Explicit formulas in the theory of automorphic forms

Moreno, Carlos

doi:10.1007/bfb0063065

Cited by 58 publications

(38 citation statements)

References 24 publications

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“…In [LiuWangYe], (6.1) was proved as a consequence of a stronger weighted prime number theorem for a Rankin-Selberg L-function, and hence require a zero-free region of the classical type (cf. [Mor1], [Mor2], [GelLapSar], and [Sar]). This is the reason why we have to assume that at least one of π and π is self-contragredient in (6.1).…”

Section: Rankin-selbergmentioning

confidence: 99%

Zeros of Automorphic L-Functions and Noncyclic Base Change

Liu

Number Theory

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Abstract.Let π be an automorphic irreducible cuspidal representation of GL m over a Galois (not necessarily cyclic) extension E of Q of degree . Assume that π is invariant under the action of the Galois group Gal E/Q . We computed the n-level correlation of normalized nontrivial zeros of L(s, π) and proved that it is equal to the n-level correlation of normalized nontrivial zeros of a product of distinct L-functions L(s, π 1 ) · · · L(s, π ) attached to cuspidal representations π 1 , . . . , π of GL m over Q. This is done unconditionally for m = 1, 2 and for m = 3, 4 with the degree having no prime factor ≤ (m 2 + 1)/2. In other cases, the computation is under a conjecture of bounds toward the Ramanujan conjecture over E, and a conjecture on convergence of certain series over prime powers (Hypothesis H over E and Q). The results provide an evidence that π should be (noncyclic) base change of distinct cuspidal representations π 1 , . . . , π of GL m (Q A ), if it is invariant under the Galois action. A technique used in this article is a version of Selberg orthogonality for automorphic L-functions (Lemma 6.2 and Theorem 6.4), which is proved unconditionally, without assuming π and π 1 , . . . , π being self-contragredient. Introduction.According to Langlands' functoriality conjecture, the L-function attached to an automorphic irreducible cuspidal representation π of GL m over a number field E should equal a product of L(s, π j ) for certain cuspidal representations π j of GL m j over Q. Arthur and Clozel [ArtClo] proved that this is indeed the case when E is a cyclic Galois extension of Q and π is stable under Gal E/Q . In fact in this case, π is the base change of exactly nonequivalent cuspidal representations π Q , π Q ⊗ η E/Q , . . . , π Q ⊗ η

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Section: Rankin-selbergmentioning

confidence: 99%

Zeros of Automorphic L-Functions and Noncyclic Base Change

Liu

Number Theory

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“…, |t| ≤ 1, (4.4) for some effectively computable positive constants c 3 and c 4 , if at least one of π and π is self-contragredient (Moreno [20] [21], Sarnak [25], and Gelbart, Lapid, and Sarnak [2]). Now we prove a weighted prime number theorem in the diagonal case.…”

Section: Rs2mentioning

confidence: 99%

“…Moreno [20] avoided GRC by an averaging technique, while others restricted themselves to the case of holomorphic cusp forms where GRC is known (Ichihara [5]), or to the Selberg class where GRC is assumed (Kaczorowski and Perelli [11]). …”

Section: Introductionmentioning

confidence: 99%

Perron’s Formula and the Prime Number Theorem for Automorphic L-Functions

Liu¹,

Ye²

2007

Pure and Applied Mathematics Quarterly

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In this paper the classical Perron's formula is modified so that it now depends no longer on sizes of individual terms but on a sum over a short interval. When applied to automorphic L-functions, this new Perron's formula may allow one to avoid estimation of individual Fourier coefficients, without assuming the Generalized Ramanujan Conjecture (GRC). As an application, a prime number theorem for Rankin-Selberg L-functions L(s, π ×π ) is proved unconditionally without assuming GRC, where π and π are automorphic irreducible cuspidal representations of GL m (Q A ) and GL m (Q A ), respectively.

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“…d(n) = number of divisors of n; this is the RamanujanPetersson conjecture [30]. The Dirichlet series Lk(s) was studied extensively in Rankin [31], Goldstein [9] has proven Merten-type conditions for Lk(s), Moreno [24], [25] has proven von Mangoldt formulas for Lk(s) in a more general setting, see e.g., Langlands [20]. From the functional equation for the cusp form / we have the following; cf., Barrucand [ In particular, if A is not divisible by four, s = A/2 is a root of Lk(s).…”

Section: Sl(z) ¿Functions Rankin's Bookmentioning

confidence: 99%

On zeros of Mellin transforms of 𝑆𝐿₂(𝑍) cusp forms

Ferguson¹,

Major²,

Powell³

et al. 1984

Math. Comp.

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Abstract. Wc compute zeros of Meilin transforms of modular cusp forms for Si.2(Z). Such Meilin transforms arc eigenforms of Hecke operators. We recall that, for all weights k and all dimensions of cusp forms, the Meilin transforms of cusp forms have infinitely many zeros of the form A/2 + tf-~\ , i.e., infinitely many zeros on the critical line.A new basis theorem for the space of cusp forms is given which, together with the Selberg trace formula, renders practicable the explicit computations of the algebraic Fourier coefficients of cusp eigenforms required for the computations of the zeros. ■The first forty of these Meilin transforms corresponding to cusp eigenforms of weight k ^ 50 and dimension ^ 4 are computed for the sections of the critical strips, a + t\f-\ , k; -1 < la < k• + 1, -40 < t sj 40. The first few zeros lie on the respective critical lines A/2 + if-1 and are simple. A measure argument, depending upon the Riemann hypothesis for finite fields, is given which shows that Hasse-Weil L-functions (including the above) lie among Dirichlet series which do satisfy Riemann hypotheses (but which need not have functional equations nor analytic continuations).

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Explicit formulas in the theory of automorphic forms

Cited by 58 publications

References 24 publications

Zeros of Automorphic L-Functions and Noncyclic Base Change

Zeros of Automorphic L-Functions and Noncyclic Base Change

Perron’s Formula and the Prime Number Theorem for Automorphic L-Functions

On zeros of Mellin transforms of 𝑆𝐿₂(𝑍) cusp forms

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