The relative trace formula in analytic number theory. (English) Zbl 1483.11101
Müller, Werner (ed.) et al., Relative trace formulas. Proceedings of the Simons symposium, Schloss Elmau, Germany, April 22–28, 2018. Cham: Springer. Simons Symp., 51-73 (2021).
This paper is a survey on asymptotic arithmetic estimates (bounds or true asymptotic) obtained by applying suitable group theoretical identities, in particular derived through relative trace formulæ and Kuznetsov-Bruggeman formula. There are many applications in analytic number theory and theory of automorphic forms. This survey is supplemented by about hundred bibliographical references.
The paper begins with the Poisson summation formula, interpreted as a trace formula: it induces original solutions for two problems: the Polya-Vinogradov inequality and the remainder bound for the number of lattice points in growing plane disks. These both asymptotically bounds are established in the Euclidean geometry framework while plane hyperbolic geometry, its spectral properties as well those of the modular surface \(\mathrm{SL}_2(\mathbb Z)\backslash \mathbb H^2\) (cusp forms, Hecke operators) are of special relevance to many of the problems quoted here. The formula (KB) established by N. V. Kuznetsov [Math. USSR, Sb. 39, 299–342 (1981); translation from Mat. Sb., N. Ser. 111(153), 334–383 (1980; Zbl 0427.10016)] and R. W. Bruggeman [Invent. Math. 45, 1–18 (1978; Zbl 0351.10019)] is crucial for many problems considered here. Although similar to the Selberg formula, as it relates certain spectral data to some integrals, it is quite different and worth to be quoted here. \[ \begin{split} 2\pi\sum_{\varphi\in\mathcal B} \frac{\lambda_\varphi(n)\lambda_\varphi(m)}{L(1,\mathrm{Ad}^2\varphi)} h(t_\varphi(n))+ \int_{\mathbb R} \frac{\sigma_{\mathrm{i}t}(n)\sigma_{\mathrm{i}t}(m)} {|\zeta(1+2\mathrm{i}t)|^2}h(t)\mathrm{d}t\\ =\delta_{m,n} \int_{\mathbb R}h(t)\tanh(\pi t)t\mathrm{d}t +\sum_c\frac 1cS(n,m,c)\widehat h_{\pm}\left(\frac{nm}{c^2}\right)\tag{KB} \end{split} \] Here, \(m,n\) are non zero integers, \(h\) is a sufficiently nice test function with \(\widehat h_\pm\) a certain integral transform of \(h\) given by a Bessel kernel depending on the sign \(\pm=\mathrm{sgn}(nm)\), \(\mathcal B\) is an orthonormal basis of joint cuspidal eigenfunctions, \(\sigma_{\mathrm{i}t}(n)=\sum_{ad=|n|}(a/d)^{\mathrm{i}t}\) is the Hecke eigenvalue of the corresponding Eisenstein series \[ S(n,m,c)=\sum_{\substack{d\pmod{c}\\ (d,c)=1}}\exp \left(\frac{2\mathrm{i}\pi(nd+m\overline{d})}c\right) \] Moreover, for a cuspidal form \(\varphi\), the \(\lambda_\varphi(n),n\in\mathbb Z\) are the Fourier coefficients in the expansion \[ \varphi(x+\mathrm{i}y)=\varphi_0+\sum_{n\not=0} \frac{\lambda_\varphi(n)}{\sqrt{|n|}} {\exp(2\mathrm{i}\pi nx)} W_{\mathrm{i}t_\varphi(n)}(4\pi |n|y), \] where \(W_\tau(y)\) is the standard Whittaker function and \(\lambda_\varphi=t_\varphi^2+1/4\). For the proof, see also H. Iwaniec and E. Kowalski [Analytic number theory. Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1059.11001), p. 360, Prop. 14.5] for a similar formula on holomorphic cusp forms.
Let us sketch the problems listed by the author, with a short discussion for each one.
For the entire collection see [Zbl 1469.11002].
