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Mar Curco Iranzo: Rational torsion of generalized modular Jacobians

Time: Wed 2022-03-30 13.15

Location: Zoom, meeting ID: 694 6016 6420 (password required)

Lecturer: Mar Curco Iranzo (Utrecht)

Abstract:

Consider a prime number p and the Jacobian attached to the modular curve \(X_0(p)\), \(J_0(p)\). In his famous paper “Modular curves and the Eisenstein ideal” (1977), Mazur computed the structure of the group \(J_{0}(p)(Q)_{tor}\), proving thus the so-called Ogg’s Conjecture. Ever since, many authors have been working on the proof of Ogg’s Conjecture for arbitrary levels N.  

On the other hand, consider the generalized Jacobian \(J_0(N)_m\) of the modular curve \(X_0(N)\) of level N, with respect to the modulus m consisting of all cusps on the modular curve. These objects are as interesting as usual jacobians in the study of arithmetic geometry of modular curves, however they have not been studied as much.

In the talk we will see how we can use results in Ogg’s Conjecture to compute the rational torsion subgroup of \(J_0(N)_m\), i.e., we will study the equivalent question to Ogg’s Conjecture for the generalized Jacobian case.

When the level N is divisible by exactly two odd primes p and q odd we will determine the group structure of the rational torsion of the Jacobian \(J_0(N)_m\) up to p- and q-primary torsion.

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