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

A Difference Version of Nori's Theorem

We consider (Frobenius) difference equations over (F_q(s,t), phi) where phi fixes t and acts on F_q(s) as the Frobenius endomorphism. We prove that every semisimple, simply-connected linear algebraic group G defined over F_q can be realized as a difference Galois group over F_{q^i}(s,t) for some i in N. The proof uses upper and lower bounds on the Galois group scheme of a Frobenius difference equation that are developed in this paper. The result can be seen as a difference analogue of Nori's Theorem which states that G(F_q) occurs as (finite) Galois group over F_q(s).Comment: 29 page

Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms

We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.Comment: 39 page

Concerted electron-proton transfer in the optical excitation of hydrogen-bonded dyes

The simultaneous, concerted transfer of electrons and protons—electron-proton transfer (EPT)—is an important mechanism utilized in chemistry and biology to avoid high energy intermediates. There are many examples of thermally activated EPT in ground-state reactions and in excited states following photoexcitation and thermal relaxation. Here we report application of ultrafast excitation with absorption and Raman monitoring to detect a photochemically driven EPT process (photo-EPT). In this process, both electrons and protons are transferred during the absorption of a photon. Photo-EPT is induced by intramolecular charge-transfer (ICT) excitation of hydrogen-bonded-base adducts with either a coumarin dye or 4-nitro-4′-biphenylphenol. Femtosecond transient absorption spectral measurements following ICT excitation reveal the appearance of two spectroscopically distinct states having different dynamical signatures. One of these states corresponds to a conventional ICT excited state in which the transferring H+ is initially associated with the proton donor. Proton transfer to the base (B) then occurs on the picosecond time scale. The other state is an ICT-EPT photoproduct. Upon excitation it forms initially in the nuclear configuration of the ground state by application of the Franck–Condon principle. However, due to the change in electronic configuration induced by the transition, excitation is accompanied by proton transfer with the protonated base formed with a highly elongated +H─B bond. Coherent Raman spectroscopy confirms the presence of a vibrational mode corresponding to the protonated base in the optically prepared state

ANOTEONSIMPLEa-POINTS OF L-FUNCTIONS

Abstract. We prove, subject to certain hypotheses, that a positive proportion of the a-points of the Riemann zeta-function and Dirichlet L-functions with primitive characters are simple and discuss corresponding results for other functions in the Selberg class. We also prove an unconditional result of this type for the a-points in fixed strips to the right of the line ℜs =1/2. 1