; see Theorem C.5.7 of [Con14] for a modern roof. Set A : θ E,l (Frob q ). By Proosition 2.8, the vector sace V E,l has dimension N E and we have (2.2) det(i AT) L(T/q, E) (mod l). Since A belongs to O(V l ), we find that T N E det(i AT 1 ) det( A) det(i AT). By (2.2) and the functional equation in Theorem 2.2, we also have T N E det(i AT 1 ) ɛ E det(i AT). Comaring these two equations, we deduce that det( A) ( 1) N E det(a) agrees with ɛ E. This roves art (i). Now suose that det(i A) 0. Since det(i + ( A)) 0, Lemma 2.13 and (2.2) give us s( A) 2 N E det(i A) ( l )2 2 N E L(1/q, E) ( l )2. Therefore, s( A) 2 N E q g 1+χ E c E ( l )2 by Corollary 2.6 (as noted in 2.5, l q g 1+χ E c E ). This roves (ii). Before roving arts (iii) and (iv), we need the following lemma. 11

12 Lemma Let E /K be the quadratic twist of E/K by a non-square β in k. Take any closed oint x of C. (i) The curves E and E have the same Kodaira symbol at x. (ii) We have a x (E ) ( 1) deg x a x (E). (iii) The integer c x (E)c x (E )γ x (E) deg x is a square. Proof. First suose that deg x is even. Since β is a square in x, we find that E and E are isomorhic over K x. All the arts of the lemma are now immediate. Now suose that deg x is odd. Let x be the valuation ring of K x and let π be a uniformizer. Tate s algorithm, as resented in [Sil94, IV 9] or [Tat75], starts with a Weierstrass equation (2.3) y 2 + a 1 xy + a 3 y x 3 + a 2 x 2 + a 4 x + a 6 for E over the local field K x with a i x. The algorithm then changes coordinates several times which imoses various conditions on the owers of π dividing the coefficients a i ; these conditions in loc. cit. are boxed (similar remarks hold for Subrocedure 7). By comleting the square in (2.3), we find that this ellitic curve is isomorhic to the one defined by the equation (2.4) y 2 x 3 + a 2 x2 + a 4 x + a 6, with a 2 a 2 + a 2 1 /4, a 4 a 4 + a 1 a 3 /2 and a 6 a 6 + a 2 3 /4. Using that q is odd, we find that a 2, a 4 and a 6 belong to x and that the conditions in Tate s algorithm are reserved. So, we may thus always assume in any alication of Tate s algorithm that we have a Weierstrass equation of the form (2.4). If E over K x is given by (2.4), then E over K x has a Weierstrass equation y 2 x 3 + a 2 βx2 + a 4 β 2 x + β 3 a 6. Alying Tate s algorithm, it is now easy to see that E and E have the same Kodaira symbol and to determine the ossibilities for c x (E)c x (E ). Let κ be the Kodaira symbol of E and E at x. If κ {I 0, II, II }, then c x (E)c x (E ) 1. If κ I n with n > 0, then c x (E)c x (E ) gcd(2n) n (recisely one of curves E and E has slit reduction at x; this uses that β is a non-square in x since deg x is odd). If κ {III, III }, then c x (E)c x (E ) 2 2. If κ {IV, IV }, then c x (E)c x (E ) If κ I n with n odd, then c x (E)c x (E ) If κ I n with n > 0 even, then c x(e)c x (E ) {2 2, 4 2 }. Finally, if κ I 0, then c x (E)c x (E ) {1 2, 2 2, 4 2 }. In all these cases, we find that integer c x (E)c x (E )γ x (E) is a square. Since deg x is odd, we conclude that c x (E)c x (E )γ x (E) deg x c x (E)c x (E )γ x (E) (γ x (E) (deg x 1)/2 ) 2 is a square. It remains to verify that a x (E ) a x (E). This is immediate if E, and hence E, has additive reduction at x since a x (E ) a x (E) 0. If E, and hence E, has multilicative reduction at x, then one has slit reduction and the other non-slit reduction (since β is not a square in x ), so a x (E ) a x (E). Finally suose that E, and hence E, has good reduction at x. Fix a Weierstrass model y 2 f (x) for E x / x with a cubic f x [x]; the equation βy 2 f (x) is a model of E x / x. Take any a x. If f (a) is a non-zero square in x, then there are two oint in E x ( x ) with x-coordinate a. If f (a) is a non-square in x, then there are two oint on E x ( x) with x-coordinate a. If f (a) 0, then E x ( x ) and E x ( x) both have one oint with x-coordinate a. Remembering the oints at infinite, we find that E x ( x ) + E x ( x) 2q deg x + 2 and hence a x (E ) a x (E). Now suose that det(i + A) 0. By Lemma 2.13 and (2.2), we have s(a) 2 N E det(i + A) ( l )2 2 N E L( 1/q, E) ( l )2. Let E be an ellitic curve over K that is a quadratic twist of E by a non-square in k. Lemma 2.14(ii) imlies that L(T, E ) L( T, E) and hence L(1/q, E ) L( 1/q, E). By Corollary 2.6, we deduce that L( 1/q, E) q g 1+χ E c E ( ) 2. We have χ E χ E since E and E have the same Kodaira symbols by Lemma 2.14(i). By Lemma 2.14(iii), the integer c E c E γ E x C c x(e)c x (E )γ x (E) deg x is a square. 12

