Research Proposal Chitrabhanu Chaudhuri [email protected] My area of research broadly falls under Complex Algebraic Geometry. Algebraic geometry is a branch of mathematics where one studies the geometry of zero sets of a collection of polynomials over a certain field, (C in my case), using algebraic techniques. These zero sets are called algebraic varieties. More specifically my research concerns Moduli of Algebraic curves. The Moduli Space of algebraic curves has been a classical object of study within Algebraic geometry and has connections with various other fields of mathematics, including but not limited to enumerative geometry [KM94], number theory [KM85], mathematical physics [GK98] and dynamical systems [EMM15]. I am currently pursuing several projects all revolving around the central topic of Moduli of Curves: 1. Arakelov Geometry of Modular curves, 2. Enumerative Geometry of Curves, 3. Topological study of the moduli of curves. I shall describe each of them in a separate section below. 1. Arakelov Geometry of Modular Curves This is ongoing joint work with Debargha Banerjee and Diganta Borah. We are interested in finding the minimal regular model and stable arithmetic self-intersection of certain modular curves with arithmetic applications in mind. 1.1. Arakelov Goemetry. Let K=Q be a number field and Ok be its ring of integers. An arith- metic surface X over OK is a regular scheme of dimension 2 with a proper map to X! Spec OK , defined in [Liu02]. The algebraic intersection theory doesn't work very well in this situation since X is not proper. Arakelov [Ara75] introduced an intersection pairing suited to this situation for the special case K = Q. Deligne [Del87] later generalised this intersection theory to arbitrary number fields. Faltings [Fal84] studied this theory extensively, proving results like the Riemmann-Roch theorem, Hodge index theorem and Noether's formula in this situation. Faltings's motivation for studying Arakelov intersection pairing was to prove results about arithmetic surfaces (or number fields), that were known for algebraic surfaces (or function fields), in particular Mordell conjec- ture which is now known as Faltings's theorem. Although Faltings's proof of Mordell conjecture did not involve Arakelov theory, but Vojta [Voj91] gave a proof using Arakelov's pairing. Faltings [Fal91]later proved a generalisation of Mordell conjecture using his results in Arakelov theory. In this section we shall use the notation h·; ·iAr for the Arakelov intersection pairing. 1.2. Modular curves. First let us discuss certain finite index subgroups of SL(2; Z), known as congruence subgroups. Fix a positive integer N called the level and let a b Γ (N) = 2 SL(2; ) j c ≡ 0 (mod N) 0 c d Z a b Γ (N) = 2 SL(2; ) j c ≡ 0 (mod N); a; d ≡ 0 (mod N) 1 c d Z 1 0 Γ(N) = A 2 SL(2; ) j A ≡ (mod N) : Z 0 1 The modular group SL(2; Z) acts on the upper half plane H = fx + iy 2 C j y > 0g through M¨obiustransformations hence so does a subgroup Γ ⊂ SL(2; Z). The quotient Y (Γ) = ΓnH is a 1 2 non-compact Riemann surface. Y (Γ) can be compactified by adding finitely many points called cusps and the compactification is denoted by X(Γ). Being a compact Riemann surface, X(Γ) is an algebraic curve over C. In fact for congruence subgroups X(Γ) is an algebraic curve over Q. We abbreviate X0(N) = X(Γ0(N)), X1(N) = X(Γ1(N)) and X(N) = X(Γ(N)). In fact, X(Γ) are moduli spaces of generalised elliptic curves for congruence subgroups Γ and are closely connected to modular forms for the corresponding subgroups. A modern reference for modular curves and modular forms is [DS05]. 1.3. Stable Arithmetic self-intersection. Let X be a smooth projective geometrically connected curve over a number field K. A regular model of X over OK is an arithmetic surface X! Spec OK with generic fiber isomorphic to X. If genus of X is atleast 1, there is a minimal (in the sense of birational geometry) regular model X which is unique upto isomorphism, but not functorial in X. Such a model is called semi-stable if all the scheme theoretic fibers of π : X! Spec OK are reduced and have only nodal singularities. There is a distinguished line bundle on X the relative dualizing sheaf !X . This line bundle !X when restricted to any fiber of π is the canonical line bundle of that curve. If X is semi-stable, the quantity 2 h!X ;!X iAr !X = [K : Q] 0 0 is independent of K in the following sense: let K =K be a finite extension and X = X ×Spec OK Spec OK0 , then h! 0 ;! 