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Elliptic curves over the rational numbers with semi-abelian reduction and two-division points

We classify elliptic curves over the rationals whose N\'eron model over the integers is semi-abelian, with good reduction at p=2, and whose Mordell--Weil group contains an element of order two that stays non-trivial at p=2. Furthermore, we describe those curves where the element of order two is narrow, or where another element of order two exists, and also express our findings in terms of Deligne--Mumford stacks of pointed curves of genus one.


Introduction
Fontaine [4] proved that there are no abelian varieties A Q = 0 over the field of rationals that have good reduction everywhere. The case of elliptic curves E Q is due to Ogg [14], who attributed the result to Tate. His proof proceeds by showing that any Weierstraß equation y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 with integral coefficients has discriminant ∆ = ±1. Ogg actually classified, up to isomorphism, those E Q where bad reduction occurs only at p = 2. It turns out that there are twelve cases, all of the form y 2 = x 3 + mx 2 + nx with discriminant ∆ = ±2 v , for certain coefficients −12 ≤ m, n ≤ 12 and exponents 6 ≤ v ≤ 15. One observes that there is always a 2-division point, with coordinates x = y = 0.
Similar questions where treated in many subsequent papers. We just mention a few: Ogg himself analyzed elliptic curves with discriminant ±2 ν · 3 and ±2 ν · 3 2 in [15]. The case ∆ = ±2 ν · p was studied by Ivorra [11], those with ∆ = p 1 p 2 by Sadek [17]. Ogg's results where extended to imaginary quadratic number fields by Setzer [20], Stroecker [21] and Kida [12]. Zhao [26] studied elliptic curves E K with good reduction everywhere over real quadratic number fields, and Takeshi [22] investigated the case of cubic number fields. The reduction behavior of an elliptic curve E Q with a 2-division point was studied by Hadano [7].
The goal of this paper is to classify elliptic curves E Q that have a 2-division point P ∈ MW(E/Z), and satisfy certain additional conditions. Here E → Spec(Z) is the Néron model, and MW(E/Z) = E(Z) = E(Q) is the Mordell-Weil group, which is a finitely generated abelian group. My personal motivation is to study the arithmetic of families of elliptic surfaces, in particular Enriques surfaces [18], but the results seem to be of independent interest. The additional conditions considered here are as follows: We stipulate that the relative group scheme E is semi-abelian, that good reduction occurs at p = 2, and that the 2-division point P ∈ E(Z) stays non-trivial in E(F 2 ).
It turns out that there are infinitely many isomorphism classes. We show that they correspond to pairs of integers (m, n) with n odd and gcd(4m + 1, n) = 1, via the Weierstraß equations y 2 + xy = x 3 + mx 2 + nx (see Theorem 2.1). We then describes the situation when there is further 2-division, which is our first main result: Theorem. (See Theorem 3.6) The following are equivalent: (i) There is another element Q = P of order two in MW(E/Z).
(iii) For every prime p > 0, the fiber E ⊗ F p has even Kodaira symbol.
I find it quit remarkable that the existence of Q = P can be seen as a condition on relative group schemes, or Kodaira symbols, or the arithmetic of the numbers m and n. If we further stipulate that P ⊂ E is narrow, that is, the section passes through the same irreducible components of the closed fibers as the zero-section O ⊂ E, the list becomes finite. The second main results of the paper is: Theorem. (See Theorem 5.1) Up to isomorphism, there are exactly two elliptic curves E Q such that its Néron model E has the following properties: (i) The structure morphism E → Spec(Z) is semi-abelian.
(ii) The closed fiber E ⊗ F 2 is an elliptic curve.
(iii) There is a narrow element P ∈ MW(E/Z) of order two whose image in E(F 2 ) is non-zero, and another element Q = P of order two. These curves are given by y 2 + xy = x 3 + 4nx 2 + nx for the coefficient n = ±1.
Taking the quotient by the subgroup scheme µ 2 ⊂ E yields two other elliptic curves, which must be of the form y 2 +xy = x 3 +m x 2 +n x. Somewhat surprisingly, they can be characterized via elements R ∈ MW(E/Z) of order four, which is the content of the third main result (see Theorem 5.2). We also reformulate some of our findings in terms of of Z-valued objects in the Deligne-Mumford stackM g,r of stable pointed curves of genus g = 1.
The paper is organized as follows: Section 1 contains basic facts on Néron models of elliptic curves, and elementary computations with Weierstraß equations. In Section 2, we show how certain geometric assumptions on Néron models lead to our Weierstraß equations. A detailed analysis of the resulting Weierstraß models is given in Section 3, which also contains the result on the existence of an additional element of order two. In Section 4 we apply Velu's formula to form some quotients by 2-division points. Section 5 contains our classification results that use additional narrowness assumptions. The final Section 6 gives reinterpretations in terms of Deligne-Mumford stacks.
Acknowledgement. This research was conducted in the framework of the research training group GRK 2240: Algebro-geometric Methods in Algebra, Arithmetic and Topology, which is funded by the Deutsche Forschungsgemeinschaft.

