The surface of Mars contains abundant sinuous ridges that appear similar to river channels in planform, but they stand as topographic highs. Ridges have similar curvature-to-width ratios as terrestrial meandering rivers, which has been used to support the hypothesis that ridges are inverted channels that directly reflect channel geometry. Anomalously wide ridges, in turn, have been interpreted as evidence for larger rivers on Mars compared to Earth. However, an alternate hypothesis is that ridges are exhumed channel-belt deposits—a larger zone of relatively coarse-grained deposits formed from channel lateral migration and aggradation. Here, we measured landform wavelength, radius of curvature, and width to compare terrestrial channels, terrestrial channel belts, and martian ridges. We found that all three landforms follow similar scaling relations, in which ratios of radius of curvature to width range from 1.7 to 7.3, and wavelength-to-width ratios range from 5.8 to 13. We interpret this similarity to be a geometric consequence of a sinuous curved line of finite width. Combined with observations of ridge-stacking patterns, our results suggest that wide ridges on Mars could indicate fluvial channel belts that formed over significant time rather than anomalously large rivers.

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We prove a characteristic p version of a theorem of Silverman on integral points in orbits over number fields and establish a primitive prime divisor theorem for polynomials in this setting. We provide some applications of these results, including a finite index theorem for arboreal representations coming from quadratic polynomials over function fields of odd characteristic.

Half-integer weight Hecke operators and their distinct properties play a major role in the theory surrounding partition numbers and Dedekind's eta-function. Generalizing the work of Ono in [K. Ono, The partition function and Hecke operators, Adv. Math. 228 (2011) 527-534], here we obtain closed formulas for the Hecke images of all negative powers of the eta-function. These formulas are generated through the use of Faber polynomials. In addition, congruences for a large class of powers of Ramanujan's Deltafunction are obtained in a corollary. We further exhibit a fast calculation for many large values of vector partition functions.

Given a family of abelian varieties over a quasiprojective smooth curve T 0 over a global field and a point P on the generic fiber, we show that the Néron-Tate canonical height hX t (Pt) of Pt along each fiber is exactly equal to a Weil height h M (t) given by an adelic metrized line bundle M on the unique smooth projective curve T containing T 0 . As a consequence, we show that a conjecture of Zhang on the finiteness of small-height specializations of P is equivalent to M being big.