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We present the Lyman-$a$ (Lya) Luminosity Function (LF) at $2.05<z<3.75$,estimated from a sample of 67 Lya-emitter (LAE) candidates in the J-PASPathfinder surveys: miniJPAS and J-NEP. These two surveys cover a totaleffective area of $\sim 1.14$ deg$^2$ with 54 Narrow Band (NB) filters acrossthe optical range, with typical limiting magnitudes of $\sim 23$. This set ofNBs allows to probe Lya emission in a wide and continuous range of redshifts.We develop a method for detecting Lya emission for the estimation of the Lya LFusing the whole J-PAS filter set. We test this method by applying it to theminiJPAS and J-NEP data. In order to compute the corrections needed to estimatethe Lya LF and to test the performance of the candidates selection method, webuild mock catalogs. These include representative populations of Lya Emittersat $1.9<z<4.5$ as well as their expected contaminants, namely low-$z$ galaxiesand $z<2$ QSOs. We show that our method is able to provide the Lya LF at theintermediate-bright range of luminosity ($\rm 10^{43.5} erg\,s^{-1} \lesssimL_{Lya} \lesssim 10^{44.5} erg\,s^{-1}$). The photometric information providedby these surveys suggests that our samples are dominated by bright,Lya-emitting Active Galactic Nuclei. At $L_{{\rm Ly}a}<10^{44.5}$erg\,s$^{-1}$, we fit our Lya LF to a power-law with slope $A=0.70\pm0.25$. Wealso fit a Schechter function to our data, obtaining: Log$(\Phi^* /\text{Mpc$^{-3}$})=-6.30^{+0.48}_{-0.70}$, Log$(L^*/ \rmerg\,s^{-1})=44.85^{+0.50}_{-0.32}$, $a=-1.65^{+0.29}_{-0.27}$. Overall, ourresults confirm the presence of an AGN component at the bright-end of the LyaLF. In particular, we find no significant contribution of star-forming LAEs tothe Lya LF at Log$(L_{\rm Lya}$ / erg s$^{-1}$)>43.5. This work serves as aproof-of-concept for the results that can be obtained with the upcoming datareleases of the J-PAS survey.

The miniJPAS & J-NEP surveys: Identification and characterization of the Ly$\alpha$ Emitter population and the Ly$\alpha$ Luminosity Function