Theta functions, fourth moments of eigenforms, and the sup-norm problem III
In the prequel, a sharp bound in the level aspect on the fourth moment of Hecke-Maaß forms with an inexplicit (in fact exponential) dependency on the eigenvalue was given. In this paper, we develop further the framework of explicit theta test functions in order to capture the eigenvalue more precisely. We use this to reduce a sharp hybrid fourth moment bound to an intricate counting problem. Unconditionally, we give a hybrid bound, which is sharp in the level aspect and with a slightly larger than convex dependency on the eigenvalue.
Rethinking metallic luster and iridescence in animal coloration
Some structural colors in nature are frequently described as metallic. For example, hummingbird plumage, jewel beetles and Morpho butterflies have this attribute. While much attention has been paid to describing the often-shifting hues of these structural colors, there has been little interest in explaining why they appear metallic. In this paper, we argue that the metallic luster (the metallic appearance or sheen) of some structural colors arises in part from a combination of two factors: a colored specular reflection and a very low diffuse reflection. Reflections with these characteristics are found in metals and are distinct from other material reflections in nature. We propose that metallic luster can be classified based on these two reflectance properties (colored specular reflection and low diffuse reflection). We also suggest that some of the ambiguity surrounding the term "iridescent structural color" can be traced to the frequent confounding of metallic luster with a common definition of iridescence: a shift of peak spectral wavelength (often referred to as hue) with viewing angle. We show using optical models and cross-polarization imaging of bird plumage that two types of structural colors that are often classified as "iridescent" and "non-iridescent" both display iridescence—but only one type has metallic luster. By considering metallic luster and iridescence separately, we simultaneously clarify terminology in structural colors and open up many new lines of inquiry regarding the perception of metallic luster in animals.
Appendix to Sums of cusp form coefficients along quadratic sequences
Let $f(z)=\sum A(n) n^{\frac{k-1}{2}}e(nz)$ be a cusp form of weight $k \ge 3$ on $\Gamma_0(N)$ with character $\chi$. By studying a certain shifted convolution sum, we prove that $\sum_{n\le X}A(n^2+h)$$=c_{f,h}X$$+O_{f,h,\epsilon}(X^{\frac{3}{4}+\epsilon})$ for $\epsilon>0$, which improves a result of Blomer from 2008 with error $X^{\frac{6}{7}+\epsilon}.$ This includes an appendix due to Raphael S. Steiner, proving stronger bounds for certain spectral averages.
Kloosterman sums do not correlate with periodic functions
We provide uniform bounds for sums of Kloosterman sums in all arithmetic progressions. As a consequence, we find that Kloosterman sums do not correlate with periodic functions.
Chowla and Sarnak conjectures for Kloosterman sums
We formulate several analogues of the Chowla and Sarnak conjectures, which are widely known in the setting of the Möbius function, in the setting of Kloosterman sums. We then show that for Kloosterman sums, in some cases, these conjectures can be established unconditionally.
Small diameters and generators for arithmetic lattices in $\mathrm{SL}_2(\mathbb{R})$ and certain Ramanujan graphs
We show that arithmetic lattices in $\mathrm{SL}_2(\mathbb{R})$, stemming from the proper units of an Eichler order in an indefinite quaternion algebra over $\mathbb{Q}$, admit a 'small' covering set. In particular, we give bounds on the diameter if the quotient space is co-compact. Consequently, we show that these lattices admit small generators. Our techniques also apply to definite quaternion algebras where we show Ramanujan-strength bounds on the diameter of certain Ramanujan graphs without the use of the Ramanujan bound.
Theta functions, fourth moments of eigenforms, and the sup-norm problem II
joint with Ilya Khayutin and Paul D. Nelson.
