Sub-convexity bound for $$GL(3) \times GL(2)$$ L-functions: the depth aspect

Abstract

Let f be an Hecke eigenform for the group Γ 0 (q) and χ d be a primitive quadratic character of conductor |d|. In this article, we prove an asymptotic for the second moment of the derivative of L(s, f ⊗ χ 8d ) at the central point 1/2, which was previously known under GRH by Petrow [8].

Subconvexity bound for ${\rm GL}(2)$ $L$-functions: $t$-aspect

“…This section will provide a rough idea of the steps involved in the proof of our main result (1). By using the approximate functional equation (see Lemma4.1) for the Lfunction L 1 2 + it, π × f , we see that our main object of study will become the sum S r (N) given in (11). To get our desired result, we need to analyze this sum S r (N) and show some cancellations.…”

“…Our next step is to apply the the GL(3) and GL(2)-Voronoi summation formulas to sum over n and m, respectively. We will proceed by assuming (a + bq, p 1 q) = 1, and the otherwise case can be done in a similar way as in [11]. We have the following lemma.…”

“…In this article, we yet again explore the subconvexity problem (ScP) for the degree six Rankin-Selberg GL(3) × GL(2) L-functions using Munshi's delta method [20]. This theme was initiated by Munshi [17] and was explored further by the second author and the third author along with Sharma, Mallesham, and various others (see [9], [10], [11], [12], [22]) to resolve subconvexity for these L-functions in various aspects (t, twist and spectral aspect). Apparently, the delta method approach so far has been more effective for L-functions associated with forms admitting a varying GL(1) factor, barring a few results (see [9], [10], [12]), where spectral parameters of a higher degree form vary, or levels of two higher degree forms vary simultaneously.…”

“…This theme was initiated by Munshi [17] and was explored further by the second author and the third author along with Sharma, Mallesham, and various others (see [9], [10], [11], [12], [22]) to resolve subconvexity for these L-functions in various aspects (t, twist and spectral aspect). Apparently, the delta method approach so far has been more effective for L-functions associated with forms admitting a varying GL(1) factor, barring a few results (see [9], [10], [12]), where spectral parameters of a higher degree form vary, or levels of two higher degree forms vary simultaneously. In the latter case, the authors produce subconvexity in some relative range of the levels and their method does not allow them to fix one of the forms.…”