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1. CJM Online first
Twists of Shimura Curves Consider a Shimura curve $X^D_0(N)$ over the rational
numbers. We determine criteria for the twist by an Atkin-Lehner
involution to have points over a local field. As a corollary we give a
new proof of the theorem of Jordan-LivnÃ© on $\mathbf{Q}_p$ points
when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$
points when $p\mid N$. We also give congruence conditions for roots
modulo $p$ of Hilbert class polynomials.
Keywords:Shimura curves, complex multiplication, modular curves, elliptic curves Categories:11G18, 14G35, 11G15, 11G10 |
2. CJM 2009 (vol 62 pp. 456)
The ChowlaâSelberg Formula and The Colmez Conjecture In this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.
Categories:11G15, 11F41, 14K22 |
3. CJM 2005 (vol 57 pp. 1102)
Power Residues of Fourier Coefficients of Modular Forms Let $\rho \colon G_{\Q} \to \GL_{n}(\Ql)$ be a motivic $\ell$-adic Galois
representation. For fixed $m > 1$ we initiate an investigation of the
density of the set of primes $p$ such that the trace of the image of an
arithmetic Frobenius at $p$ under $\rho$ is an $m$-th power residue
modulo $p$. Based on numerical investigations with modular forms we
conjecture (with Ramakrishna) that this density equals $1/m$ whenever the
image of $\rho$ is open. We further conjecture that for such $\rho$ the set
of these primes $p$ is independent of any set defined by Cebatorev-style
Galois-theoretic conditions (in an appropriate sense). We then compute these
densities for certain $m$ in the complementary case of modular forms of
CM-type with rational Fourier coefficients; our proofs are a combination of
the Cebatorev density theorem (which does apply in the CM case) and
reciprocity laws applied to Hecke characters. We also discuss a potential
application (suggested by Ramakrishna) to computing inertial degrees at $p$
in abelian extensions of imaginary quadratic fields unramified away from $p$.
Categories:11F30, 11G15, 11A15 |