2 Arakelov Geometry of Modular Curves X0(p ) PMNTS 2019 Chitrabhanu Chaudhuri IISER Pune 15 February, 2019 Bogomolov's Conjecture Today I shall talk about a work I jointly did with Debargha Banerjee and Diganta Borah. Let X be a smooth projective geometrically connected curve over a number field K with genus g > 1. Let J be the Jacobian of X . Fix an embedding φ : X ! J for some degree 1 divisor of X . The Bogomolov's conjecture, now a theorem due to Ullmo and Zhang says that there is an > 0 such that fx 2 X (K) j hNT(φ(x)) < g is finite; hNT denotes the Neron-Tate height on J. We investigate a certain invariant called the the stable arithmetic 2 self-intersection number for certain modular curves X0(p ). This invariant, using Zhangs work, will give us an explicit value for such an epsilon. 2 C. Chaudhuri (IISER Pune) Modular Curves X0(p ) 15 February, 2019 2 / 26 Modular Curves We are interested in congruence subgroups of the modular group PSL(2; Z) of the form a b Γ (N) = 2 PSL(2; ) j c ≡ 0 (mod N) : 0 c d Z Such a group Γ acts on the upper half plane by M¨obiustransformations and the quotient Y (Γ) = H=Γ is a Riemann surface. Usually Y (Γ) is not compact but of finite type. 1 By adding finitely many points p1;:::; pm 2 P (Q) we can compactify Y (Γ) and get a compact Riemann surface: X (Γ) = Y (Γ) [ fp1;:::; pmg: The points p1;:::; pm are known as the cusps of Γ. X (Γ) is a projective algebraic curve over C and its genus is denoted by gΓ. 2 C. Chaudhuri (IISER Pune) Modular Curves X0(p ) 15 February, 2019 3 / 26 Toy Example z The group Γ0(2) is generated by τ(z) = z + 1 and γ(z) = 2z+1 . It is thus easy to see that the fundamental domain in H for Γ0(2) is: ∼ 1 Γ0(2) has 2 cusps corresponding to 0 and 1 and X0(2) = X (Γ0(2)) = P . 2 C. Chaudhuri (IISER Pune) Modular Curves X0(p ) 15 February, 2019 4 / 26 Modular Interpretation In fact X0(N) is an algebraic curve over Q, and hence defined over any number field K. X0(N)(K), the K points of X0(N), are in bijection with isomorphism classes of pairs (E; C) where E is an elliptic curve over K, and C ⊂ E(K) is a cyclic subgroup of order N. Hence these objects are important in arithmetic geometry. We shall denote by gN the genus of X0(N). This can be calculated using the Riemann Hurwitz formula. 2 C. Chaudhuri (IISER Pune) Modular Curves X0(p ) 15 February, 2019 5 / 26 Minimal Regular Model A regular model XOK of X0(N) over OK is a regular scheme of relative dimension 1 over Spec OK such that the generic fiber XK is isomorphic to X0(N)(K). XOK is then an arithmetic surface in the sense of Liu. If the genus gN > 1, there is a unique regular model, X0(N)OK , which is minimal in the sense of birational morphisms. That is any proper birational morphism from X0(N)OK is necessarily an isomorphism. Equivalently X0(N)OK does not have any prime vertical divisor that can be blown down without introducing a singularity. 2 C. Chaudhuri (IISER Pune) Modular Curves X0(p ) 15 February, 2019 6 / 26 Arakelov Geometry Let X! Spec Z be an arithmetic surface. The usual algebraic intersection of divisors is not very well suited for this situation. However, Arakelov discovered an intersection theory that takes into account the “fiber at infinity". Deligne generalised this theory to arithmetic surfaces X! Spec OK , where K is a number field. Faltings extensively studied this intersection theory. His goal was to prove results about arithmetic surfaces or number fields that were already known for complex surfaces or function fields. In particular this theory was used to give a new proof of Mordell's conjecture (Vojta) and Serge Lang's generalisation of that conjecture (Faltings). Gillet and Soule later generalised Arakelov's work to higher dimensional arithmetic varieties. 