Preliminaries
Here we discuss basic facts on Néron models of elliptic curves and make some useful elementary computations with Weierstraß equations. Let R be a Dedekind ring, which we regard as a ground ring, F = Frac(R) its field of fractions, and (1) be an elliptic curve with coefficients a i ∈ F . The homogenization of the Weierstraß equation yields a closed embedding E F ⊂ P 2 F , where the origin O F ∈ E F corresponds to the point (0 : 1 : 0) ∈ P 2 (F ).
Let X be the resolution of singularities for the closure of the subset E F ⊂ P 2 R . The closure of the origin defines a section O ⊂ X , and the resulting homogeneous spectrum is called the Weierstraß model of the elliptic curve E Q . Note that it does not depend on the chosen Weierstraß equation (1). Each fiber of Y → Spec(R) is either an elliptic curves, or a twisted from of the rational nodal curve or the rational cuspidal curve. The locus of non-smoothness Sing(Y /R), which is defined by the first Fitting ideal for the sheaf Ω 1 Y /R (compare [3], Section 2), comprises finitely many closed points.
The minimal resolution of singularities X → Y is called the minimal model of the elliptic curve E F . The locus of smoothness E = Reg(X/R) is the Néron model of the elliptic curve E F . It has the property that the restriction map E(R) → E(Q) is bijective. The resulting abelian group is called the Mordell-Weil group. The Néron model E → Spec(R) acquires in a canonical way the structure of a relative group scheme. We write O ⊂ E for the zero-section. For each point a ∈ Spec(R), the fiber E a = E ⊗ κ(a) is an algebraic group scheme. We write E 0 a ⊂ E a for the connected component of the origin, and Φ a = E a /E 0 a for the resulting group scheme of components. If the fiber E a is an elliptic curve, one says that E Q has good reduction at a ∈ Spec(R). If E 0 a is a twisted form of G m ⊗ κ(a), one says that E Q has multiplicative reduction. One says that the Néron model E is semi-abelian if the elliptic curve E Q has good reduction or multiplicative reduction at all closed points.
An element P ∈ MW(E/R) is called narrow if its images in Φ a vanishes, for all closed points a ∈ Spec(R). In other words, the section P, O ⊂ E pass through the same irreducible components inside each closed fiber E a . The resulting subgroup There is an open covering D(f 1 )∪. . .∪D(f n ) of Spec(R) such that each restriction Y ⊗ R[1/f i ] is defined by some equation (1) with coefficients a i ∈ R[1/f i ] as a family of cubic curves in P 2 ⊗R[1/f i ]. In general, it is impossible to find a global embedding Y ⊂ P 2 . However, making a substitution x = u 2 x and y = u 3 y with u ∈ F × we obtain a Weierstraß equation (1) with coefficients a i ∈ R. Such a Weierstraß equation is called globally minimal if the resulting family of cubics is isomorphic to Y → Spec(R).
If the Dedekind ring R is local, that is, a discrete valuation ring, the Tate Algorithm [23] produces a minimal Weierstraß equation. Every round of the algorithm consists of twelve steps, each involving a change of coordinates x = u 2 x + r and y = u 3 y + su 2 x + t, with u ∈ R × and r, s, t ∈ R, to produces new equations. If the situation val(a i ) ≥ i arises, one replaces the coefficients by a i /π i , where π ∈ R is the uniformizer, and starts a new round. If not, the Weierstraß equation is minimal and indeed describes Y . It the ring R is factorial, one actually may run the Tate Algorithm locally and thus obtains a globally minimal Weierstraß equation. Proof. The problem is local, so we may assume that R is a discrete valuation ring, say with uniformizer π ∈ R. Seeking a contradiction, we assume that the Weierstraß equation is not minimal. Then there is a change of coordinates x = u 2 x + r, y = u 3 y +usx +t over the field F = Frac(R) such that the new Weierstraß equation has coefficients in R, with val(∆ ) = val(∆) − 12. Then val(c 4 ) = val(c 4 ) − 4. In particular ∆, c 4 ∈ m R , contradiction. Thus the equation is minimal. According to [2], Proposition 5.1 the closed fiber is elliptic or multiplicative. In other words, the Kodaira symbol is I v for some v ≥ 0. By the first step in the Tate Algorithm, we actually have v = val(∆).
We now state two observations for Weierstraß equations and the resulting families of cubics Y ⊂ P 2 that are valid over arbitrary ground rings R. Proof. It suffices to treat the case that the ground ring R = k is an algebraically closed field, say of characteristic p ≥ 0. According to [2], Proposition 5.1 the cubic Y ⊂ P 2 is the rational cuspidal curve if and only if the ∆ = c 4 = 0. Using for example [9], we compute for our Weierstraß equation the values (2) ∆ = a 2 4 ((4a 2 + 1) 2 − 64a 4 ) and c 4 = (4a 2 + 1) 2 − 48a 4 . These two equations combined give the relation (3) ∆ = a 2 4 (c 4 − 16a 4 ). Obviously, the vanishing of 4a 2 + 1 and a 4 implies the vanishing of ∆ and c 4 . Conversely, suppose that ∆ = c 4 = 0. We conclude a 4 = 0 in characteristic p = 2 from (3), and for p = 2 from the first equation in (2). The second equation finally gives 4a 2 + 1 = 0. Lemma 1.3. Let a 2 , a 4 , a 2 , a 4 ∈ R. Suppose the two Weierstraß equations y 2 + xy = x 3 + a 2 x 2 + a 4 x and y 2 + xy = x 3 + a 2 x 2 + a 4 x are related by a change of coordinates that respects P = (0, 0). Suppose the ring R has the property that 2 ∈ R is a regular element and R × ∩ (1 + 2R) = {±1}. Then the equations coincide, and the change of coordinates is either the identity or the sign involution x = x , y = −y − x .
