Semi-stable abelian varieties with good reduction outside 15 Ren´e Schoof Universit`a di Roma “Tor Vergata” Serre’s Conjecture In 2004 Khare, Wintenberger and independently Dieulefait proved Serre’s Conjecture: THEOREM. Let F be a finite field with q elements and let q ρ : Gal(Q/Q) −→ GL (F ) 2 q be a continuous, irreducible and odd representation. Then ρ is modular. odd and modular ODD: the determinant of ρ(ι) is −1 for one (or for every) complex conjugation ι in Gal(Q/Q). MODULAR: there exists integers k ≥ 2, N ≥ 1 and a normalized eigenform f = (cid:80) a qn of weight k with respect to the modular k≥1 n group Γ (N) such that for every prime l not dividing qN, the 1 characteristic polynomial of ρ(φ ) is equal to l X2−a X +χ(p)pk−1, in F [X]. p q Here φ denotes the Frobenius element of a prime lying over l and l χ is a suitable character of (Z/NZ)∗ that depends on f. Dieulefait In 2010 Dieulefait gave a new proof of Serre’s conjecture for representations of odd level. His proof is based on results of Kisin’s and is in some sense simpler. It involves an intricate induction with respect to the weight k and the level N. The induction takes off at level 3 and weights 2, 4 and 6. These cases are taken care of by the following three theorems. Semi-stable abelian varieties THEOREM 1. There do not exist any semi-stable abelian varieties over Q with good reduction outside 3. THEOREM 2. There do not exist any semi-stable abelian √ varieties over Q( 5) with good reduction outside 3. THEOREM 3. Up to isogeny, the only simple semi-stable abelian variety over Q with good reduction outside 15 is the elliptic curve given by the equation Y2+XY +Y = X3+X2. X (15) 0 Theorem 1 was already proved by Brumer and Kramer in 2001. Each of Theorems 2 and 3 imply Theorem 1. Today’s lecture is devoted to Theorem 3. The elliptic curve E given Y2+XY +Y = X3+X2 has conductor 15. It is isogenous to the Jacobian J (15) of the 0 modular curve X (15). 0 It follows that the Galois representation given by the 3-torsion points of any simple semi-stable abelian variety over Q with good reduction outside 15, is a 2-dimensional modular representation in the sense of Serre’s conjecture. Group schemes Let A be a semi-stable abelian variety over Q with good reduction outside 15. And let p be prime not equal to 3 or 5. The fact that A has good reduction outside 15, implies that for each n ≥ 1, the pn-torsion points of A are the points of a group scheme A[pn] over the ring Z[ 1 ] that is finite and flat. 15 It follows that the Galois action on A[pn] is unramified at primes l (cid:54)= 3,5,p. In other words, for every σ in the inertia group of any prime lying over l, one has σ = id as automorphisms of A[pn](Q). Group schemes The fact that A has semi-stable reduction at the primes lying over 3 and 5 implies that for every σ in the inertia group of any prime lying over 3 or 5, one has (σ−id)2 = 0 as an endomorphism of A[pn](Q). This is a consequence of Grothendieck’s semi-stable reduction Theorem (SGA 7). a category Let p be a prime not equal to 3 or 5. By C we denote the category of commutative finite and flat group schemes over Z[ 1 ] that have p-power order. Moreover each object 15 G of C has the property that for every σ in the inertia group of any prime lying over 3 or 5, one has (σ−id)2 = 0 as an endomorphism of G(Q). objects Z p (cid:37) (cid:38) Z[ 1 ] Q 15 p (cid:38) (cid:37) Z[ 1 ] 15p An object of C is determined by 1) A group scheme G over Z[ 1 ]. It is ´etale. 1 15p 2) A group scheme G over the complete local ring Z . 2 p It can be described in terms of Dieudonn´e modules with extra structure. ∼ 3) A compatibility isomorphism G1/Qp = G2/Qp.