Kenneth Kramer

Non-existence of certain semistable abelian varieties

Abstract: We show that there is no semistable abelian variety defined over ℚ with bad reduction at exactly one prime p≤ 7.

Semistable Abelian Varieties with Small Division Fields

The conjecture of Shimura-Taniyama-Weil, now proved through the work of Wiles and disciples, is only part of the Langlands program. Based on a comparison of the local factors ([And], [Seri]), it also predicts that the L-series of an abelian surface defined over ℚ should be the L-series of a Hecke eigen cusp form of weight 2 on a suitable group commensurable with Sp4 (ℤ). The only decisive examples are related to lifts of automorphic representations of proper subgroups of Sp4, for example the beautiful work of Yoshida ([Yos], [BSP]).

Certain abelian varieties bad at only one prime

An abelian surface

Residue properties of certain quadratic units

A_{/{\mathbb Q}}

The rank of elliptic curves

of prime conductor

The conductor of an abelian variety

N

Modularity of a family of elliptic curves

is favorable if its 2-division field

On the 2-Sylow subgroup of the Hilbert kernel of

F

K_{2}

is an

of number fields

{\mathcal S}_5