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Using Langlands's beyond endoscopy idea, we study the Asai $L$-function associated to a real quadratic field $\Bbb{K}/\Bbb{Q}$. We prove that the Asai $L$-function associated to a cuspidal automorphic representation over $\Bbb{K}$ has analytic continuation to the complex plane with at most a simple pole at $s=1$. We then show if the $L$-function has a pole then the representation is a base change from $\Bbb{Q}$. While this result is known using integral representations from the work of Asai and Flicker, the approach here uses novel analytic number techniques and gives a deeper understanding of the geometric side of the relative trace formula. Moreover, the approach in this paper will make it easier to grasp more complicated functoriality via beyond endoscopy. We also describe a connection of Langlands's technique to the distinguishing of representations via periods.

Quadratic base change and the analytic continuation of the Asai L-function: A new trace formula approach