arXiv subjects

Paul D. Nelson

Publications and source records attributed to Paul D. Nelson.

29 records · Page 2Linked to original sources

arXiv2018

Subconvex bounds on GL(3) via degeneration to frequency zero

For a fixed cusp form $π$ on $\operatorname{GL}_3(\mathbb{Z})$ and a varying Dirichlet character $χ$ of prime conductor $q$, we prove that the subconvex bound \[ L(π\otimes χ, \tfrac{1}{2}) \ll q^{3/4 - δ} \] holds for any $δ< 1/36$. This improves upon the earlier bounds $δ< 1/1612$ and $δ< 1/308$ obtained by Munshi using his $\operatorname{GL}_2$ variant of the $δ$-method. The method developed here is more direct. We first express $χ$ as the degenerate zero-frequency contribution of a carefully chosen summation formula à la Poisson. After an elementary "amplification" step exploiting the multiplicativity of $χ$, we then apply a sequence of standard manipulations (reciprocity, Voronoi, Cauchy--Schwarz and the Weil bound) to bound the contributions of the nonzero frequencies and of the dual side of that formula.

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arXiv2018

Some analytic aspects of automorphic forms on GL(2) of minimal type

Let $π$ be a cuspidal automorphic representation of $PGL_2(\mathbb{A}_\mathbb{Q})$ of arithmetic conductor $C$ and archimedean parameter $T$, and let $ϕ$ be an $L^2$-normalized automorphic form in the space of $π$. The sup-norm problem asks for bounds on $\| ϕ\|_\infty$ in terms of $C$ and $T$. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the $L^2$-mass $|ϕ|^2 (g) \, d g$ of $ϕ$. All previous work on these problems in the conductor-aspect has focused on the case that $ϕ$ is a newform. In this work, we study these problems for a class of automorphic forms that are not newforms. Precisely, we assume that for each prime divisor $p$ of $C$, the local component $π_p$ is supercuspidal (and satisfies some additional technical hypotheses), and consider automorphic forms $ϕ$ for which the local components $ϕ_p \in π_p$ are "minimal" vectors. Such vectors may be understood as non-archimedean analogues of lowest weight vectors in holomorphic discrete series representations of $PGL_2(\mathbb{R})$. For automorphic forms as above, we prove a sup-norm bound that is sharper than what is known in the newform case. In particular, if $π_\infty$ is a holomorphic discrete series of lowest weight $k$, we obtain the optimal bound $C^{1/8 -ε} k^{1/4 - ε} \ll_ε |ϕ|_\infty \ll_ε C^{1/8 + ε} k^{1/4+ε}$. We prove also that these forms give analytic test vectors for the QUE period, thereby demonstrating the equivalence between the strong QUE and the subconvexity problems for this class of vectors. This finding contrasts the known failure of this equivalence for newforms of powerful level.

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arXiv2018

Analytic isolation of newforms of given level

We describe a method for understanding averages over newforms on $Γ_0(q)$ in terms of averages over all forms of some level. The method is simplest when $q$ is divisible by the cubes of its prime divisors.

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arXiv2017

Quantum variance on quaternion algebras, II

A method for determining quantum variance asymptotics on compact quotients attached to non-split quaternion algebras is developed in general and applied to "microlocal lifts" in the non-archimedean setting. The results obtained are in the spirit of recent work of Sarnak--Zhao. The arguments involve a careful analytic study of the theta correspondence, the interplay between additive and multiplicative harmonic analysis on quaternion algebras, the equidistribution of translates of elementary theta functions, and the Rallis inner product formula.

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arXiv2016

Quantum variance on quaternion algebras, I

We determine the quantum variance of a sequence of families of automorphic forms on a compact quotient arising from a non-split quaternion algebra. Our results compare to those obtained by Luo--Sarnak, Zhao, and Sarnak--Zhao on the modular curve, whose method required a cusp. Our method uses the theta correspondence to reduce the problem to the estimation of metaplectic Rankin--Selberg convolutions. We apply it here to the first non-split case.

