In this talk I shall discuss the conjecture of the existence of a subglacial lake at Svalbard: why it was raised, in which physical environment, why it is worth to be investigated. With the aim to provide a numerical validation, I have proposed an algorithm based on the classical conservation laws for the system 'atmosphere, glacier, subglacial lake and bedrock'. Actually, numerical tests confirm the likelihood of the conjectured lake. Critical aspects of the adopted mathematical numerical modelling will be presented.