Tesis doctoral de Santiago Molina Blanco
Given an order r in an imaginary quadratic field k, the propose of this thesis is to study the specialization of the set cm(r) of heegner points at the special fibers of the shimura curve x0(d,n), focusing our attention on the bad reduction case. for a given prime p, the set of singular points and irreducible components, if p is a prime of bad reduction, or the set of supersingular points, otherwise, are characterized by isomorphism classes of oriented eichler orders. The algebraic interpretation of these sets of geometric objects together with the characterization of the set cm(r) as optimal embeddings provides that the specialization map, although its geometric nature, induces a map where both the source and the target are pure algebraic objects. along this work, we prove that the necessary and sufficient condition for a point p in cm(r) to specialize to a singular point of the singular special fiber at p corresponds to verify if p ramifies in k or not. Moreover, we give an algebraic interpretation of the geometric map discussed above. We can summarize the algebraic nature of the specialization of the points in cm(r) by means of the following table: singular pts irred. Comp. P in cm(r) singular spec.? Algebraic int. (Singular) algebraic int. (Non-singular) p divides d pic(d/p,np) pic(d/p,n) 2 p ramifies in k ?S :cmd,n(r)?Cmd/p,np(r) ?C :cmd,n(r)?Cmd/p,n(r) 2 q divides n pic(dq,n/q) pic(d,n/q) 2 q ramifies in k ?S :cmd,n(r)?Cmdq,n/q(r) ?C :cmd,n(r)?Cmd,n/q(r) 2 in this table, each of the sets pic(d,n) denote the set of isomorphism classes of oriented eichler orders of level n in a quaternion algebra of discriminant d, and each of the sets cmd,n(r) denote the set of optimal embeddings of r into any of the orders in pic(d,n). Composing with the natural projection cmd,n(r)?Pic(d,n), that maps any optimal embedding to the isomorphism class of its target, and identifying each point in cm(r) with its corresponding optimal embedding in cmd,n(r), one obtains the singular point or the irreducible component where the point lies. the construction of such algebraic maps allows us to compute explicitly the specialization of such heegner points by means of the software magma, for example. When the shimura curve is hyperelliptic, thus defined by an equation of the form y 2 =p(x), controlling the reduction of certain heegner points we obtain certain information about the discriminants and the resultants of the polynomials involved in the factorization of p(x) over the rationals. Combining this data with the splitting fields of such polynomials, obtained by means of complex multiplication theory, and their leading coefficients, obtained via gross-zagier theory, we are able to find p(x), namely, an explicit equation for the shimura curve. These methods are completely different from those used for the classical modular situation where cusps and fourier coefficients are available. next, we present a list with the equations that we have obtained: x(39,1) x(55,1) x(35,1)/w5 x(51,1) /w17 x(57,1) /w3 x(65,1) /w13 x(65,1) /w5 x(69,1) /w23 x(85,1) /w5 x(85,1) /w85 y 2 = -(7x 4 + 79x 3 + 311x 2 + 497x + 277) (x 4 + 9x 3 + 29x 2 + 39x + 19) y 2 = -(3x 4 – 32x 3 + 130x 2 – 237x + 163) (x 4 – 8x 3 + 34x 2 – 83x + 81) y2 = -x (9x + 4) (4x + 1) (172×3 + 176×2 + 60x + 7) y2 = -x (7×3 + 52×2 + 116x + 68) (x – 1) (x + 3) y 2 = -(x – 9) (x 3 -19x 2 + 119x – 249) (7x 2 – 104x + 388) y 2 = -(x 2 – 3x + 1) (7x 4 – 3x 3 – 32x 2 + 25x – 5) y2 = -(x2 + 7x + 9) (7×4 + 81×3 + 319×2 + 508x + 268) y2 = -x (x + 4) (4×4 – 16×3 + 11×2 + 10x + 3) y2 = -(3×2 – 41x + 133) (x4 – 23×3 + 183×2 – 556x + 412) y2 = (x2 – 3x + 1) (x4 + x3 – 15×2 + 20x – 8) the last part of the thesis is devoted to the study of three more applications of our theoretical results on the specialization of heegner points: 1. The distribution of the specialization of points in cm(r) among the set of irreducible components and singular points of a singular special fiber of the shimura cure. 2. The computation of the group of automorphisms of a shimura curve. 3. The image of the degree zero divisors with support in cm(r) in the group of connected components of the reduction of the jacobian of x0(d,n) and its relation with the birch and swinnerton-dyer conjecture.
Datos académicos de la tesis doctoral «Specialization of heegner points and applications«
- Título de la tesis: Specialization of heegner points and applications
- Autor: Santiago Molina Blanco
- Universidad: Politécnica de catalunya
- Fecha de lectura de la tesis: 07/07/2010
Dirección y tribunal
- Director de la tesis
- Josep Gonzalez Rovira
- Tribunal
- Presidente del tribunal: pilar Bayer isant
- kenneth a. Ribet (vocal)
- pierre Parent (vocal)
- ignacio Sols lucia (vocal)