Identities for Anderson generating functions for Drinfeld modules

Anderson generating functions are generating series for division values of points on Drinfeld modules, and they serve as important tools for capturing periods, quasi-periods, and logarithms. They have been fundamental in recent work on special values of positive characteristic L-series and in transcendence and algebraic independence problems. In the present paper we investigate techniques for expressing Anderson generating functions in terms of the defining polynomial of the Drinfeld module and determine new formulas for periods and quasi-periods.Comment: 18 page

Linear independence of Gamma values in positive characteristic

We investigate the arithmetic nature of special values of Thakur's function field Gamma function at rational points. Our main result is that all linear independence relations over the field of algebraic functions are consequences of the known relations of Anderson and Thakur arising from the functional equations of the Gamma function.Comment: 51 page

The Semisimplicity Conjecture for A-Motives

We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V_p(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Akio Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G_p(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G_p(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V_p(M). The second states that the connected component of G_p(M) is reductive if M is semisimple and has a separable endomorphism algebra.Comment: 47 page

Variations of training load, monotony, and strain and dose-response relationships with maximal aerobic speed, maximal oxygen uptake, and isokinetic strength in professional soccer players

This study aimed to identify variations in weekly training load, training monotony, and training strain across a 10-week period (during both, pre- and in-season phases); and to analyze the dose-response relationships between training markers and maximal aerobic speed (MAS), maximal oxygen uptake, and isokinetic strength. Twenty-seven professional soccer players (24.9±3.5 years old) were monitored across the 10-week period using global positioning system units. Players were also tested for maximal aerobic speed, maximal oxygen uptake, and isokinetic strength before and after 10 weeks of training. Large positive correlations were found between sum of training load and extension peak torque in the right lower limb (r = 0.57, 90%CI[0.15;0.82]) and the ratio agonist/antagonist in the right lower limb (r = 0.51, [0.06;0.78]). It was observed that loading measures fluctuated across the period of the study and that the load was meaningfully associated with changes in the fitness status of players. However, those magnitudes of correlations were small-to-large, suggesting that variations in fitness level cannot be exclusively explained by the accumulated load and loading profile