Elliptic curves, up to isomorphism, are in bijec tion with. They, in turn, are used to construct the extension of the. We consider the moduli spaces n of curvesc of genus 2 with the property. Is there a description of the moduli space of elliptic. The complex analytic structure and the relation to teichmuller space is further discussed in. The moduli space of curves alessio corti october 27, 1997 this is a write up of my lecture in the cambridge \geometry seminar, an introduction to the construction and. Line bundles on moduli space of elliptic curves and modular forms. Moreover, i believe that the period map is locally an isomorphism in this case, hence different picard numbers occur. Elliptic curves in moduli space of stable bundles of rank 3. Moduli for pairs of elliptic curves with isomorphic ntorsion by david carlton submitted to the department of mathematics on april 3, 1998 in partial fulfillment of the. The moduli space of elliptic curves m1,1 and its delignemumford compacti. The moduli stack of elliptic curves is denoted by m ell or by m 1,1, which is a special case of the moduli stack m g,n of genus g curves with n marked points. Our main goal is to pose some questions and conjectures about these families, guided by the principle of \unlikely intersections from arithmetic geometry, as in za.
An elliptic curve over cis a riemann surface of genus one with a marked point. Unfortunately, this space is not like anything familiar. We have already seen in elliptic curves what an elliptic curve looks like when graphed in the plane, where and are real. For instance, the higherdimensional analogue of the moduli space of elliptic curves discussed above is the moduli space of abelian varieties, such as the siegel modular variety. You easily can get the dimension of the corresponding moduli space from this. These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at zhejiang university in july.
An introduction to moduli spaces of curves and its. These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at zhejiang university in july, 2008. The construction of moduli spaces and geometric invariant. Moduli of abelian covers of elliptic curves internet archive.
Intuitively speaking, we can describe an elliptic curve over a scheme s as an algebraic family of elliptic curves, one for each point of s. I then proved in a fair amount of detail that the moduli problem of elliptic curves with. The situation is quite similar to the relation one of us found between the bailyborel compacti. Is there a description of the moduli space of elliptic surfaces. Christophe breuil, brian conrad, fred diamond, and richard taylor introduction in this paper, building on work of wiles wi and of wiles and one of us r. An elliptic curve over c is the quotient of a one dimensional vector space v over c by a lattice. C is called the coarse moduli spaceof elliptic curves. Denote the moduli stack over specz of smooth elliptic curves with n marked points and r nonzero tangent vectors by m1.
Moreover, two elliptic curves are isomorphic if and only if they have the same jinvariant, so the isomorphism class of an elliptic curve is determined by a single complex number. Modular compactifications of the space of pointed elliptic curves ii. An elliptic curve over c is a riemann surface e of genus one, equipped with a marked point e. When studying geometric objects, it is desirable to classify them according to different criteria in order to be able to distinguish the equivalent classes in this category. This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. We study the multiplicationbyp map on an elliptic curve, which gives a strati. Uln,d the modulispace of sequivalence classes of semistable vector bundles of rank n and degree d resp. Oct 02, 20 in view of the above, we complete the proof of the theorem on the moduli of elliptic curves. I know two general methods for constructing arithmetic models of the compactifications of modular curves. Dial m1,1 for moduli an elliptic curve is a smooth genus 1 curve. In the very interesting but long and hardreading hence not so. We will let x 1n denote the smooth projective curve which contains y 1n as a dense zariski open subset. There is a one to one correspondence between elliptic curves and lattices.
Title the moduli space of once punctured elliptic curves with. Moduli spaces of hecke modi cations for rational and. We now want to construct the moduli space of elliptic curves. A moduli stack of elliptic curves is a moduli stack of elliptic curves, hence. The orbifold euler characteristic of the moduli space of curves was originally computed in. I have often heard the words moduli stack of elliptic curves, but i have nowhere found a fromscratch definition of this object. In higher dimensions, moduli of algebraic varieties are more difficult to construct and study. The moduli space of curves of genus two covering elliptic curves. Pdf we show that the moduli space of rational elliptic surfaces admitting a section is locally a complex hyperbolic variety of dimension eight. I will discuss \classical examples and some more recent results cubic fourfolds, holomorphic symplectic varieties, fano fourfolds. We introduce a sequence of isolated curve singularities, the elliptic mfold points, and an associated sequence of stability conditions, generalizing the usual definition of delignemumford stability.
The arithmetic study of the moduli spaces began with jacobis fundamenta nova in 1829, and the modern theory was erected by eichlershimura, igusa, and delignerapoport. The moduli stack of elliptic curves is a smooth separated delignemumford stack of finite type over specz, but is not a scheme as elliptic curves have nontrivial automorphisms. Computing genus 2 curves from invariants on the hilbert. Finally, i give edidins construction of the stack of elliptic curves, but do not prove that it is in. Over the complex numbers, an elliptic curve is a torus. Feb 16, 2017 we now want to construct the moduli space of elliptic curves.
This is the first lecture on the arithmetic moduli theory of elliptic curves. Moduli spaces such as the moduli of elliptic curves which we discuss below play a central role in a variety of areas that have no immediate link to the geometry being classi. The moduli space of rational elliptic surfaces 3 involving a hyperbolic hermitian lattice over the eisenstein ring. This allows the construction of iterated integrals involving.
