Murmurations in the depth aspect
We compute the murmuration density function for the family of Hecke forms of weight $k$ and prime power level $N=\ell^a$, with $\ell$ a fixed odd prime and $a\to \infty$.
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arXiv subjects
Claire Burrin
Publications and source records attributed to Claire Burrin.
A Human Story of Curiosity and Relevance
Mathematics is changing. Computers are verifying proofs, checking calculations, and exploring complex structures that would overwhelm human effort. Yet curiosity-driven research is where tomorrow's breakthroughs are quietly prepared. In this invited article for the VSH-AEU Bulletin, these themes are illustrated through anecdotes centered on a mathematical object central to the author's work: the lattice.
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Modular systoles are extremal for the crossing number
We study crossing numbers for systoles of congruence surfaces. Taken as a family of curves on a family of surfaces, we show that the growth rate of their intersection is optimally small among all sets of curves of the same cardinality lying on the same topological surface.
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The Brjuno and Wilton Functions
The Brjuno and Wilton functions bear a striking resemblance, despite their very different origins; while the Brjuno function $B(x)$ is a fundamental tool in one-dimensional holomorphic dynamics, the Wilton function $W(x)$ stems from the study of divisor sums and self-correlation functions in analytic number theory. We show that these perspectives are unified by the semi-Brjuno function $B_0(x)$. Namely, $B(x)$ and $W(x)$ can be expressed in terms of the even and odd parts of $B_0(x)$, respectively, up to a bounded defect. Based on numerical observations, we further analyze the arising functions $\Delta^+(x) = B^+(x) - 2B_0^+(x)$ and $\Delta^-(x) = W^-(x) - 2B_0^-(x)$, the first of which is H\"older continuous whereas the second exhibits discontinuities at rationals, behaving similarly to the classical popcorn function.
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Equidistribution, covering radius, and Diophantine approximation for rational points on the sphere
We study the distribution of rational points of fixed height on the sphere at shrinking scales. For the two-dimensional sphere, we prove an unconditional variance estimate for primitive square-level Linnik sets, essentially matching the random-model prediction. We obtain almost-everywhere equidistribution in caps down to the optimal scale $R\gg n^{-1/2+\delta}$, pointwise equidistribution down to $R\gg n^{-1/4+o(1)}$, and Wasserstein equidistribution down to the optimal bound. We also derive applications to covering, intrinsic Diophantine approximation, and Linnik's conjecture on sums of two squares and a mini-square.
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Equidistribution of continuous low-lying pairs of horocycles via Ratner's theorem
We record an alternative proof of a recent joint equidistribution result of Blomer and Michel, based on Ratner's topological rigidity theorem. This approach has the advantage of extending to non-uniform lattices.
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On the crossing number of arithmetic curve systems
We show that the family of systoles of hyperbolic surfaces associated with congruence lattices in $\mathrm{SL}_2(\mathbb{Z})$ have asymptotically minimal crossing number.
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Rational points on ellipsoids and modular forms
The theory of modular forms and spherical harmonic analysis are applied to establish new best bounds towards the counting and equidistribution of rational points on spheres and other higher dimensional ellipsoids, in what may be viewed as a textbook display of the 'unreasonable effectiveness' of modular forms.
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Pairs in discrete lattice orbits with applications to Veech surfaces
Let $\Lambda_1$, $\Lambda_2$ be two discrete orbits under the linear action of a lattice $\Gamma<\mathrm{SL}_2(\mathbb{R})$ on the Euclidean plane. We prove a Siegel$-$Veech-type integral formula for the averages $$ \sum_{\mathbf{x}\in\Lambda_1} \sum_{\mathbf{y}\in\Lambda_2} f(\mathbf{x}, \mathbf{y}) $$ from which we derive new results for the set $S_M$ of holonomy vectors of saddle connections of a Veech surface $M$. This includes an effective count for generic Borel sets with respect to linear transformations, and upper bounds on the number of pairs in $S_M$ with bounded determinant and on the number of pairs in $S_M$ with bounded distance. This last estimate is used in the appendix to prove that for almost every $(\theta,\psi)\in S^1\times S^1$ the translations flows $F_\theta^t$ and $F_\psi^t$ on any Veech surface $M$ are disjoint.
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Windings of prime geodesics
The winding of a closed oriented geodesic around the cusp of the modular orbifold is computed by the Rademacher symbol, a classical function from the theory of modular forms. In this article, we introduce a new construction of winding numbers to record the winding of closed oriented geodesics about a prescribed cusp of a general cusped hyperbolic orbifold. For various arithmetic families of surfaces, this winding number can again be expressed by a Rademacher symbol, and access to the spectral theory of automorphic forms yields statistical results on the distribution of closed (primitive) oriented geodesics with respect to their winding.
