We state a conjecture on the reduction modulo defining characteristic of unipotent representation finite reductive group.

We shall call a simple abelian variety A/Q modular if it is isogenous (over Q) to a factor of the Jacobian of a modular curve. In this paper we shall call a representation ρ̄ : GQ→GL2(F̄l) modular if there is a modular abelian variety A/Q, a number field F/Q of degree equal to dimA, an embedding OF ↪→ End(A/Q) and a homomorphism θ : OF→F̄l such that ρ̄ is equivalent to the action of GQ on the ker θ...

(i) Since 32 = 9 = 7 · 1 + 2, then we clearly have 32 mod 7 = 9 mod 7 = 2. (ii) The solution is easy found by inspection: We have 4 · 4 mod 15 = 16 mod 15 = 1, and thus 4−1 mod 15 = 4. (iii) I have seen many complicated solutions here, but the solution is very simple, and uses the fact that 32 mod 8 = 9 mod 8 = 1. Then, we use the fact (discussed in class) that a · b mod n = (a mod n) · (b mod ...

Let $A$ be a right coherent ring and $\mathcal{X}$ contravariantly finite subcategory of ${\rm{mod}}\mbox{-}A$ containing projectives. In this paper, we construct recollement abelian categories $({\rm{mod}}_{0}\mbox{-}\mathcal{X}, {\rm{mod}}\mbox{-}\mathcal{X}, {\rm{mod}}\mbox{-}A)$, where ${\rm{mod}}_{0}\mbox{-}\mathcal{X}$ is full ${\rm{mod}}\mbox{-}\mathcal{X}$ consisting all functors vanish...

Example 1.1. Let m = 7. The following table shows that the first time a nonzero number mod 7 has a power congruent to 1 varies. While Fermat’s little theorem tells us that a6 ≡ 1 mod 7 for a 6≡ 0 mod 7, we see in the table that the exponent 6 can be replaced by a smaller positive exponent for 1, 2, 4, and 6. k 1 2 3 4 5 6 1k mod 7 1 2k mod 7 2 4 1 3k mod 7 3 2 6 4 5 1 4k mod 7 4 2 1 5k mod 7 5 ...

Purpose. To study whether a corneal thickness segmentation model, consisting in a central circular zone of 1 mm radius centered at the corneal apex (zone I) and five concentric rings of 1 mm width (moving outwards: zones II to VI), could boost the diagnostic accuracy of Heidelberg Retina Tomograph's (HRT's) MRA and GPS. Material and Methods. Cross-sectional study. 121 healthy volunteers and 125...