Proof. Write x = u 2 x + r and y = u 3 y + su 2 x + t. Comparing coefficients for a 1 = a 1 = 1 gives u = 1 + 2s, which means u = s = −1 or u = 1, s = 0. Composing with the sign involution if necessary, we may assume that the latter holds. Since P = (0, 0) is fixed, we also have r = t = 0.

From geometry to equations
Let E Q be an elliptic curve over the field of rationals Q, and E → S be the Néron model over S = Spec(Z). Write O ⊂ E for the zero-section, X → S for the minimal model, and Y → S for the Weierstraß model. The goal of this section is to establish the following: Theorem 2.1. Suppose the elliptic curve E Q satisfies the following three conditions: There is an element P ∈ MW(E/Z) of order two whose image in the group E(F 2 ) remains non-zero.
Then there are integers m, n ∈ Z with n odd and gcd(4m + 1, n) = 1 such that the Weierstraß model is given by y 2 + xy = x 3 + mx 2 + nx, and the section P ⊂ Y is defined by the equations x = y = 0. Moreover, these integers m, n are unique, and we have P ∩ O = ∅.
The proof is given at the end of this section, after a few preparations. Since the ring Z is factorial, the Weierstraß model is a closed subscheme Y ⊂ P 2 , and the affine scheme Y O ⊂ A 2 is defined by a Weierstraß equation with coefficients a i ∈ Z. Any two such Weierstraß equations differ by a change of coordinates x = u 2 x + r, y = u 3 y + usx + t, with r, s, t ∈ Z and u = ±1, as described in [23], Section 2.
Lemma 2.2. If the closed fiber E ⊗ F 2 is an ordinary elliptic curve, there is a Weierstraß equation with a 1 = 1 and a 3 = 0.
Proof. Let us recall from [10], Chapter 3, §6 that up to isomorphism, there are five elliptic curves E 1 , . . . , E 5 over the prime field k = F 2 . The groups E i (k) are cyclic, and all 1 ≤ n ≤ 5 occur as orders. Let us choose indices in a natural fashion, so that |E i (k)| = i. To be explicit, we write which are supersingular and have invariant j = 0. We are more interested in the ordinary elliptic curves, which have j = 1 and are given by Since we are in characteristic p = 2 and the unit group F × 2 is trivial, the coefficient a 1 ∈ F 2 of the Weierstraß equations is invariant under every change of coordinates. For our Néron model E → Spec(Z) of the elliptic curve E Q , this means that a 1 ∈ Z is odd. A change of coordinates y = y + sx achieves a 1 = 1. A further change of coordinate y = y + t then yields a 3 = 0 as well. Proposition 2.3. If conditions (ii) and (iii) from Theorem 2.1 hold, one may choose a Weierstraß equation so that P ⊂ Y is given by P = (0, 0), and the coefficients satisfy a 1 = 1 and a 3 = a 6 = 0. Furthermore, a 4 ∈ Z then must be odd.
Proof. We saw in the previous proof that the supersingular elliptic curves over F 2 contain no element of order two, hence we may find a Weierstraß of the form y 2 +xy = x 3 + a 2 x 2 + a 4 x + a 6 . This form is preserved under each change of coordinates x = x − 2t, y = y + t. Since the subscheme P ⊂ Y is disjoint from the zerosection, it has coordinates P = (m, n) for some integers m, n ∈ Z. So after a change of coordinates, we may assume n = 0. The sign involution on E is given by −(x, y) = (x, −y − x). For the element P ∈ MW(E/Z) of order two, this means (m, 0) = (m, −m), thus m = 0. Summing up, x = y = 0 is a solution for our Weierstraß equation, and this means a 6 = 0. Seeking a contradiction, we suppose that a 4 is even. Then the fiber Y ⊗ F 2 is given by y 2 + xy = x 3 + x 2 or y 2 + xy = x 3 . Both curves are singular at the point (0, 0), contradiction.
Proposition 2.4. Suppose Y is given by a Weierstraß equation of the form y 2 +xy = x 3 + mx 2 + nx. Let p > 0 be a prime. Then the fiber E ⊗ F p is additive if and only if we have p | gcd(4m + 1, n).
Proof. Consider the ideal a = (4m+1, n). According to Lemma 1.2, the fiber E ⊗F p is additive if and only if a ⊂ (p), or equivalently p | gcd(4m + 1, n).
