Invariant dimensions and maximality of geometric monodromy action

Let X be a smooth separated geometrically connected variety over F_q and f:Y-> X a smooth projective morphism. We compare the invariant dimensions of the l-adic representation V_l and the F_l-representation \bar V_l of the geometric \'etale fundamental group of X arising from the sheaves R^wf_*Q_l and R^wf_*Z/lZ respectively. These invariant dimension data is used to deduce a maximality result of the geometric monodromy action on V_l whenever \bar V_l is semisimple and l is sufficiently large. We also provide examples for \bar V_l to be semisimple for l>>0

On the rationality of algebraic monodromy groups of compatible systems

Let E be a number field and X be a smooth geometrically connected variety defined over a characteristic p finite field F_q. Given an n-dimensional pure E-compatible system of semisimple \lambda-adic representations \rho_\lambda of the fundamental group \pi_1(X) with connected algebraic monodromy groups G_\lambda, we construct a common E-form G of all the groups G_\lambda. In the absolutely irreducible case, we construct a common E-form i:G->GL_{n,E} of all the tautological representations G_\lambda->GL_{n,E_\lambda} and a G-valued adelic representation \rho_A^G of \pi_1(X) such that their composition is isomorphic to the product representation of all \rho_\lambda. Moreover, if X is a curve and the (absolute) outer automorphism group of G^der is trivial, then the \lambda-components of \rho_A^G form an E-compatible system of G-representations. Analogous rationality results in characteristic zero, predicted by the Mumford-Tate conjecture, are obtained under some conditions including ordinariness.Comment: 35 pages. Thm. 1.1(ii) is improved so that G sits in GL_{n,E

Specialization of monodromy group and l-independence

Let $E$ be an abelian scheme over a geometrically connected variety $X$ defined over $k$, a finitely generated field over $\mathbb{Q}$. Let $\eta$ be the generic point of $X$ and $x\in X$ a closed point. If $\mathfrak{g}_l$ and $(\mathfrak{g}_l)_x$ are the Lie algebras of the $l$-adic Galois representations for abelian varieties $E_{\eta}$ and $E_x$, then $(\mathfrak{g}_l)_x$ is embedded in $\mathfrak{g}_l$ by specialization. We prove that the set $\{x\in X$ closed point $| (\mathfrak{g}_l)_x\subsetneq \mathfrak{g}_l\}$ is independent of $l$ and confirm Conjecture 5.5 in [2].Comment: 4 page

Consumer myopia, compatibility and aftermarket monopolization

In this paper, I show that the standard Bertrand competition argument does not apply when firms compete for myopic consumers who optimize period-by-period. I develop the model in the context of aftermarket. With overlapping-generations of consumers, simultaneous product offerings in the primary market and aftermarket establishes a price floor for the primary good. This constraint prevents aftermarket rents from being dissipated by the primary market competition. Duopoly firms earn positive profits despite price competition with undifferentiated products. Nonetheless, government interventions to reinforce aftermarket competition such as a standardization requirement may lead to the partial collapse of the primary market.aftermarket, Bertrand competition, bounded rationality, standardization.

Modeling Reverse-Phi Motion-Selective Neurons in Cortex: Double Synaptic-Veto Mechanism

Reverse-phi motion is the illusory reversal of perceived direction of movement when the stimulus contrast is reversed in successive frames. Livingstone, Tsao, and Conway (2000) showed that direction-selective cells in striate cortex of the alert macaque monkey showed reversed excitatory and inhibitory regions when two different contrast bars were flashed sequentially during a two-bar interaction analysis. While correlation or motion energy models predict the reverse-phi response, it is unclear how neurons can accomplish this. We carried out detailed biophysical simulations of a direction-selective cell model implementing a synaptic shunting scheme. Our results suggest that a simple synaptic-veto mechanism with strong direction selectivity for normal motion cannot account for the observed reverse-phi motion effect. Given the nature of reverse-phi motion, a direct interaction between the ON and OFF pathway, missing in the original shunting-inhibition model, it is essential to account for the reversal of response. We here propose a double synaptic-veto mechanism in which ON excitatory synapses are gated by both delayed ON inhibition at their null side and delayed OFF inhibition at their preferred side. The converse applies to OFF excitatory synapses. Mapping this scheme onto the dendrites of a direction-selective neuron permits the model to respond best to normal motion in its preferred direction and to reverse-phi motion in its null direction. Two-bar interaction maps showed reversed excitation and inhibition regions when two different contrast bars are presented