The paper begins with the Poisson summation formula, interpreted as a trace formula: it induces original solutions for two problems: the Polya-Vinogradov inequality and the remainder bound for the number of lattice points in growing plane disks. These both asymptotically bounds are established in the Euclidean geometry framework while plane hyperbolic geometry, its spectral properties as well those of the modular surface \(\mathrm{SL}_2(\mathbb Z)\backslash \mathbb H^2\) (cusp forms, Hecke operators) are of special relevance to many of the problems quoted here. The formula (KB) established by N. V. Kuznetsov [Math. USSR, Sb. 39, 299–342 (1981); translation from Mat. Sb., N. Ser. 111(153), 334–383 (1980; Zbl 0427.10016)] and R. W. Bruggeman [Invent. Math. 45, 1–18 (1978; Zbl 0351.10019)] is crucial for many problems considered here. Although similar to the Selberg formula, as it relates certain spectral data to some integrals, it is quite different and worth to be quoted here. \[ \begin{split} 2\pi\sum_{\varphi\in\mathcal B} \frac{\lambda_\varphi(n)\lambda_\varphi(m)}{L(1,\mathrm{Ad}^2\varphi)} h(t_\varphi(n))+ \int_{\mathbb R} \frac{\sigma_{\mathrm{i}t}(n)\sigma_{\mathrm{i}t}(m)} {|\zeta(1+2\mathrm{i}t)|^2}h(t)\mathrm{d}t\\ =\delta_{m,n} \int_{\mathbb R}h(t)\tanh(\pi t)t\mathrm{d}t +\sum_c\frac 1cS(n,m,c)\widehat h_{\pm}\left(\frac{nm}{c^2}\right)\tag{KB} \end{split} \] Here, \(m,n\) are non zero integers, \(h\) is a sufficiently nice test function with \(\widehat h_\pm\) a certain integral transform of \(h\) given by a Bessel kernel depending on the sign \(\pm=\mathrm{sgn}(nm)\), \(\mathcal B\) is an orthonormal basis of joint cuspidal eigenfunctions, \(\sigma_{\mathrm{i}t}(n)=\sum_{ad=|n|}(a/d)^{\mathrm{i}t}\) is the Hecke eigenvalue of the corresponding Eisenstein series \[ S(n,m,c)=\sum_{\substack{d\pmod{c}\\ (d,c)=1}}\exp \left(\frac{2\mathrm{i}\pi(nd+m\overline{d})}c\right) \] Moreover, for a cuspidal form \(\varphi\), the \(\lambda_\varphi(n),n\in\mathbb Z\) are the Fourier coefficients in the expansion \[ \varphi(x+\mathrm{i}y)=\varphi_0+\sum_{n\not=0} \frac{\lambda_\varphi(n)}{\sqrt{|n|}} {\exp(2\mathrm{i}\pi nx)} W_{\mathrm{i}t_\varphi(n)}(4\pi |n|y), \] where \(W_\tau(y)\) is the standard Whittaker function and \(\lambda_\varphi=t_\varphi^2+1/4\). For the proof, see also H. Iwaniec and E. Kowalski [Analytic number theory. Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1059.11001), p. 360, Prop. 14.5] for a similar formula on holomorphic cusp forms.
Let us sketch the problems listed by the author, with a short discussion for each one.
- ●
- Statistics of eigenvalues: vertical Sato-Tate laws, density results for Maaß forms violating the Ramanujan conjecture, large sieve estimates.
- ●
- Arithmetic applications (Kloosterman sums, primes and arithmetic functions): Linnik’ problem (cancellation in sums of Kloosterman sums), shifted convolution sums, primes in long arithmetic progressions, the first case of Fermat’s last theorem, proportion of zeta zeros on the critical line, the largest prime factor of \(n^2+1\), equidistribution of roots of quadratic congruences, prime geodesic theorem.
- ●
- Applications to \(L\)-functions: symmetric type of \(L\)-functions; spectral expansion of the fourth moment of the Riemann zeta function, beyond endoscopy, non-vanishing of \(L\)-function.
- ●
- Applications to \(L\)-functions with subconvexity and equidistribution: sums of three squares, equidistribution of Heegner points, Maaß distribution problems.
For the entire collection see [Zbl 1469.11002].
Reviewer: Laurent Guillopé (Nantes)
MSC:
| 11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |
| 11F03 | Modular and automorphic functions |
| 11F30 | Fourier coefficients of automorphic forms |
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
Keywords:
trace formula; cusp forms; modular group; Fourier coefficient; Bruggeman formula; Kuznetsov formula; Kloosterman sums; \(L\)-functions; equidistributionReferences:
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