We define two new invariants for genus 2 curves as values of modular functions on the hilbert moduli space and show how to compute them. Modular compactifications of the space of pointed elliptic. Elliptic curves a complex elliptic curve is a smooth plane cubic curve e, that is. In section 3 we recall the case of elliptic curves.
In view of the above, we complete the proof of the theorem on the moduli of elliptic curves. C has a maximal mapf of degreen to an elliptic curvee. For elliptic curves they correspond respectively to the j and. A familyof elliptic curves over a base space b is a fibration x. Points of the moduli space should corre spond to isomorphism classes of elliptic curves. Elliptic curves over schemes the notion of elliptic curves over arbitrary schemes is indispensable for the topic of moduli spaces. Moduli for pairs of elliptic curves with isomorphic ntorsion. This curve has a natural model y 1n q, which for n3 is a ne moduli scheme for elliptic curves with a point of exact order n. Tw, we will prove the following two theorems see x2. This is the problem underlying siegel modular form theory.
Line bundles on moduli space of elliptic curves and. We x an algebraic number that is the jinvariant of an elliptic curve without complex multiplication. Our goal will be to understand the space of all elliptic curves, the socalled moduli space m1,1. We prove that the number of jinvariants with complex multiplication such that j. Integrality properties in the moduli space of elliptic curves schmid, stefan. Hurwitz spaces and moduli spaces of marked elliptic curves. A moduli stack of elliptic curves is a moduli stack of elliptic curves, hence a stack. We wish to construct a moduli space for elliptic curves. Don zagier, john harer, the euler characteristic of the moduli space of curves, inventiones mathematicae 1986 volume.
I also briefly discuss the fibers in bad characteristic. Moduli of elliptic curves peter bruin 12 september 2007 1. Here the n marked points and the anchor points of the r tangent vectors are distinct. I began by explaining why the natural moduli problem for elliptic curves is not representable by a scheme. The construction of moduli spaces and geometric invariant theory. The construction of moduli spaces and geometric invariant theory by dinamo djounvouna in algebraic geometry, classi. These notes are based on four lectures at kawa 2015, in. Elliptic curves in moduli space of stable bundles of rank. These describe the moduli space of rational elliptic surfaces unless i stupidly overlooked som parts. Elliptic curves we start with onedimensional analogs of k3 surfaces, namely elliptic curves. Moduli spaces of hecke modi cations for rational and elliptic. By torelli maps n is viewed as a subset of the moduli spacea 2 of principally polarized abelian surfaces.
A heisenberg level structure on an elliptic curve is a kind of marking of the linear action by thenoncommutativeheisenberg group on the space of global sections of a line bundle. Worse still, the curves with jinvariants j 0 and j 1728 have extra. Introduction the purpose of these notes is to provide a quick introduction to the moduli of elliptic curves. We study holomorphic involutions on elliptic curves in section 3. Integrality properties in the moduli space of elliptic curves. The notion of elliptic curves over arbitrary schemes is indispensable for the topic of moduli spaces. In the thesis at hand we discuss two problems of integral points in the moduli space of elliptic curves. Introduction to riemann surfaces, moduli spaces and its. In order to do this we will need to first understand the meaning of the following statement. In this paper we consider the case of genus 1, and describethe moduli space using natural a extension ofigetaconstruction, that is, we make a completelist of once puctured elliptic curves with lagrangian sublatticessee \s 2. This is a brief introduction to the theory of moduli and mirror families of k3 surfaces based on lectures given at the summer workshop on moduli in hamburg, august 20. On arithmetic curves in the moduli spaces of curves. Modular forms arise naturally as holomorphic sections of powers of the hodge bundle over the orbifold m1,1.
Involutions of higgs moduli spaces over elliptic curves. The moduli space of elliptic curves theories and theorems. The prank strata of the moduli space of hyperelliptic curves jeffrey d. Their goal is to introduce and motivate basic concepts and constructions such as orbifolds and stacks important in the study of moduli spaces of curves and abelian varieties through the example of elliptic curves. Uln,d the moduli space of sequivalence classes of semistable vector bundles of rank n and degree d resp. There are many excellent and thorough references on the subject, ranging from the slightly archaic igu59 and shi94 to the more di. The arithmetic study of the moduli spaces began with jacobis fundamenta nova in 1829, and the modern theory was erected. The primary example is the moduli space of elliptic curves. Moduli space of elliptic curves with heisenberg level. I then proved in a fair amount of detail that the moduli problem of elliptic curves with full level 3 structure is representable by a scheme. As we will see below, however, c lacks an important additional property and cannot be considered the true moduli space of elliptic curves. We give a new method for generating genus 2 curves over a finite field with a given number of points on the jacobian of the curve. The prank strata of the moduli space of hyperelliptic curves.
We also include a proof that the hyperbolic postcritically nite maps are zariski dense in the moduli. Towards defining a stack of elliptic curves william stein. A natural conclusion is that c is the moduli space of elliptic curves. In mathematics, the moduli stack of elliptic curves is an algebraic stack.