Proof of Theorem 2.1: Suppose we have an elliptic curve E Q whose Néron model E → Spec(R) satisfies conditions (i)-(iii). According to Proposition 2.3, we may choose a Weierstraß equation of the form y 2 + xy = x 3 + mx 2 + nx. In light of Proposition 2.4 we have gcd(4m + 1, n) = 1. The coefficients m, n ∈ Z are unique by Lemma 1.3. The Weierstraß model Y ⊂ P 2 is the zero-locus for the homogeneous equation Y 2 Z + XY Z = X 3 + mX 2 Z + nXZ 2 . The images of O and P on the Weierstraß model are given by the equations X = Z = 0 and X = Y = 0, which are clearly disjoint. Thus O, P ⊂ E are disjoint.

Analysis of the Weierstraß equation
Let m, n ∈ Z be two integers with n odd and gcd(4m+1, n) = 1. We now consider the family of cubics Y ⊂ P 2 defined by the Weierstraß equation  (5) is globally minimal, and the family of cubics Y ⊂ P 2 coincides with the Weierstraß model for the elliptic curve E Q .
Let E → Spec(Z) be the Néron model, with zero-section O ⊂ E. The section of the Weierstraß model Y given by x = y = 0 induces a section P ⊂ E. The following is an immediate converse for Theorem 2.1: Proposition 3.2. The relative group scheme E → Spec(Z) is semi-abelian, the element P ∈ MW(E/Z) has order two, and the section P ⊂ E is disjoint from the zero-section. The closed fiber E ⊗ F 2 is an ordinary elliptic curve, which contains two rational points if m is odd, and four rational points if m is even.
Proof. The sign involution on E is given by (x, y) → (x, −y − x), hence the element P ∈ MW(E/Z) has order two. Obviously, the images (0 : 1 : 0) and (0 : 0 : 1) on the Weierstraß model Y ⊂ P 2 of the sections O, P ⊂ E are disjoint, and the closed fiber E ⊗ F p belongs to the ordinary elliptic curves in (4). The Néron model E is semi-abelian by Proposition 2.4. Obviously, the close fiber E ⊗ F 2 occurs in our list of ordinary elliptic curves (4), and the assertion on E(F 2 ) is immediate.  (5) and the zero locus 2y + x = 0 and y = 3x 2 + 2mx + n for the partial derivatives. Hence Sing(Y /Z) is the spectrum of the ring (6) Z[1/2, x, y]/(2y + x, −8y 3 + (4m + 1)y 2 − 2ny, 12y 2 − (4m + 1)y + n). The intersection with the zero-locus for x = y = 0 is the spectrum of Z[1/2]/(n). This ring coincides with Z/nZ, because n is odd. The assertion follows.
As an immediate consequence, we get: Fix some prime p > 0. The fiber E ⊗ F p has Kodaira symbol I v for some integer v ≥ 0. Let us make some general observations on the multiplicative case v ≥ 1. The zero component G 0 of the algebraic group scheme G = E ⊗ F p is either the multiplicative group G m,Fp or its quadratic twist G m,Fp with respect the involution λ → 1/λ and the cyclic extension F p ⊂ F p 2 . In the letter case, the underlying scheme is isomorphic to P 1 Spec(F p 2 ), and one says that the fiber is twisted. For v = 1 this means that X ⊗ F p is obtained from the projective line P 1 by replacing an F p 2 -valued point by some rational point. Now suppose v ≥ 2, and decompose X ⊗ F p = C 0 ∪ C 1 ∪ . . . ∪ C w into irreducible components, with C 0 = P 1 ⊗ F p . In the untwisted case X ⊗ F p is actually a cycle of v projective lines. In the twisted case C 1 ∪. . .∪C w becomes after renumeration a chain of curves with function fields F p 2 (T ) or F p (T ). For v odd we have w = (v − 1)/2, the C 1 , . . . , C w−1 are projective lines over F p 2 , and the terminal component C w is obtained from its normalization P 1 ⊗ F p 2 by replacing an F p 2 -valued point by some F p -valued point. For v even we have w = (v − 2)/2 and as terminal component C w = P 1 ⊗ F p , otherwise the situation is the same. Note that the underlying scheme Let us now unravel what this concretely means for our Weierstraß equation (5). Here the Legendre symbol ( d p ) = ±1 is useful, which gives the class of of an integer d in F × p /F ×2 p , where p > 0 is an odd prime not dividing d.
Let us say that the fiber E ⊗ F p has even Kodaira symbol if the Kodaira symbol is I v for some even integer v ≥ 0. This means that either E ⊗ F p is an elliptic curve, or that the component group scheme Φ p has even order. For each integer r ≥ 1, write E[r] for the kernel of the multiplication map r : E → E, which is defined by Note that the formation of kernels commutes with base-change, because fiber products do so. Moreover, E[r] inherits the structure of relative group scheme, whose structure morphism is separated and of finite type. We now come to our first main result: Theorem 3.6. The following are equivalent: (i) There is another element Q = P of order two in MW(E/Z).
(iii) For every prime p > 0, the fiber E ⊗ F p has even Kodaira symbol.
