Rankin Selberg L-functions Research Articles

A note on the standard zero-free region for [formula omitted]-functions

In this short note, we establish a standard zero-free region for a general class of L-functions for which their logarithms have coefficients with nonnegative real parts, including the Rankin–Selberg L-functions for unitary cuspidal automorphic representations.

On Möbius Functions from Automorphic Forms and a Generalized Sarnak’s Conjecture

Abstract In this paper, we consider generalized Möbius functions associated with two types of L-functions: Rankin–Selberg L-functions of symmetric powers of distinct holomorphic cusp forms and L-functions derived from Maass cusp forms. We show that these generalized Möbius functions are weakly orthogonal to bounded sequences. As a direct corollary, a generalized Sarnak’s conjecture holds for these two types of Möbius functions.

Exponential sums with the Dirichlet coefficients of Rankin–Selberg L-functions

Exponential sums with the Dirichlet coefficients of Rankin–Selberg L-functions

On extra zeros of 𝑝-adic Rankin–Selberg 𝐿-functions

A version of the extra-zero conjecture, formulated by the first named author, is proved for p p -adic L L -functions associated with Rankin–Selberg convolutions of modular forms of the same weight. This result provides an evidence in support of this conjecture in the noncritical case, which remained essentially unstudied.

On the Simultaneous Sign Changes of Coefficients of Rankin–Selberg L-Functions over a Certain Integral Binary Quadratic Form

In this paper, we consider the simultaneous sign changes of coefficients of Rankin–Selberg L-functions associated to two distinct Hecke eigenforms supported at positive integers represented by some certain primitive reduced integral binary quadratic form with negative discriminant D. We provide a quantitative result for the number of sign changes of such sequence in the interval (x, 2x] for sufficiently large x.

On the asymptotics of coefficients of Rankin–Selberg L-functions

On the asymptotics of coefficients of Rankin–Selberg L-functions

Rank growth of elliptic curves over 𝑁-th root extensions

Fix an elliptic curve E E over a number field F F and an integer n n which is a power of 3 3 . We study the growth of the Mordell–Weil rank of E E after base change to the fields K d = F ( d 2 n ) K_d = F(\!\sqrt [2n]{d}) . If E E admits a 3 3 -isogeny, then we show that the average “new rank” of E E over K d K_d , appropriately defined, is bounded as the height of d d goes to infinity. When n = 3 n = 3 , we moreover show that for many elliptic curves E / Q E/\mathbb {Q} , there are no new points on E E over Q ( d 6 ) \mathbb {Q}(\sqrt [6]d) , for a positive proportion of integers d d . This is a horizontal analogue of a well-known result of Cornut and Vatsal [Nontriviality of Rankin-Selberg L-functions and CM points, L-functions and Galois representations, vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 121–186]. As a corollary, we show that Hilbert’s tenth problem has a negative solution over a positive proportion of pure sextic fields Q ( d 6 ) \mathbb {Q}(\sqrt [6]{d}) . The proofs combine our recent work on ranks of abelian varieties in cyclotomic twist families with a technique we call the “correlation trick”, which applies in a more general context where one is trying to show simultaneous vanishing of multiple Selmer groups. We also apply this technique to families of twists of Prym surfaces, which leads to bounds on the number of rational points in sextic twist families of bielliptic genus 3 curves.

On the Local Maxima Behaviour of Hecke Eigenvalues and Its Applications

Let Hk denote the space of primitive holomorphic cusp forms of even integral weight k for the full modular group Γ = SL(2, ℤ). Denote by $${\lambda _{{\rm{sy}}{{\rm{m}}^m}f}}(n)$$ the nth normalized coefficient of the Dirichlet expansion of the mth symmetric power L-function associated to f. In this paper, we establish the asymptotic formulas of sums of pairwise maxima concerning the normalized coefficients of symmetric power L-functions. We also establish similar results for the normalized coefficients of Rankin-Selberg L-functions L(symif × symjf, s) and L(symif × symjg, s) attached to f and g, respecitvely. As applications, we also consider the proportion of the sign changes among the difference of normalized coefficients associated with symmetric power L-functions and Rankin-Selberg L-functions attached to f and g, respectively.

On the Rankin–Selberg L-function related to the Godement–Jacquet L-function

On the Rankin–Selberg L-function related to the Godement–Jacquet L-function

Large oscillations of the argument of Rankin–Selberg L-functions

Large oscillations of the argument of Rankin–Selberg L-functions

On the Beilinson–Bloch–Kato conjecture for Rankin–Selberg motives

In this article, we study the Beilinson–Bloch–Kato conjecture for motives associated to Rankin–Selberg products of conjugate self-dual automorphic representations, within the framework of the Gan–Gross–Prasad conjecture. We show that if the central critical value of the Rankin–Selberg L-function does not vanish, then the Bloch–Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch–Kato Selmer group constructed from a certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanishing of the central critical first derivative of the Rankin–Selberg L-function, then the Bloch–Kato Selmer group is of rank one.