2 C. Chaudhuri (IISER Pune) Modular Curves X0(p ) 15 February, 2019 7 / 26 Arakelov's Intersection Pairing Let X! Spec OK be an arithmetic surface with generic fiber X = XK . For each embedding σ : K ! C we have a connected Riemann surface Xσ = X ×Spec K,σ Spec C: Collectively we denote G X1 = Xσ: σ:K!C For two divisors C; D on X the Arakelov pairing is given by hC; Di = hC; Difin + hC; Di1: 2 C. Chaudhuri (IISER Pune) Modular Curves X0(p ) 15 February, 2019 8 / 26 Where X hC; Difin = log jOX ;x =(Cx ; Dx )j x2X (2) is the usual algebraic intersection as in Liu. Moreover, in the special case when C and D have no common prime divisors X X σ hC; Di1 = − nα,σmβ,σgcan(Pα,σ; Qβ,σ); σ:K!C α,β P σ where Cσ = α nα,σPα,σ and Dσ = mβ,σQβ,σ and gcan is the canonical Green's function on Xσ. 2 C. Chaudhuri (IISER Pune) Modular Curves X0(p ) 15 February, 2019 9 / 26 Relative Dualizing Sheaf Let X be a smooth projective curve over K of genus gX > 1 and XOK the minimal regular. There is a relative dualizing sheaf !X as defined in Liu for instance. The quantity 2 h!X ;!X i !X = [K : Q] is independent of the field K if X has semi-stable reduction over OK and is an invariant of X . We call this the stable arithmetic self-intersection number of X . The arithmetic significance of this invariant was given by Szpiro. He demonstrated that its strict positivity is equivalent to Bogomolov's conjecture 2 C. Chaudhuri (IISER Pune) Modular Curves X0(p ) 15 February, 2019 10 / 26 Stable Self Intersection The self intersection numbers are usually difficult to calculate and requires knowledge of the geometry of minimal regular model of X . Some work has been done for modular curves and Fermat curves as well as curves of low genus. X (N) Abbes-Ullmo gave explicit bounds for !2 in terms of N, 0 X0(N) when N is square free and co-prime to 6. X (N) Mayer proved an asymptotic formula for !2 when N 1 X1(N) square free with two distinct prime factors q; r ≥ 5. X (N) Certain upper bound known due to K¨uhn. Fp Curilla and K¨uhngave upper-bounds for the stable arithmetic self intersection number for Fermat curves of prime exponent p. 2 C. Chaudhuri (IISER Pune) Modular Curves X0(p ) 15 February, 2019 11 / 26 Main Result 2 We investigate the modular curves X0(p ). Probably the reason why they had not been tackled before is that the geometry of the minimal regular model is more complicated in this case. 2 We obtain an asymptotic expression for ! 2 = ! 2 for primes p > 13. p X0(p ) Theorem (Banerjee, Borah,|) 2 2 p log p 2 ! 2 = 2g 2 log p + + o(p log p ). p p 8 In comparison Mayer shows that the asymptotic expression for !2 in the case of X1(N) is 3gN log N + o(gN log N). We shall first mention applications of this result, then give an outline of our proof. 2 C. Chaudhuri (IISER Pune) Modular Curves X0(p ) 15 February, 2019 12 / 26 Effective Bogomolov S.W. Zhang introduced another pairing, called the admissible pairing, for divisors on arithmetic surfaces. This pairing is closely related to the Arakelov pairing. 2 Let !a denote the admissible self-intgersection of the relative dualizing sheaf. Using the geometry of the special fiber the admissible self intersection can be calculated from the Arakelov self intersection. Zhang showed that the in Bogomolov's conjecture can be explicitly 2 written in terms !a . 2 C. Chaudhuri (IISER Pune) Modular Curves X0(p ) 15 February, 2019 13 / 26 Using Zhang's work we can prove the following effective version of Bogomolov's conjecture. Theorem For a sufficiently large prime p, the set 1 x 2 X (p2)( ) j h (φ(x)) < − log(p2) 0 Q NT 2 is finite whereas 2 2 x 2 X0(p )(Q) j hNT(φ(x)) ≤ (1 + ) log(p ) is infinite. 2 C. Chaudhuri (IISER Pune) Modular Curves X0(p ) 15 February, 2019 14 / 26 2 Semistable model for X0(p ) From the work of Deligne and Mumford it is clear that for any Arithmetic surface X! Spec K there is a finite extension K 0=K so that 0 X ×Spec OK Spec OK is semistable. This means that the special fibers of X × Spec OK OK 0 , if any, are all reduced and have nodal singularities. The following theorem is due to Edixhoven, although it is not made explicit in his paper. Theorem (Edixhoven) Q[x] 2 Let K = , then the minimal regular model of X0(p ) over x(p2−1)=2 − p OK is semistable. 2 C. Chaudhuri (IISER Pune) Modular Curves X0(p ) 15 February, 2019 15 / 26 Stable models for these curves over Qp were already computed by Edixhoven. However, the stable models are not regular. 2 We outline the procedure to obtain the minimal regular model X0(p )OK . 2 We start with the minimal regular model X0(p )Z over the ring of integers constructed by Edixhoven. 2 For any prime q 6= p, the fiber X0(p )Fq is smooth. So there is only one special fiber, the fiber over (p) 2 Spec Z.