Armand Brumer

Paramodular abelian varieties of odd conductor

A precise and testable modularity conjecture for rational abelian surfaces A with trivial endomorphisms, End_Q A = Z, is presented. It is consistent with our examples, our non-existence results and recent work of C. Poor and D. S. Yuen on weight 2 Siegel paramodular forms. We obtain fairly precise information on ell-division fields of semistable abelian varieties A, mainly when A[ell] is reducible, by considering extension problems for groups schemes of small rank. Our general results imply, for instance, that the least prime conductor of an abelian surface is 277.

The number of elliptic curves over ℚ with conductorNwith conductorN

We prove that the number of elliptic curves E/ℚ with conductorN isO(N1/2+ε). More generally, we prove that the number of elliptic curves E/ℚ with good reduction outsideS isO(M1/2+ε), whereM is the product of the primes inS. Assuming various standard conjectures, we show that this bound can be improved toO(Mc/loglogM).

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.

The class group of all cyclotomic integers

For each nz 1, let &,=e2ni’n and let O,=Z[[,,] be the ring of integers in the cyclotomic field K, of nth roots of unity. We denote 0, = li,m On = Un On the ring of all cyclotomic integers, the direct limit being taken with respect to the natural inclusions 0,-O, for n dividing m. For any commutative ring A, the group of isomorphism classes of projective modules of rank 1 is denoted by Pit(A) as usual. For instance, Pic(OJ is the ideal class group of On while Pic(Om) = lim Pic(OJ. The aim of this note is to prove the following plausible result which does not seem to be in the literature.

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

The average rank of elliptic curves I

A_{/{\mathbb Q}}

The rank of elliptic curves

of prime conductor

The conductor of an abelian variety

N

Arithmetic of division fields

is favorable if its 2-division field