13 Therefore, L( 1/q, E) q g 1+χ E c E γ E ( ) 2. Since l q g 1+χ E c E γ E, we conclude that s(a) 2 N E q g 1+χ E c E γ E ( l )2. This comletes the roof of (ii). Finally, suose that det(i ± A) 0. By (ii) and (iii), we have s( I) s( A) s(a) (2 N E q g 1+χ E c E ) (2 N E q g 1+χ E c E γ E ) ( l )2 γ E ( l )2. This roves art (iv) since disc(v E,l ) s( I). 3. FAMILIES OF QUADRATIC TWISTS 3.1. Setu. Let R be either a finite field whose characteristic is greater than 3 or a ring of the form [S 1 ] with S a finite set of rimes containing 2 and 3. Fix a Weierstrass equation (3.1) y 2 x 3 + a 2 (t)x 2 + a 4 (t)x + a 6 (t) with a i R[t] such that its discriminant R[t] is non-zero. Assume that the j-invariant J(t) of the ellitic curve over F(t) defined by (3.1), where F is the quotient field of R, has non-constant j-invariant. When R [S 1 ], we will allow ourselves to reeatedly enlarge the finite set S so that various roerties hold. For examle if R has characteristic 0, we shall assume that (t) 0 (mod ) for all rimes / S. We now consider quadratic twists by degree 1 olynomials. Define the R-scheme M Sec R[u, (u) 1 ]. Let k be any finite field that is an R-algebra (i.e., a finite extension of the field R or a finite field whose characteristic does not lie in S). Take any m M(k), i.e., an element m k with (m) 0, and let E m be the ellitic curve over k(t) defined by the Weierstrass equation (3.2) (t m)y 2 x 3 + a 2 (t)x 2 + a 4 (t)x + a 6 (t). We will rove the following in 3.5 Lemma 3.1. After ossibly increasing the finite set S when R [S 1 ], the multiset Kod(E m ) and the Kodaira symbol of E m at are indeendent of the choice of k and m. After ossibly increasing the set S when R has characteristic 0, we shall assume that the conclusions of Lemma 3.1 hold. Let Φ be the common multiset of Kodaira symbols from Lemma 3.1; the assumtion that J(t) is non-constant ensures that Φ is non-emty. Let κ be the common Kodaira symbol at of Lemma 3.1. The integers N Em, χ Em, Em and γ Em can be determined directly from Kod(E m ) Φ, so they are indeendent of of k and m; denote their common values by N, χ, and γ, resectively. Define the integer B Em : x b x(e m ) deg x, where the sum is over the closed oints of 1 k Sec k[t] and the b x (E m ) are defined in 2.2. Since B Em can be determined directly from Φ and κ, we find that it is indeendent of k and m; denote this common integer by B. Finally, take any rime l 6 that is not the characteristic of R. If R has characteristic 0, we relace S by S {l} Main reresentation. Fix notation and assumtions as in 3.1. The goal of this section is to rove the following roosition which give a reresentation of the étale fundamental grou of M that encodes the L-functions of the various quadratic twists E m. 13

14 Proosition 3.2. After ossibly relacing S by a larger finite set of rimes when R has characteristic 0, there is an N-dimensional orthogonal sace V l over l and a continuous reresentation θ l : π 1 (M) O(V l ) such that for any R-algebra k that is a finite field of order q and any m M(k), the following hold: (a) det(i θ l (Frob m )T) L(T/q, E m ) (mod l), (b) det(θ l (Frob m )) ( 1) N ɛ Em, (c) if det(i θ l (Frob m )) 0, then s( θ l (Frob m )) 2 N q 1+χ c Em ( l )2, (d) if det(i + θ l (Frob m )) 0, then s(θ l (Frob m )) 2 N q 1+χ c Em γ ( l )2, (e) if det(i ± θ l (Frob m )) 0, then disc(v l ) γ ( l )2. We now construct a lisse sheaf of l -modules that will give rise to the reresentation θ l of Proosition 3.2. We have already defined the R-scheme M Sec A, where A : R[u, (u) 1 ]. Set C 1 M ; it is a smooth roer curve of genus 0 over M that can be obtained by extending 1 M Sec A[t]. Define U Sec A[t, (t u) 1, (t) 1 ]; it is an oen M-subscheme of C. After ossibly enlarging the set S, we may assume that the closed subscheme D : C U of C is étale over M. The Weierstrass equation (t u)y 2 x 3 + a 2 (t)x 2 + a 4 (t)x + a 6 (t). defines an ellitic curve E U. Let : E[l] be the l-torsion subscheme of E. The morhism E U allows us to view as a lisse l -sheaf on U. Define the sheaf : R 