In light of (2) the discriminant of our Weierstraß equation becomes (7) ∆ = d 2 (4m + 1 + 16d) 2 (4m + 1 + 32d) 2 , and we conclude with Proposition 3.5 that all closed fibers E ⊗F p have even Kodaira symbol. We now show (i)⇒(iv). Let Q = P be another element of order two. Replacing Q by Q + P if necessary, we may assume that its image in the group E(F 2 ) is non-zero. According to Theorem 2.1, the family of cubics Y ⊂ P 2 admits a description with another Weierstraß equation for some unique integers m , n ∈ Z with n odd and gcd(4m + 1, n ) = 1, where in the new coordinates the closed subscheme Q ⊂ Y is defined by x = y = 0. The two Weierstraß equations are related by a change of variables x = u 2 x + r, y = u 3 y + su 2 x + t with u = ±1 and r, s, t ∈ Z. Composing with the sign involution if necessary, we may assume that u = 1. We now use [23], equations (2.2) to compare coefficients. The condition a 1 = a 1 = 1 yields s = 0, and a 3 = a 3 = 0 gives r = −2t. In turn, we get n = a 4 = a 4 + 2ra 2 − t + 3r 2 = n − 4tm − t + 12t 2 .
Dividing by 4d = 0 yields the desired equation n + 4dm + 16d 2 + d = 0. The integer d must be odd, because n is odd.
We now come to the implication (iii)⇒(ii). To simplify notation, write G = E [2] and S = Spec(Z). Since E is semi-abelian, the multiplication map r : E → E is quasi-finite and flat for all integers r = 0, according to [1], Section 7.2, Lemma 2. Being a base-change, the structure morphism G → S is quasi-finite and flat. Each geometric fiber for E → S is either an elliptic curve, or G m × Z/vZ with an even integer v ≥ 1. It follows that the fibers G ⊗ F p are finite group schemes of constant order four. By Zariski's Main Theorem, G is an open subscheme of some finite Z-scheme W . Replacing W by the closure of G me may assume that G ⊂ W is schematically dense. Then the structure morphism W → S, which is finite and flat, has deg(W/S) = 4. Hence for each prime p > 0, the inclusion G ⊗ F p ⊂ W ⊗ F p is an equality. The Nakayama Lemma gives G = W , so the structure morphism G → Spec(Z) is finite and flat of degree four.
The section P ⊂ E yields a homomorphism Z/2Z → E[2] of group schemes over S. This is a monomorphism, because the intersection P ∩ O is empty. The quotient H exists as a group scheme over Z, its formation commutes with base-change, and the structure morphism H → S is free of rank two. By the Tate-Oort Classification ( [24], Corollary to Theorem 3), we either have H = µ 2 or H = Z/2Z. In the latter case, the group G ⊗ F 2 would beétale, contradicting the fact that multiplication by r = 2 on the elliptic curve E ⊗ F 2 is notétale.
Thus we have a short exact sequence 0 → Z/2Z → G → µ 2 → 0. The group scheme µ 2 is the spectrum of the ring Z[T ]/(T 2 −1), which sits in a cartesian diagram According to Minkowski's Theorem ( [13], Chapter III, Theorem 2.17), the scheme S = Spec(Z) is simply-connected, in the sense that its algebraic fundamental group π alg 1 (S, Ω) vanishes ( [6], Exposé V). The same holds for the connected union µ 2 = S ∪ S. In particular, theétale covering pr : G → µ 2 admits a section s : µ 2 → G, and we may assume that the restriction s ⊗ F 2 sends the neutral element to the neutral element. The expression s(xy)/s(x)s(y), where we write the group laws in multiplicative fashion, defines a morphism of schemes ψ : µ 2 → Z/2Z. It factors either over the zero-section or the non-zero section, because both µ 2 and S are connected. Base-changing to F 2 , we see that ψ must be the zero-map. In turn, the morphism s : µ 2 → G of schemes is a homomorphism of group schemes, thus splits the extension of group schemes. Summing up, we have E[2] = Z/2Z ⊕ µ 2 . It remains to verify the coordinates for the two-torsion sections, assuming that the equivalent conditions (i)-(iv) hold. By definition we have P = (0, 0). In light of the equation n = −d(4m + 1 + 16d) we make the change of coordinates x = x + 4d and y = y − 2d, which transforms the old Weierstraß equation into y 2 + x y = x 3 + m x 2 + n x , where (8) m = m + 12d and n = n + 8dm + 2d + 48d 2 = d(4m + 1 + 32d).
The two-torsion section x = y = 0 in the new Weierstraß equation corresponds to the two-torsion section Q ⊂ Y defined by x = 4d and y = −2d. In other words, Q = (4d, −2d). The coordinates for P + Q follow from the group law for elliptic curves.

