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1. The modular curve X(5) has genus 0: it is the Riemann sphere with 12 cusps located at the vertices of a regular icosahedron.
2. There is an explicit classical model for X 0 (N), the classical modular curve; this is sometimes called the modular curve.
3. Hecke operators may be studied geometrically, as correspondences connecting pairs of modular curves.
4. The zeta-function of a modular curve is a product of the Mellin transforms (cf.
5. It is important to note that the classical modular curves are part of the larger theory of modular curves.
6. In this lecture our goal is simply to introduce the notion of a modular curve, beginning with the canonical example X(1).
7. We can now dene the modular curve X(1) = H/, which contains all the points in Y (1), plus the cusp at innity.
8. Abstract We obtain dening equations of modular curves X0(N ), X1(N ), and X(N ) by explicitly constructing modular functions using generalized Dedekind eta functions.
9. Dening equations of modular curves Let (cid:1) be a congruence subgroup of SL2(R).
10. The classical modular curves X((cid:1)) are dened to be the quotients of the extended upper half-plane H = {(cid:1) C : Im (cid:1) > 0} Q {} by the action of (cid:1).
11. It turns out that a modular curve has the structure of a compact Riemann surface.
12. In this post, we’ll show how we can use the Riemann-Hurwitz formula to derive the genus of the modular curve .
13. We describe an algorithm for constructing plane models of the modular curve X1(N ) and discuss the resulting equations when N 51.
14. Let X be the reduction of the modular curve X(p) in characteristic ℓ (with ℓ≠p).
15. Realising a modular curve as a Riemann surface 4.3.
16. In order to do so, we introduce modular curves.
17. A modular curve with respect to is the quotient space of orbits under the action of on H. We will see that every modular curve is in fact a Riemann surface.
18. We work with compact modular curves and establish a weight 2k modular form as a k-fold dierential form on the associated modular curve.
19. By construction, these modular curves provide covers (atlases) for the moduli stack of elliptic curves ℳ ell ( ℂ ) \mathcal{M}_{ell}(\mathbb{C}) over the complex numbers.
20. Abstract: In this talk we would like to review the basic theory of modular curves.
21. In this talk we would like to review the basic theory of modular curves.
22. Then I will focus on the Jacobian associated to a modular curve and how they can be described.
23. In this talk I explain how to get explicit equations for a model of some kind of modular curves.
24. Modular curves like X0(N) and X1(N) appear very frequently in arithmetic geometry.
25. We wish to give such a moduli description for two other modular curves, denoted here by Xnsp(p) and Xnsp+(p) associated to non-split Cartan subgroups and their normaliser in GL2(𝔽p).
26. Introduction We seek an arithmetic construction of the theory of modular curves and modular forms (including Hecke operators, q-expansions, and so forth).
27. Universal elliptic curves over ane modular curves will not admit Weierstrass models globally.
28. This thesis deals with the connections between modular curves and adle rings.
29. H To any such we may associate the noncompact modular curve /.
30. H SL2(Z) be a To any such we may associate the noncompact modular curve /.
31. The Question Do noncongruence modular curves also have a moduli interpretation?
32. Modular curves are of central interest for both the theoretical and compu- tational investigation of elliptic curves.
33. In order to apply this algorithm, one must precompute a large number of explicit models for modular curves.
34. One approach to the problem of computing models for modular curves is to produce a basis for the space of weight two cusp forms.
35. This gives an intuitive method for relating the Hecke module, de(cid:12)ned as a subgroup of the divisor group of a modular curve, with the space of modular forms of weight two.
36. There are many ways to dene modular curves; well dene them by their moduli structure, since this is the description which will be useful for point counting.
37. Modular Curves September 4, 2013 The rst examples of Shimura varieties we encounter are the modular curves.
38. In this lecture we review the basics of modular curves, beginning with the complex theory and progressing towards modular curves over number elds.
39. 1 Modular curves as complex manifolds 1.1 Lattices and the upper half plane H is the upper half plane, a complex manifold.
40. 2 Modular curves over number elds We have just seen that for any nite-index subgroup SL2(Z), the quotient \H is a compact Riemann surface and therefore corresponds to a smooth projective curve X().
41. We show how the Langlands-Kottwitz method can be used to determine the local factors of the Hasse-Weil zeta-function of the modular curve at places of bad reduction.
42. The most successful tool used to obtain these equations is the canonical embedding, combined with the fact that the dierentials on a modular curve correspond to the weight 2 cusp forms.
43. 0 (p) having genus 2 or 3, and for the genus 4 and 5 curves X + Heights of modular curves are studied and a discussion is given of the size of coecients occurring in equations for X0(N ).
44. The star of this thesis is the modular curve X0(N ) and we will examine its life from several dierent angles.
45. The modular curve X0(N ) is very important as it is one of the objects which links the world of elliptic curves with the world of modular forms.

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• [{'LOWER': 'modular'}, {'LEMMA': 'curve'}]