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arXiv2016

Microlocal lifts and quantum unique ergodicity on $GL(2,\mathbb{Q}_p)$

We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of $GL(2,\mathbb{Q}_p)$ for automorphic forms belonging to the principal series. We interpret this conclusion in terms of the equidistribution of eigenfunctions on covers of a fixed regular graph or along nested sequences of regular graphs. Our results are the first of their kind on any p-adic arithmetic quotient. They may be understood as analogues of Lindenstrauss's theorem on the equidistribution of Maass forms on a compact arithmetic surface. The new ingredients here include the introduction of a representation-theoretic notion of "p-adic microlocal lifts" with favorable properties, such as diagonal invariance of limit measures, the proof of positive entropy of limit measures in a p-adic aspect, following the method of Bourgain--Lindenstrauss, and some analysis of local Rankin--Selberg integrals involving the microlocal lifts introduced here as well as classical newvectors. An important input is a measure-classification result of Einsiedler--Lindenstrauss.

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arXiv2013

Bounds for Rankin--Selberg integrals and quantum unique ergodicity for powerful levels

Let f be a classical holomorphic newform of level q and even weight k. We show that the pushforward to the full level modular curve of the mass of f equidistributes as qk -> infinity. This generalizes known results in the case that q is squarefree. We obtain a power savings in the rate of equidistribution as q becomes sufficiently "powerful" (far away from being squarefree), and in particular in the "depth aspect" as q traverses the powers of a fixed prime. We compare the difficulty of such equidistribution problems to that of corresponding subconvexity problems by deriving explicit extensions of Watson's formula to certain triple product integrals involving forms of non-squarefree level. By a theorem of Ichino and a lemma of Michel--Venkatesh, this amounts to a detailed study of Rankin--Selberg integrals int|f|^2 E attached to newforms f of arbitrary level and Eisenstein series E of full level. We find that the local factors of such integrals participate in many amusing analogies with global L-functions. For instance, we observe that the mass equidistribution conjecture with a power savings in the depth aspect is equivalent to the union of a global subconvexity bound and what we call a "local subconvexity bound"; a consequence of our local calculations is what we call a "local Lindelof hypothesis".

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arXiv2012

Stable averages of central values of Rankin-Selberg L-functions: some new variants

As shown by Michel-Ramakrishan (2007) and later generalized by Feigon-Whitehouse (2008), there are "stable" formulas for the average central L-value of the Rankin-Selberg convolutions of some holomorphic forms of fixed even weight and large level against a fixed imaginary quadratic theta series. We obtain exact finite formulas for twisted first moments of Rankin-Selberg L-values in much greater generality and prove analogous "stable" formulas when one considers either arbitrary modular twists of large prime power level or real dihedral twists of odd type associated to a Hecke character of mixed signature.

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arXiv2012

Evaluating modular forms on Shimura curves

Let f be a newform, as specified by its Hecke eigenvalues, on a Shimura curve X. We describe a method for evaluating f. The most interesting case is when X arises as a compact quotient of the hyperbolic plane, so that classical q-expansions are not available. The method takes the form of an explicit, rapidly-convergent formula that is well-suited for numerical computation. We apply it to the problem of computing modular parametrizations of elliptic curves, and illustrate with some numerical examples.

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arXiv2012

Mass equidistribution of Hilbert modular eigenforms

Let F be a totally real number field, and let f traverse a sequence of non-dihedral holomorphic eigencuspforms on GL(2)/F of weight (k_1,...,k_n), trivial central character and full level. We show that the mass of f equidistributes on the Hilbert modular variety as max(k_1,...,k_n) tends to infinity. Our result answers affirmatively a natural analogue of a conjecture of Rudnick and Sarnak (1994). Our proof generalizes the argument of Holowinsky-Soundararajan (2008) who established the case F = Q. The essential difficulty in doing so is to adapt Holowinsky's bounds for the Weyl periods of the equidistribution problem in terms of manageable shifted convolution sums of Fourier coefficients to the case of a number field with nontrivial unit group.

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arXiv2012

Equidistribution of cusp forms in the level aspect

Let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing squarefree level q tending to infinity. We prove that the pushforward of the mass of f to the modular curve of level 1 equidistributes with respect to the Poincaré measure. Our result answers affirmatively the squarefree level case of a conjecture spelled out by Kowalski, Michel and Vanderkam (2002) in the spirit of a conjecture of Rudnick and Sarnak (1994). Our proof follows the strategy of Holowinsky and Soundararajan (2008) who show that newforms of level 1 and large weight have equidistributed mass. The new ingredients required to treat forms of fixed weight and large level are an adaptation of Holowinsky's reduction of the problem to one of bounding shifted sums of Fourier coefficients (which on the surface makes sense only in the large weight limit), an evaluation of the p-adic integral needed to extend Watson's formula to the case of three newforms where the level of one divides but need not equal the common squarefree level of the other two, and some additional technical work in the problematic case that the level has many small prime factors.

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