Quotients by 2-division points
We keep the notation as in the previous section, and study the Néron model E for the Weierstraß equation y 2 + xy = x 3 + mx 2 + nx, where n = −d(4m + 1 + 16d) for some odd integer d with gcd(4m + 1, d) = 1. We saw in Theorem 3.6 that there are two elements P, Q ∈ MW(E/Z) of order two whose images in E(F 2 ) keep order two. Moreover, the sum P + Q generates a copy of the multiplicative group scheme µ 2 ⊂ E. The quotient E/µ 2 is a relative group scheme over the ring Z whose structure morphism is separated and of finite type. It comes with a canonical section of order two that is disjoint from the zero-section, because E[2]/µ 2 = Z/2Z. From this one may deduce that Theorem 2.1 applies to the generic fiber (E/µ 2 ) Q . In fact, it is possible to infer directly the equation for its Weierstraß model: The elliptic curve (E/µ 2 ) Q is given by the globally minimal Weierstraß equation Proof. Applying Velu's Formula [25] with the subgroup in (E/µ 2 ) Q generated by the 2-division point we get a Weierstraß equation y 2 + xy = x 3 + mx 2 + a 4 x 2 + a 6 with new coefficients A change of coordinates x = x −2t, y = y +t with t = − 1 2 m−4d− 1 8 transforms this into y 2 + xy = x 3 + (4m + 24d + 3 4 )x 2 + 16d 2 x. A further change of coordinates with x = x , y = y + 1 2 x gives y 2 + 2xy = x 3 + (4m + 24d)x 2 + 16d 2 x, which immediately leads to the desired Weierstraß equation.
The elements P, Q ∈ MW(E/Z) may pass through different components of the fibers E ⊗ F p . We may easily unravel the situation, by considering their difference, which is also their sum: Lemma 4.2. The element P +Q ∈ MW(E/Z) has non-trivial class in the component group scheme Φ p if and only if the prime p divides (4m + 1 + 16d)(4m + 1 + 32d).
The translation action of µ 2 on the Néron model E extends to an action of the minimal model X. The induced action on the fibers X ⊗ F p may or may not be free.
Proposition 4.3. The induced action of µ 2 ⊗ F 2 on the fiber X ⊗ F p is not free if and only if p | d. In this case, the locus of fixed points coincides with Sing(X ⊗ F p ). Moreover, the minimal resolution of singularities for the quotient scheme X/µ 2 is the minimal model of the elliptic curve (E/µ 2 ) Q .
Proof. The action is non-free if and only if E ⊗ F p is multiplicative, and P + Q ≡ O in the component group scheme Φ p . In this case, the µ 2 -action stabilizes every irreducible component and fixes precisely the singularities on X ⊗ F p . According to equations (7), the discriminant ∆ of our Weierstraß equation is the square of d(4m + 1 + 16d)(4m + 1 + 32d). Thus the fiber E ⊗ F p is multiplicative if and only if p divides this number. By Lemma 4.2, we have P + Q ≡ 0 in the component group scheme Φ p if and only if p does not divide (4m + 1 + 16d)(4m + 1 + 32d). The first assertion follows.
The fixed points a ∈ X for the µ 2 -action yield rational double points of type A 1 on the quotient scheme X/µ 2 . Let X → X/µ 2 be the resolution of singularities. The three schemes in X → X/µ 2 ← X are Gorenstein. The dualizing sheaf ω X/Z is trivial, hence the dualizing sheaves on X/µ 2 and X are numerically trivial. It follows that X → Spec(Z) contains no (−1)-curve, thus X is the minimal model of (E/µ 2 ) Q .

Classification of curves with additional sections
We now classify elliptic curves E Q whose Néron model E is semi-abelian and whose Mordell-Weil group MW(E/Z) has certain elements of order two. Our first result is: Theorem 5.1. Up to isomorphism, there are exactly two elliptic curves E Q such that its Néron model E has the following properties: (i) The structure morphism E → Spec(Z) is semi-abelian.
(ii) The closed fiber E ⊗ F 2 is an elliptic curve.
(iii) There is a narrow element P ∈ MW(E/Z) of order two whose images in E(F 2 ) are non-zero, and another element Q = P of order two. These elliptic curve are given by the global minimal Weierstraß equation The former has invariant j = 20346417/289 and the only singular fiber occurs at p = 17. The latter has j = 13997521/225, with singular fibers at p = 3 and p = 5. All these singular fibers have Kodaira symbol I 2 .
Proof. Suppose E Q satisfies the conditions (i)-(iii). According to Theorem 2.1, the Weierstraß model Y is given by the equation y 2 + xy = x 3 + mx 2 + nx where n is odd and gcd(4m + 1, n) = 1, and the element P ∈ MW(E/Z) is given by x = y = 0. Since this element is narrow, we must have n = ±1, by Corollary 3.4. Since there is another 2-division point Q = P , Theorem 3.6 tells us that n = −d(4m + 1 + 16d) for some integer d. We thus have d = ±1 and 4m + 16d = 0, thus m = −4d.
Conversely, we have to check that the two equations have the stated properties. The equations are relatively minimal by Proposition 3.1, and the Néron model E satisfies conditions (i)-(iii) according to Proposition 3.2. The description of the singular fibers follow from Proposition 3.5. One may compute their j-invariant with [9].