For an $L^2$-normalized holomorphic newform $f$ of weight $k$ on a hyperbolic surface of volume $V$ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over $\mathbb{Q}$, we prove the sup-norm estimate $\Vert \mathrm{Im}(\cdot)^{\frac{k}{2}}f \Vert_{\infty} \ll_{\epsilon} (kV)^{\frac{1}{4}+\epsilon}$ with absolute implied constant. For a cuspidal Maaß newform $\varphi$ of eigenvalue $\lambda$ on such a surface, we prove that $\Vert \varphi \Vert_{\infty} \ll_{\lambda,\epsilon} V^{\frac{1}{4}+\epsilon}$. We establish analogous estimates in the setting of definite quaternion algebras.
Theta functions, fourth moments of eigenforms, and the sup-norm problem I
joint with Ilya Khayutin.
We give sharp point-wise bounds in the weight-aspect on fourth moments of modular forms on arithmetic hyperbolic surfaces associated to Eichler orders. Thereby, we strengthen a result of Xia and extend it to co-compact lattices. We realize this fourth moment by constructing a holomorphic theta kernel on $\mathbf{G} \times \mathbf{G} \times \mathbf{SL}_{2}$, for $\mathbf{G}$ an indefinite inner-form of $\mathbf{SL}_2$ over $\mathbb{Q}$, based on the Bergman kernel, and considering its $L^2$-norm in the Weil variable. The constructed theta kernel further gives rise to new elementary theta series for integral quadratic forms of signature $(2,2)$.
Sup-norm of Hecke-Laplace eigenforms on $S^3$
We prove sub-convex bounds on the fourth moment of Hecke-Laplace eigenforms on $S^3$. As a corollary, we get a bound on the sup-norm on an individual eigenform, which constitutes an improvement over what is achievable through employing the Iwaniec-Sarnak amplifier.
The harmonic conjunction of automorphic forms and the Hardy-Littlewood circle method
Ph.D. thesis at the University of Bristol.
The delta symbol circle method is applied to the problem of equidistribution of rational points on shrinking sets of the 3-sphere. This leads to a correlation sum of Kloosterman sums and an exponential function also known as the twisted Linnik–Selberg conjecture, which is further analysed by means of the Kuznetsov trace formula. Furthermore, the circle method in the guise of Wooley’s efficient congruencing machinery is employed in an effective manner to obtain effective bounds on Vinogradov’s mean value theorem.
On a twisted version of Linnik and Selberg’s conjecture
We generalise the work of Sarnak-Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik-Selberg Conjecture.
A conjectural extension of Hecke's converse theorem
We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main idea is to linearize the Euler product, replacing it by twists by Ramanujan sums. We provide evidence for the conjecture, including proofs of some special cases and under various additional hypotheses.
Twisted Linnik implies optimal covering exponent for $S^3$
We show that a twisted variant of Linnik's conjecture on sums of Kloosterman sums leads to an optimal covering exponent for $S^3$.
Near counterexamples to Weil’s converse theorem
We show that in Weil's converse theorem the functional equations of multiplicative twists for at least the first $\sqrt{\frac{p-24}{3}}$ moduli are needed in order to prove the modularity for $\Gamma_0(p)$.
Effective Vinogradov’s mean value theorem via efficient boxing
We combine Wooley's efficient congruencing method with earlier work of Vinogradov and Hua to get effective bounds on Vinogradov's mean value theorem.
Supnorm of modular forms of half-integral weight in the weight aspect
We bound the supnorm of half-integral weight Hecke eigenforms in the Kohnen plus space of level $4$ in the weight aspect, by combining bounds obtained from the Fourier expansion with the amplification method using a Bergman kernel.
Uniform bounds on sup-norms of holomorphic forms of real weight
We establish uniform bounds for the sup-norms of modular forms of arbitrary real weight $k$ with respect to a finite index subgroup $\Gamma$ of $\mathrm{SL}_2(\mathbb{Z})$. We also prove corresponding bounds for the supremum over a compact set. We achieve this by extending to a sum over an orthonormal basis $\sum_j y^k |f_j(z)|^2$ and analysing this sum by means of a Bergman kernel and the Fourier coefficients of Poincaré series. As such our results are valid without any assumption that the forms are Hecke eigenfunctions. Under some weak assumptions we further prove the right order of magnitude of $\sup_{z \in \mathbb{H}} \sum_j y^k |f_j(z)|^2$.