We break the convexity bound in the t-aspect for L-functions attached to cusp forms f for GL2(k) over arbitrary number fields k. The argument uses asymptotics with error term with a power saving, for second integral moments over spectral families of twists L(s,f⊗χ) by Grossencharacters χ, from our previous paper on integral moments.

We obtain second integral moments of automorphic L-functions on adele groups GL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL1 and GL2. This requires reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL2(ℚ) or GL2(ℚ(i)) with a single, coherent, global object that makes sense over a number field. This is the first expression of integral moments in adele-group terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers ℚ, we recover the classical results on moments.

We consider twisted multivariable zeta series associated to polynomials of several variables. We introduce a new class of polynomials, namely HDF, that contains strictly non-degenerate and hypoelliptic polynomials. For polynomials belonging to the HDF class, we show that we can extend holomorphically our series to $\mathbb{C}^T$. Then, thanks to a new principle called ‘the Exchange Lemma’, we give very simple formulae for the values of our series at $T$-tuples of negative integers. Finally, we make the $p$-adic interpolation of those values. Thus, we have generalized the results of Cassou-Noguès (that she used to construct the $p$-adic $L$-functions of totally real fields) in two ways: we consider multivariable series and our series are associated to more general polynomials. In addition, our proof is completely different.

We consider the Dirichlet series Z(P,A;s) = [sum ]$_m∈ A ∩ Z^n$P$^-s$(m) (s ∈ C) where P ∈ R[X$_1$, …, X$_n$] and A is an open semi-algebraic subset of R$^n$. We will say that Z(P,A;s) exists if this multiple series is absolutely convergent. In this paper we study the existence and several properties of meromorphic continuations of such series, under certain assumption on P and A. As an application, we show the existence of a finite asymptotic expansion of the counting function with support in A: N$_p$(A,t):= [sharp ] m ∈ A ∩ Z$^n$ | P(m) [les ] t} when t → +∞.