Let E be the Néron model for the Weierstraß equation y 2 +xy = x 3 +4nx 2 +nx with n = ±1 as in the preceding theorem. After replacing Q by Q + P , we may assume that both P, Q ∈ MW(E/Z) stay non-trivial in E(F 2 ). Consider the subgroup scheme µ 2 ⊂ E generated by the sum P + Q. The induced µ 2 -action on the minimal model X is free, according to Proposition 4.3, and the quotient X/µ 2 arises from the Weierstraß equation y 2 + xy = x 3 ± 2x 2 + x, by Proposition 4.1. It is somewhat surprising that these two curves can be characterize in terms of 4-division points: Theorem 5.2. Up to isomorphism, there are exactly two elliptic curves E Q such that its Néron model E has the following properties: The fiber E ⊗ F 2 is an ordinary elliptic curve.
(iii) There is a narrow element R ∈ MW(E/Z) of order four whose image in E(F 2 ) keeps order four. These elliptic curve are given by the global minimal Weierstraß equation The former has invariant j = 35937/17, and the only singular fiber appears at p = 17. The latter has j = −1/15, with singular fibers at p = 3 and p = 5. All these singular fibers have Kodaira symbol I 1 Proof. Suppose first that E Q satisfies conditions (i)-(iii). The element P = 2R in MW(E/Z) is narrow, has order two, and remains non-zero in E(F 2 ). Consider the group G = Z/2Z. It acts via translation by P on the Néron model E, with induced actions on the minimal model X and the Weierstraß model Y . The G-action on is free on the open subset E ⊂ X, and fixes the complement Sing(X/Z) = X E. This complement is finite, because E → Spec(Z) is semi-abelian, and disjoint from the fiber X ⊗ F 2 , because E ⊗ F 2 is an elliptic curve. It follows that for each point a ∈ Sing(X/Z), the image b ∈ X/G yields a rational double point of type A 1 . Let X → X/G be the minimal resolution of singularities. As in the proof for Proposition 4.3, one sees that X is the minimal model of the elliptic curve (E/G) Q . Write E for its Néron model.
We see that E → Spec(Z) is semi-abelian, and for each closed fiber E ⊗ F p with Kodaira symbol I v , the closed fiber E ⊗ F p has Kodaira symbol I 2v . Moreover, the image P ⊂ E of R ⊂ E, which is the quotient of the closed subscheme R∪(R+P ) ⊂ E by the G-action, defines a narrow element P ∈ MW(E /Z) of order two whose image in E (F 2 ) is non-zero. According to Theorem 5.1, we may assume that E is given by a Weierstraß equations of the form for some sign n = ±1, such that P ⊂ E is given by the equations x = y = 0. According to Theorem 3.6, we have another element Q ∈ MW(E /Z) of order two that is disjoint from the zero-section, namely Q = (−4d, 2d) with the value d = −n. In turn, P + Q is the third element of order two, which must hit the zero-section, and generates a subgroup scheme µ 2 ⊂ E . Since R ⊂ E is disjoint from the zerosection, we infer that µ 2 = E[2]/G. In turn, we get an identification of elliptic curves E Q = (E/E [2]) Q = (E /µ 2 ) Q . The latter quotient can be computed with Proposition 4.1, and is given by the Weierstraß equation Conversely, if E Q is given by one of the Weierstraß equation y 2 + xy = x 3 + 2nx 2 + nx with n = ±1, then conditions (i)-(ii) hold by Proposition 3.2. One easily sees that R = (−n, n) has order four in in the groups E(Q) and E(F 2 ). The statements on the singular fibers follow from Proposition 3.5. Since all fibers are irreducible, every element in MW(E/Z) is narrow. The j-invariant is easily computed with [9].

Reinterpretation in terms of stacks
In this final section we reformulate our main results in terms of stacks. For details on the theory of stacks, we refer to the monographs of Laumon and Moret-Bailly [8] and Olsson [16].
Let g, r ≥ 0 be integers with g + r ≥ 2, and M g,r be the Deligne-Mumford stack over the base ring Z whose fiber categories M g,r (R) are tuples (C, σ 1 , . . . , σ r ) as follows: C is a finitely presented flat proper R-scheme whose fibers are smooth curves with h 0 (O Ca ) = 1 and h 1 (O Ca ) = g, and σ 1 , . . . , σ n are pairwise disjoint sections for the structure morphism f : C → Spec(R). This has a natural compactification M g,r , where the fibers C a have at most normal crossing singularities, the sections pass through the smooth locus, and the automorphism group is finite. The structure morphismM g,r → Spec(Z) is proper, and the inclusion M g,r ⊂M g,r is open. By abuse of notation, we write O C (σ i ) for the invertible sheaf attached to the image of the section σ i : Spec(R) → C, which is an effective Cartier divisor.