1 π (j ( )) of l -modules on M, where j : U C is the inclusion morhism and π: C M is the structure morhism. The Weil airing gives an alternating airing l (1). The cu roduct and this airing on gives a symmetric airing of sheaves on M. Lemma 3.3. The sheaf is lisse. R 2 π (j ( ) j ( )) R 2 π (j ( l (1))) l Proof. Define π π j; it is the structure morhism U M. We can identify with a subsheaf of R 1 π ( ); for examle by using the low degree terms of the Leray sectral sequence. The homomorhism R 1 π! ( ) : R 1 π (j! ( )) induced by the inclusion j! ( ) j ( ) is surjective (this uses that D M has relative dimension 0). Therefore, is the image of a homomorhism R 1 π! ( ) R 1 π ( ). It thus suffices to rove that the sheaves R 1 π! ( ) and R 1 π ( ) are lisse. Using Poincaré duality, it suffices to rove that R 1 π! ( ) and R 1 π! ( ) are lisse. The sheaves R 1 π! ( ) and R 1 π! ( ) of l -modules are lisse by Corollaire of [Lau81]; the function ϕ of loc. cit. is constant since D M is étale and the sheaves and are tamely ramified (since 2 and 3 are invertible in A). Now take any m M(k), where k is a finite field of order q that is an R-algebra. Base changing by m, we obtain from E U M an oen subvariety U m of C m 1 k and an ellitic curve E m U m. The generic fiber of E m U m is an ellitic curve defined over k(t) given by the equation (3.2) with u substituted by m which, by abuse of notation, we have already denoted by E m. Let m be a geometric oint of M lying over m obtained from an algebraic closure k of k. Let m be the fiber of at m; it is an l -vector sace that comes with a symmetric airing, from secializing the airing on. The geometric Frobenius Frob m acts on m. By roer base change, we have (3.3) m H 1 (C m, j (E m[l])) H 1 (C k, j (E m[l])) V Em,l, 14

15 where j : U m C m is the inclusion morhism; the last equality uses Remark The induced airing on m agrees with the airing on V Em,l from 2.5. With resect to (3.3), the action of Frob m on m corresonds to the action of Frob q on V E,l. Let ξ be a geometric generic oint of M. Our airing on is non-degenerate since is lisse and it is non-degenerate on the fiber m. Denote by V l the fiber of at ξ with its airing; it is an orthogonal sace over l. The lisse sheaf thus gives rise to a continuous reresentation θ l : π 1 (M, ξ) O(V l ). With m M(k) above, we find that there is an isomorhism ϕ : V l VE,l of orthogonal saces such that ϕ 1 θ E,l (Frob q ) ϕ lies in the same conjugacy class of O(V l ) as θ l (Frob m ). All the roerties of θ l given in Proosition 3.2 are now direct consequences of Proositions 2.8 and Big monodromy. Fix notation and assumtions as in 3.1. After ossibly increasing S when R has characteristic 0, let θ l : π 1 (M) O(V l ) be the reresentation of Proosition 3.2. Let Φ be the multiset consisting of Φ with one symbol κ removed. Assume further that the following conditions hold: Φ contains I n for some n 1, Φ contains more than one I 0, 6B N. The following exlicit version of a theorem of Hall [Hal08] says that the image under θ l of the geometric fundamental grou is big. Theorem 3.4 (Hall). With assumtions as above, the grou θ l (π 1 (M F )) contains Ω(V l ) and is not a subgrou of SO(V l ), where F is an algebraic closure of the quotient field F of R. In 3.4, we will sketch some of the stes in Hall roof of Theorem 3.4. The main reason for doing this is to ensure that all the conditions are exlicit (in [Hal08], one is allowed to relace the original curve by a suitably high degree twist so that the last two conditions before the statement of the theorem hold). We will also need to refer to some of the details when handling the n 5 case of Theorem Sketch of Theorem 3.4. First suose that R [S 1 ] and let be the lisse sheaf on M from 3.2. Take any / S and let θ,l : π 1 (M ) O(V l ) be the reresentation obtained by secializing θ l, equivalently, at the fiber of M above. Since the formation of commutes with arbitrary base change, the reresentation θ,l agrees with the reresentation arising from the setu with 3.1 by starting with the same Weierstrass equation excet relacing R by. For / S sufficiently large, all the conditions of Theorem 3.4 hold. Since θ,l (π 1 (M )) agrees with θ l (π 1 (M )) for all sufficiently large, it thus suffices to rove the