The Kernels of Jacobians at Eisenstein primes

Presentation (2014, December 22)

Non-optimal levels of a reducible mod l modular representation

E-print/Working paper (2014)

Let $ell>3$ be a prime and $N$ is a square-free integer prime to $\ell$. For each prime divisor $p$ of $N$, let $a_p$ is either 1 or -1. We give a sufficient criterion for the existence of a newform $f ... [more ▼]

Let $ell>3$ be a prime and $N$ is a square-free integer prime to $\ell$. For each prime divisor $p$ of $N$, let $a_p$ is either 1 or -1. We give a sufficient criterion for the existence of a newform $f$ of weight 2 for $\Gamma_0(N)$ such that the mod $\ell$ Galois representation attached to $f$ is reducible and $U_p f= a_p f$ for prime divisors $p$ of $N$. The main techniques used are level raising methods based on an exact sequence due to Ribet. [less ▲]

The 2nd MYNT

Presentation (2013, August 28)

We present the multiplicity one theorem for Eisenstein maximal ideals.

Intensive workshop on Fermat's last theorem

Presentation (2013, August)

We give 9 lectures about Wiles' proof on Fermat's Last Theorem.

Bay area algebraic number theory and arithmetic geometry day

Presentation (2013, April 06)

We discussed level-raising method for residually reducible Galois representations.

The index of an Eisenstein ideal of multiplicity one

E-print/Working paper (2013)

Mazur’s fundamental work on Eisenstein ideals for prime level has a variety of arith- metic applications. In this article, we generalize his work to arbitrary square-free level. We compute the index of an ... [more ▼]

Mazur’s fundamental work on Eisenstein ideals for prime level has a variety of arith- metic applications. In this article, we generalize his work to arbitrary square-free level. We compute the index of an Eisenstein ideal and the dimension of the m-torsion of the modular Jacobian vari- ety, where m is an Eisenstein maximal ideal. In many cases, the dimension of the m-torsion is 2, in other words, multiplicity one theorem holds. [less ▲]