We are mainly interested in the case g = 1 and r ≥ 1. Note that the objects (C, σ 1 ) from M 1,1 can be regarded as families of elliptic curves E → Spec(R), where the zero-section O ⊂ E is the image of σ 1 . Likewise, objects fromM 1,1 can be regarded as families Y → Spec(R) of cubic curves with a zero-section O ⊂ Y , locally given by Weierstraß equations where ∆, c 4 generate the unit-ideal. Given an object (C, σ 1 , . . . , σ r ) ∈M 1,r (R), the invertible sheaf O C (σ 1 ) is semiample, and the homogeneous spectrum together with the induced section σ 1 : Spec(R) → Y passing through Reg(Y /Z) defines an object (P, σ 1 ). The construction (C, σ 1 , . . . , σ r ) → (P, σ 1 ) actually yields a morphism of stacksM 1,r →M 1,1 , and composition with the j-invariant gives M 1,r j → P 1 . By abuse of notation, we write j(C) ∈ P 1 (R) for the image of an object (C, σ 1 , . . . , σ r ). If R = k is a field, this may be interpreted as a number j(C) ∈ k ∪ {∞}. Now let m, n ∈ Z be integers with n odd and gcd(4m + 1, n) = 1, and consider our Weierstraß equation y 2 + xy = x 4 + mx 2 + nx. The Weierstraß model Y of the ensuing elliptic curve E Q together with its zero-section O ⊂ Y can be regarded as an object inM 1,1 (Z). This defines a map defined on the set Ψ = {(m, n) ∈ Z 2 | n odd and gcd(4m + 1, n) = 1}. The equations x = y = 0 define an element P ∈ MW(E/Z) of order two. The strict transforms of O ∪P ⊂ E on the minimal model X yields a relatively semi-ample invertible sheaf L on X. The homogeneous spectrum Y = P (X, L ) = Proj t≥0 H 0 (X, L ⊗t ) defines a partial resolution X → Y → Y . Write O , P ⊂ Y for the strict transforms of O, P ⊂ Y . Each geometric fiber for the structure morphism Y → Spec(R) is either an elliptic curve or a cycle of rational curves with one or two irreducible components, and P , Q are contained in Reg(Y /Z). This gives a map If the element P ∈ MW(E/Z) is narrow, which means n = ±1 according to Proposition 3.4, the partial desingularization coincides with the Weierstraß model, and the map becomes (m, n) → (Y, O, P ).
Proof. Suppose first that (C, σ 1 , σ 2 ) arises from a pair (m, n) with n = ±1. Then C = Y , and the closed fiber at p = 2 has invariant j = 1. Moreover, all geometric fibers of the structure morphism C → Spec(Z) are integral. Thus the Picard scheme Pic C/Z exists ( [5], Theorem 3.1), and we have an identification E = Pic 0 C/Z of relative group schemes, given by D → O C (D − O). Since P has order two in E(Z), the invertible sheaf O C (2σ 1 − 2σ 2 ) has trivial class in Pic C/Z (Z). The Leray-Serre spectral sequence gives an exact sequence Pic(Z) −→ Pic(C) −→ Pic C/Z (Z), and the factoriality of the ring Z ensures that O C (2σ 1 − 2σ 2 ) is trivial.
Conversely, suppose that (C, σ 1 , σ 2 ) has the stated properties. Then j(C⊗F 2 ) = 1, and this ensures that the generic fiber E Q = C Q becomes an elliptic curve, with origin given by σ 1 . Let E be its Néron model. Write X and Y for the minimal model and Weierstraß model, respectively. From O C (2σ 2 ) O C (2σ 1 ) we infer that σ 1 , σ 2 : Spec(R) → C pass through the same irreducible component, in all fibers for C → Spec(Z). It follows that the fibers are irreducible. Moreover, using that the sections pass through Reg(C/Z) we infer C = Y . The image of σ 2 defines a section P ⊂ E that is disjoint from the zero-section O ⊂ E. Moreover, the element P ∈ MW(E/Z) has order two and is narrow. From Theorem 2.1 and Proposition 3.4 we see that C = Y is given by a Weierstraß equation y 2 + xy = x 3 + mx 2 ± x. The equation is unique by Lemma 1.3. Now consider the situation of Theorem 5.2, such that we have a narrow element R ∈ MW(E/Z) of order four whose image in E(F 2 ) also has order four. The resulting narrow element P = 2R has order two in both E(Q) and E(F 2 ), and the three sections O, R, P are pairwise disjoint. This gives a map Proof. It follows from Theorem 5.2 that m = 4n and n = ±1 gives a family of pointed stable curves (C, σ 1 , σ 2 , σ 3 ) with the stated properties. The two Weierstraß equations have different j-invariants, so the map in question is injective.
Conversely, suppose that we have an object (C, σ 1 , σ 2 , σ 3 ) with the properties at hand. Then C and σ 1 define an elliptic curve E Q . Write E for its Néron model, and let R, P ∈ MW(E/Z) be the elements corresponding to σ 2 , σ 3 . Set S = Spec(Z). The conditions on the invertible sheaves O C (σ i ) ensure 4R = O and 2R = P . If a geometric fiber for C → Spec(Z) contains a copy of P 1 , it must intersect some image σ i (S), by stability. Again by the condition on the invertible sheaves O C (σ i ) we infer that all geometric fibers are irreducible. It follows that C coincides with the Weierstraß model Y of the elliptic curve E Q . This ensures that the structure morphism E → S is semi-abelian. Moreover, the subschemes O, R, P ⊂ E map to the σ i (S) ⊂ Y under the canonical morphism E → Y . It follows that O, R, P are pairwise disjoint. Hence R has order four in both E(Q) and E(F 2 ). Applying Theorem 5.2, we see that the family of cubics Y = C is define by the Weierstraß equation y 2 + xy = x 3 + 4nx 2 + nx with either n = 1 or n = −1.