0 i h! ;! i X X Ar = X X Ar : [K0 : Q] [K : Q] This quantity will henceforth be called the stable arithmetic self-intersection of X. The significance of this invariant has been shown by Szpiro [Szp90], who showed that the strict positivity of this number is equivalent to Bogomolov's conjecture. 1.4. Bogomolov's conjecture. 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. Bogomolov's conjecture, now a theorem due to Ullmo [Ull98] and Zhang [Zha93] 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. This conjecture is a generalisation of the Manin- Mumford conjecture. An effective version of this conjecture is known to be true due to Zhang [Zha93], where the author demonstrates an explicit value for in terms of the stable arithmetic self-intersection of the curve X. 1.5. Survey of known results. The minimal regular models of modular curves have been com- puted in many cases. In the case of square free levels N, Deligne and Rapoport [DR73] constructed regular models of X(N);X1(N) and X0(N) as moduli spaces of generalised elliptic curves. Katz and Mazur [KM85] gave a different moduli theoretic construction for Y (N);Y1(N) and Y0(N) using Drinfeld level structures and then compactified these by adding cusps. The encyclopaedic work of Katz and Mazur is now the most comprehensive reference for this material. It turns out that when N is not square, these models are not regular. Hence for non-square free level constructing a m minimal regular model is a major task, which has been undertaken by many people, for X0(p N) see Edixhoven [Edi90], and Weinstein [Wei16] for semi-stable models for X(pmN). The stable arithmetic self-intersection has also been studied extensively in the case of modular curves. In the case of X0(N), Abbes|Ullmo have given asymptotic upper and lower bounds for !2 , when N is square-free and co-prime to 6. X0(N) 3 For the modular curves X1(N), Mayer has shown that when N is odd, square free and of the form N = N 0qr with q; r > 4 !2 = 3g log N + o(g log N); X1(N) N N where gN is the genus of X1(N). In [BC18] using results from [BBC17], we have have shown that for N = p2, 2 2 2 ! 2 = 2g 2 log p + o(g 2 log p ): X0(p ) p p Using this and results of Zhang [Zha93], we are able to give an effective version of Bogomolov's conjecture: 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. 1.6. Future research. There are several future direction in this project. 3 Stable model for X0(p ) has been computed by Coleman and McMurdy in [CM06] and [CM10] along with a study of the special fiber. So using their work we would like to carry out the program 3 of [BC18] for X0(p ). More precisely: 3 Question 1. Find the stable arithmetic self-intersection of X0(p ). The stable model of Coleman|McMurdy is not regular so the first task is to compute a minimal 3 regular model for X0(p ). The geometry of the special fiber of the stable model is much more complicated in this case and it will be a much harder task to compute the algebraic (or finite) part of the self-intersection of the relative dualizing sheaf. However, computation of the analytic (or 2 infinite) part should be analogous in flavour to the case of X0(p ). n Finding stable models of X0(p ) for n > 3 is a well known open problem as well as finding regular semi-stable models. It would be a long term goal to study the geometry of regular minimal models n for X0(p ) over the ring of integers of finite extensions of Q over which the models are semi-simple. 2. Enumerative Geometry of Curves This is a joint project with Ritwik Mukherjee and Nilkantha Das. Our varieties are over C. An algebraic curve is a projective variety of dimension 1. There are two invariants associated to an algebraic curve, the geometric genus and the arithmetic genus. When the curve is smooth these are the same, but for singular curves the two numbers may be different. In this section genus will mean geometric genus. 2.1. Curve counting. A fundamental problem in enumerative algebraic geometry is: (g) 2 Question 2.