We investigate the IceCube detection potential of very high energy neutrinos from blazars , for different classes of ” hadronic ” models , taking into account the limits imposed on the neutrino flux by the recent Fermi telescope observations .

Assuming the observed \gamma -ray emission is produced by the decay of neutral pions from proton-proton interactions , the measurement of the time-averaged spectral characteristics of blazars in the GeV energy band imposes upper limits on the time-averaged neutrino flux .
Comparing these upper limits to the 5 \sigma discovery threshold of IceCube for different neutrino spectra and different source locations in the sky , we find that several BL Lacs with hard spectra in the GeV band are within the detection potential of IceCube .
If the \gamma -ray emission is dominated by the neutral pion decay flux , none of the flat-spectrum radio quasars are detectable with IceCube .
If the primary high energy proton spectrum is very hard and/or neutrinos are produced in proton-photon , rather than proton-proton reactions , the upper limit on the neutrino flux imposed by the measured \gamma -ray spectra is relaxed and gamma-ray observations impose only lower bounds on the neutrino flux .
We investigate whether these lower bounds guarantee the detection of blazars with very hard neutrino spectra ( spectral index \Gamma _ { \nu } \sim 1 ) , expected in the latter type model .

We show that all the ” hadronic ” models of activity of blazars are falsifiable with IceCube .
Furthermore , we show that models with \gamma -ray emission produced by the decay of neutral pions from proton-proton interactions can be readily distinguished from the models based on proton-gamma interactions and/or models predicting very hard high energy proton spectra via a study of the distribution of spectral indices of \gamma -ray spectra of sources detected with IceCube .