The second moment of Rankin–Selberg L-functions

Abstract We prove an asymptotic expansion of the second moment of the central values of the $\mathrm {GL}(n)\times \mathrm {GL}(n)$ Rankin–Selberg L-functions $L(1/2,\pi \otimes \pi _0)$ for a fixed cuspidal automorphic representation $\pi _0$ over the family of $\pi $ with analytic conductors bounded by a quantity that is tending to infinity. Our proof uses the integral representations of the L-functions, period with regularised Eisenstein series and the invariance properties of the analytic newvectors.

Rankin–Selberg periods for spherical principal series

By the unfolding method, Rankin-Selberg L-functions for ${\rm GL}(n)\times{\rm GL}(m)$ can be expressed in terms of period integrals. These period integrals actually define invariant forms on tensor products of the relevant automorphic representations. By the multiplicity-one theorems due to Sun-Zhu and Chen-Sun such invariant forms are unique up to scalar multiples and can therefore be related to invariant forms on equivalent principal series representations. We construct meromorphic families of such invariant forms for spherical principal series representations of ${\rm GL}(n,\mathbb{R})$ and conjecture that their special values at the spherical vectors agree in absolute value with the archimedean local L-factors of the corresponding L-functions. We verify this conjecture in several cases. This work can be viewed as the first of two steps in a technique due to Bernstein-Reznikov for estimating L-functions using their period integral expressions.

EXTREME VALUES OF GEODESIC PERIODS ON ARITHMETIC HYPERBOLIC SURFACES

AbstractGiven a closed geodesic on a compact arithmetic hyperbolic surface, we show the existence of a sequence of Laplacian eigenfunctions whose integrals along the geodesic exhibit nontrivial growth. Via Waldspurger’s formula we deduce a lower bound for central values of Rankin-Selberg L-functions of Maass forms times theta series associated to real quadratic fields.

Relations of rationality for special values of Rankin–Selberg L-functions of GLn×GLm over CM-fields

Relations of rationality for special values of Rankin–Selberg L-functions of GLn×GLm over CM-fields

LARGE VALUES OF L-FUNCTIONS ON THE 1-LINE

AbstractWe study lower bounds of a general family of L-functions on the $1$ -line. More precisely, we show that for any $F(s)$ in this family, there exist arbitrarily large t such that $F(1+it)\geq e^{\gamma _F} (\log _2 t + \log _3 t)^m + O(1)$ , where m is the order of the pole of $F(s)$ at $s=1$ . This is a generalisation of the result of Aistleitner, Munsch and Mahatab [‘Extreme values of the Riemann zeta function on the $1$ -line’, Int. Math. Res. Not. IMRN2019(22) (2019), 6924–6932]. As a consequence, we get lower bounds for large values of Dedekind zeta-functions and Rankin-Selberg L-functions of the type $L(s,f\times f)$ on the $1$ -line.

Upper bounds of some special zeros for the Rankin-Selberg L-function

Abstract In this paper, we prove some conditional results about the order of zero at central point s = 1/2 of the Rankin-Selberg L-function L(s, πf × π͠′ f ). Then, we give an upper bound for the height of the first zero with positive imaginary part of L(s, πf × π͠′ f ). We apply our results to automorphic L-functions.

Double square moments and subconvexity bounds for Rankin-Selberg L-functions of holomorphic cusp forms

Let $f$ and $g$ be holomorphic cusp forms of weights $k_1$and $k_2$ for the congruence subgroups$\Gamma_0(~\mathcal{N}_1~)$ and $\Gamma_0(~\mathcal{N}_2~)$,respectively. In this paper the square moment of theRankin-Selberg $L$-function for $f$ and $g$ in the aspect of both weights in short intervals is bounded, when$k_1^\varepsilon\ll~k_2\ll~k_1^{1-\varepsilon}$.These bounds are the mean Lindelof hypothesis in one caseand subconvexity bounds on average in other cases. Thesesquare moment estimates also imply subconvexity bounds forindividual $L(\frac12+{\rm~i}t,f\times~g)$ for all $g$ when $f$is chosen outside a small exceptional set. In the bestcase scenario the subconvexity bound obtained reaches theWeyl-type bound proved by Lau et al. (2006) inboth the $k_1$ and $k_2$ aspects.

Critical values of L-functions for GL3 × GL1 and symmetric square L-functions for GL2 over a CM field

In article [14], an algebraicity result, an extension of Mahnkopf's result [10], for the critical values of L-functions for GL3×GL1 over a totally real number field was proved. In this paper, we again prove an extension of Mahnkopf [10], for all the critical values of certain Rankin–Selberg L-functions attached to a pair of algebraic automorphic representations on GL3×GL1 over a CM field, having cohomology with a general coefficient system.

Test vectors for Rankin–Selberg L-functions

We study the local zeta integrals attached to a pair of generic representations (π,τ) of GLn×GLm, n>m, over a p-adic field. Through a process of unipotent averaging we produce a pair of corresponding Whittaker functions whose zeta integral is non-zero, and we express this integral in terms of the Langlands parameters of π and τ. In many cases, these Whittaker functions also serve as a test vector for the associated Rankin–Selberg (local) L-function.