An explicit open image theorem for products of elliptic curves

Let $K$ be a number field and $E_1, \ldots, E_n$ be elliptic curves over $K$, pairwise non-isogenous over $\overline{K}$ and without complex multiplication over $\overline{K}$. We study the image of the adelic representation of the absolute Galois group of $K$ naturally attached to $E_1 \times \cdots \times E_n$. The main result is an explicit bound for the index of this image in $\left\{ (x_1,\ldots,x_n) \in \operatorname{GL}_2(\hat{\mathbb{Z}})^n \bigm\vert \det x_i = \det x_j \;\; \forall i,j \right\}$.Comment: 18 pages. v2: improved expositio

Bounds for Serre's open image theorem for elliptic curves over number fields

For $E/K$ an elliptic curve without complex multiplication we bound the index of the image of $\operatorname{Gal}(\bar{K}/K)$ in $\operatorname{GL}_2(\hat{\mathbb{Z}})$, the representation being given by the action on the Tate modules of $E$ at the various primes. The bound is effective and only depends on $[K:\mathbb{Q}]$ and on the stable Faltings height of $E$. We also prove a result relating the structure of subgroups of $\operatorname{GL}_2(\mathbb{Z}_\ell)$ to certain Lie algebras naturally attached to them.Comment: Final version, accepted for publication in Algebra and Number Theor

Legal Determinants of the Return on Equity

Recent work documents that better legal institutions are associated with broader equity markets. We investigate whether international differences in legal institutions also help explain the international cross-section of expected stock returns. We document three main regularities. First, total stock market returns are positively correlated with overall measures of the quality of institutions, such as judicial efficiency and rule of law, but have no relationship with measures of shareholder rights, controlling for risk. Second, dividend yields and earning-price ratios also correlate positively with judicial efficiency and rule of law, but negatively with shareholder rights' protection, controlling for risk and expected earnings growth. Thirdly, the excess return on new issues is negatively associated with the quality of accounting standards. We interpret the positive effect of the overall quality of institutions on equity returns as capturing the resulting curtailment of private benefits and increase of profitability, under imperfect international integration of stock markets. The negative impact of shareholders' legal protection and of accounting standards can instead be seen as resulting from the implied reduction in shareholders' auditing and monitoring costs.law, enforcement, shareholder protection, corporate governance, return on equity

Law and Equity Markets: a Simple Model

We analyze how the law and its enforcement affect equity market equilibrium. Improvements in the legal system, while invariably associated with broader equity markets, have different effects on equity returns depending on the institutional change considered and on the degree of international stock market segmentation. The model is useful to interpret the results of recent empirical work, such as La Porta et al. (1997) and Lombardo and Pagano (1999). In particular, it can rationalize the observed cross-country pattern, whereby better institutions are associated both with broader equity markets and higher risk-adjusted returns on equity.law, enforcement, shareholder protection, corporate governance, return on equity