theorem in the case where R is a finite field. Now assume that R is a finite field k whose characteristic is at least 5. We now describe the setu and key results of Hall from 6 of [Hal08]. Set C 1 k and denote its function field by K : k(t). Let E 1 be the ellitic curve over K defined by (3.1). For each non-zero olynomial f k[t], let E f be the ellitic curve over K obtained by taking the quadratic twist of E 1 by f. Warning: we are following Hall s notation throughout 3.4; the curve E t m is denoted elsewhere in the aer by E m. We have l 5 and l is invertible in k. The j-invariant j E1 K of E 1 is the same as the j-invariant of each E f. Therefore, E f is indeendent of f and hence agrees with. In articular, our assumtion l 6 imlies that l does not divide max{1, v x (j E1 )} for all x C. In [Hal08, 6], it is also assumed 15

16 that l is chosen so that the Galois grou of the extension K(E 1 [l])/k contains a subgrou isomorhic to SL 2 ( l ); however, this is a consequence of l 6q and Proosition 2.7. Let f C be the Néron model of E f /K and let f [l] be its l-torsion subscheme. We have V E f,l H 1 (C k, f [l]) by Remark 2.10 For each integer d 0, we let F d be the oen subvariety of d+1 consisting of tules (a k 0,..., a d ) for which the olynomial d i0 a i t i is searable of degree d and relatively rime to (t) k[t]. For each extension k /k, we will identify each oint f F d (k ) with the corresonding degree d olynomial in k [t]. Now assume that d 1. As noted in [Hal08], there is an orthogonally self-dual lisse sheaf d,l F d of l -modules such that for any finite extension k k of k and any f F d (k ), the (geometric) fiber of d,l above f is H 1 (C k, f [l]) V E f,l. Moreover, the airing on d,l agrees with the airing from 2.5 on the fibers V E f,l. Fix a olynomial g F d 1 (k). Let U g be the oen subvariety of 1 consisting of c for which k (c)g(c) 0. We view U g as a closed subvariety of F d via the closed embedding ϕ : U g F d, c (c t)g(t). We then have an orthogonally self-dual lisse sheaf ϕ ( d,l ) of l -modules on U g. In 6.3 of [Hal08], it is noted that ϕ ( d,l ) over U g,k agrees with the middle convolution sheaf MC 1 ( g [l]); this has the consequence that the sheaf ϕ ( d,l ) is geometrically irreducible and tame. We now focus on the case with d 1 and g 1. The variety U g in 1 Sec k[u] is equal to k M Sec k[u, (u) 1 ]. For each finite extension k /k and m M(k ), the (geometric) fiber of ϕ ( d,l ) above m is H 1 (C k, t m [l]) V Et m,l (which is V Em,l in the notation of 3.1). We find that the sheaf ϕ ( d,l ) over M U g is recisely our sheaf from 3.2 and they have the same airing. We record the follow consequence for θ l. Lemma 3.5. The reresentations θ l : π 1 (M) O(V l ) is geometrically irreducible and tame. Since θ l is tamely ramified, its restriction to π 1 (M k ) factors through the maximal tame quotient π t 1 (M k ) of π 1(M k ). Let Z be the set of k-oints of 1 M; it consists of the c k for which (c) 0. k For each oint c Z { }, let σ c be a generator of an inertia subgrou π t 1 (M ) at c. Choosing an k ordering of the oints Z { }, we may assume that the σ c are taken so that the roduct of the σ c, with resect to the ordering, is trivial. The grou π t 1 (M k ) is (toological) generated by {σ c : c Z}; we do not need σ since the roduct of the σ c is trivial. In articular, {θ l (σ c ) : c Z} generates the grou θ l (π 1 (M k )). We need two quick grou theory definitions. For each A O(V l ), we define dro(a) to be the codimension in V l of the subsace fixed by A. We say that an element A O(V l ) is an isotroic shear if it is non-trivial, uniotent and satisfies (A I) 2 0. Lemma 3.6. Fix a oint c Z and let κ be the Kodaira symbol of E 1 /K at t c. (i) If κ I 0, then θ l (σ c ) I. (ii) If κ I n for some n 1, then θ l (σ c ) is a reflection. (iii) If κ I 0, then θ l(σ c ) is an isotroic shear. (iv) We have dro(θ l (σ c )) 2. Proof. The is a consequence of Lemma 6.5 of [Hal08] and also its roof for art (iv). It is actually stated for E g E 1 in loc. cit., but E 1 and E 1 have the same Kodaira symbols). Remark 3.7. For later, we note that u to this oint we have not made use of the three additional assumtions from 3.3. The following grou theoretic result is a secial case of Theorem 3.1 of [Hal08] with r 2. 16