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Infinitely many twin prime polynomials of odd degree
While the twin prime conjecture is still famously open, it holds true in the setting of finite fields: There are infinitely many pairs of monic irreducible polynomials over $\mathbb{F}_q$ that differ by a fixed constant, for each $q \geq 3$. Elementary, constructive proofs were given for different cases by Hall and Pollack. In the same spirit, we discuss the construction of a further infinite family of twin prime tuples of odd degree, and its relations to the existence of certain Wieferich primes and to arithmetic properties of the combinatorial Bell numbers.
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The Manin-Drinfeld theorem and the rationality of Rademacher symbols
For any noncocompact Fuchsian group $\Gamma$, we show that periods of the canonical differential of the third kind associated to residue divisors of cusps are expressed in terms of Rademacher symbols for $\Gamma$ - generalizations of periods appearing in the classical theory of modular forms. This result provides a relation between Rademacher symbols and the famous theorem of Manin and Drinfeld. More precisely, Fuchsian groups whose Rademacher symbols are rational-valued verify the statement of Manin-Drinfeld. We then establish the rationality of Rademacher symbols for various families of Fuchsian groups.
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Translates of rational points along expanding closed horocycles on the modular surface
We study the limiting distribution of the rational points under a horizontal translation along a sequence of expanding closed horocycles on the modular surface. Using spectral methods we confirm equidistribution of these sample points for any translate when the sequence of horocycles expands within a certain polynomial range. We show that the equidistribution fails for generic translates and a slightly faster expanding rate. We also prove both equidistribution and non-equidistribution results by obtaining explicit limiting measures while allowing the sequence of horocycles to expand arbitrarily fast. Similar results are also obtained for translates of primitive rational points.
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Effective counting for discrete lattice orbits in the plane via Eisenstein series
We prove effective bounds on the rate in the quadratic growth asymptotics for the orbit of a non-uniform lattice of SL(2,R), acting linearly on the plane. This gives an error bound in the count of saddle connection holonomies, for some Veech surfaces. The proof uses Eisenstein series and relies on earlier work of many authors (notably Selberg). Our results improve earlier error bounds for counting in sectors and in smooth star shaped domains.
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Transformation laws for generalized Dedekind sums associated to Fuchsian groups
We establish transformation laws for generalized Dedekind sums associated to the Kronecker limit function of non-holomorphic Eisenstein series and their higher-order variants. These results apply to general Fuchsian groups of the first kind, and examples are provided in the cases of the Hecke triangle groups, the Hecke congruence groups $\Gamma_0(N)$, and the non-congruence arithmetic groups $\Gamma_0(N)^+$.
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Dedekind sums, reciprocity, and non-arithmetic groups
Dedekind sums, arithmetic correlation sums that arose in Dedekind's study of the modular transformation of the logarithm of the eta-function, are surprisingly ubiquitous. Their arithmetic properties attracted the attention of number theorists, combinatorists, and theoretical computer scientists alike, and they appear more broadly in geometry, topology, and physics. Accordingly, there is a similarly vast literature on variations and generalizations of Dedekind sums. This note surveys recent developments of some aspects of Dedekind sums for (non-uniform) lattices in SL_2(R), otherwise referred to as Dedekind symbols.
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Reciprocity of Dedekind sums and the Euler class
Dedekind sums are arithmetic sums that were first introduced by Dedekind in the context of elliptic functions and modular forms, and later recognized to be surprisingly ubiquitous. Among the variations and generalizations introduced since, there is a construction of Dedekind sums for lattices in $\mathrm{SL}_2(\mathbf{R}). Building upon work of Asai, we prove the reciprocity law for these Dedekind sums, based on a concrete realization of the Euler class. As an application, we obtain an explicit formula expressing Dedekind sums for Hecke triangle groups in terms of continued fractions.
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Generalized Dedekind sums and equidistribution mod 1
Dedekind sums are well-studied arithmetic sums, with values uniformly distributed on the unit interval. Based on their relation to certain modular forms, Dedekind sums may be defined as functions on the cusp set of $SL(2,\mathbb{Z})$. We present a compatible notion of Dedekind sums, which we name Dedekind symbols, for any non-cocompact lattice $\Gamma<SL(2,\mathbb{R})$, and prove the corresponding equidistribution mod 1 result. The latter part builds up on a paper of Vardi, who first connected exponential sums of Dedekind sums to Kloosterman sums.
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