Algebraic Cycles and Even Unimodular Lattices

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135450/1/jlms0209.pd

Exact holomorphic differentials on a quotient of the Ree curve

We produce several families of exact holomorphic differentials on a quotient X of the Ree curve in characteristic 3, defined by X:yq-y=xq0(xq-x)/Fq (where q0 = 3s, s ≥ 1 and q=3q02). We conjecture that they span the whole space of exact holomorphic differentials, and prove this in the cases s = 1 and s = 2, by calculating the kernel of the Cartier operator

EISENSTEIN PRIMES, CRITICAL VALUES AND GLOBAL TORSION

We consider congruences between Eisenstein series and cusp forms—of weight k, level N and character χ of conductor N—modulo large prime divisors of L(1 − k, χ−1). We show that such primes occur in the order of a “global torsion ” group attached to the cusp form f, and (under a certain hypothesis) also in the denominator of the algebraic part of the rightmost critical value L f (k − 1). These occurrences are linked by the Bloch–Kato conjecture. 1

SOME SIEGEL MODULAR STANDARD L-VALUES, AND SHAFAREVICH-TATE GROUPS

We explain how the Bloch-Kato conjecture leads us to the following conclusion: a large prime dividing a critical value of the L-function of a classical Hecke eigenform f of level 1, should often also divide certain ratios of critical values for the standard L-function of a related genus two (and in general vector-valued) Hecke eigenform F. The relation between f and F (Harder’s conjecture in the vector-valued case) is a congruence involving Hecke eigenvalues, modulo the large prime. In the scalar-valued case we prove the divisibility, subject to weak conditions. In two instances in the vector-valued case, we confirm the divisibility using elaborate computations involving special differential operators. These computations do not depend for their validity on any unproved conjecture