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Pavel Guerzhoy

Publications and source records attributed to Pavel Guerzhoy.

17 recordsLinked to original sources
arXiv2026

Non-vanishing of a certain quantity related to the p-adic coupling of mock modular forms with newforms

Several authors have recently proved results which express a cusp form as a $p$-adic limit of weakly holomorphic modular forms under repeated application of Atkin's $U$-operator. Initially, these results had a deficiency: one could not rule out the possibility when a certain quantity vanishes and the final result fails to be true. Later on, Ahlgren and Samart \cite{AS} found a method to prove that no exceptions happen in the specific case considered by El-Guindy and Ono, Hanson and Jameson, and (independently) Dicks. generalized this method to finitely many other cases. In this paper, we present a different approach which allows us to prove a similar non-vanishing result for an infinite family of similar cases. Our approach also allows us to return back to the original example considered by El-Guindy and Ono, where we calculate the (manifestly non-zero) quantity explicitly in terms of Morita's $p$-adic $Γ$-function.

math.NT↗
arXiv2026

On $U_p$-congruences for meromorphic modular forms with supersingularity

In this paper, we investigate congruences for meromorphic modular forms $F$ which have a pole at a single point $z$ in the fundamental domain of $\mathrm{SL}_2(\mathbb Z)$. For a prime $p$ with good supersingular reduction at the elliptic curve corresponding to $z$, we show that there exists a cusp form $f$ such that $F|U_p^m \equiv f|U_p^m \pmod{p^{κ_m}}$, where $κ_m=αm -β$ with $α$ only depending on the weight of $F$ and $β$ depending on $F$ and $p$ but is independent of $m$. In particular, if the space of cusp forms is trivial, then $F|U_p^m\equiv 0 \pmod{p^{κ_m}}$ vanishes $p$-adically to a high order. In order to prove these results, we use the fact that $p$ has supersingular reduction to realize $F$ as an overconvergent modular form and then utilize the theory of overconvergent forms to show the congruences.

math.NT↗
arXiv2026

Congruences for traces of singular moduli and Hurwitz - Kronecker class numbers

Traces of singular moduli were introduced and studied by Zagier in 1998. Being simultaneously the (traces of) values of a modular function ($j$-invariant) and Fourier coefficients of modular forms - which constitutes Zagier's duality - these integers are quite interesting. Since then, a substantial amount of research was devoted to various properties of these numbers, congruences in particular. We present an alternative point of view on these congruences, specifically, we view them as congruences between certain weight $3/2$ modular forms under repeated action of $U$-operator. That allows us to obtain a general result which includes some previously known results as special cases. Our approach is especially effective when the prime modulus is relatively small. In these cases, we obtain explanations for certain numerical observations and quantification of some previously known qualitative results. As an application, we obtain modulo $11$ congruences between the traces of singular moduli and class numbers of quadratic fields in the case when the twisted central special value of the $L$-function associated with the elliptic curve of conductor $11$ vanishes.

math.NT↗
arXiv2025

On the $p$-adic values of the weight two Eisenstein series for supersingular primes

The weight two Eisenstein series may be considered as the first example of a Katz $p$-adic modular form. Classically, its values are defined for the primes of ordinary reduction. We offer a modified definition which applies uniformly to all primes of good reduction, both ordinary and supersingular. We show that, in the case of complex multiplication, these $p$-adic values coincide with the algebraic value of this Eisenstein series.

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arXiv2024

Position Information Emerges in Causal Transformers Without Positional Encodings via Similarity of Nearby Embeddings

Transformers with causal attention can solve tasks that require positional information without using positional encodings. In this work, we propose and investigate a new hypothesis about how positional information can be stored without using explicit positional encoding. We observe that nearby embeddings are more similar to each other than faraway embeddings, allowing the transformer to potentially reconstruct the positions of tokens. We show that this pattern can occur in both the trained and the randomly initialized Transformer models with causal attention and no positional encodings over a common range of hyperparameters.

cs.CL↗
arXiv2024

Deep hole lattices and isogenies of elliptic curves

Given a lattice $L$ in the plane, we define the affiliated deep hole lattice $H(L)$ to be spanned by a shortest vector of $L$ and a deep hole of $L$ contained in the triangle with sides corresponding to the shortest basis vectors. We study the geometric and arithmetic properties of deep hole lattices. In particular we investigate conditions on $L$ under which $H(L)$ is well-rounded and prove that $H(L)$ is defined over the same field as $L$. For the period lattice corresponding to an isomorphism class of elliptic curves, we produce a finite sequence of deep hole lattices ending with a well-rounded lattice which corresponds to a point on the boundary arc of the fundamental strip under the action of $\operatorname{SL}_2(\mathbb{Z})$ on the upper halfplane. In the case of CM elliptic curves, we prove that all elliptic curves generated by this sequence are isogenous to each other and produce bounds on the degree of isogeny. Finally, we produce a counting estimate for the planar lattices with a prescribed deep hole lattice.

math.NT↗
arXiv2020

On sparse geometry of numbers

Let $L$ be a lattice of full rank in $n$-dimensional real space. A vector in $L$ is called $i$-sparse if it has no more than $i$ nonzero coordinates. We define the $i$-th successive sparsity level of $L$, $s_i(L)$, to be the minimal $s$ so that $L$ has $s$ linearly independent $i$-sparse vectors, then $s_i(L) \leq n$ for each $1 \leq i \leq n$. We investigate sufficient conditions for $s_i(L)$ to be smaller than $n$ and obtain explicit bounds on the sup-norms of the corresponding linearly independent sparse vectors in~$L$. This result can be viewed as a partial sparse analogue of Minkowski's successive minima theorem. We then use this result to study virtually rectangular lattices, establishing conditions for the lattice to be virtually rectangular and determining the index of a rectangular sublattice. We further investigate the $2$-dimensional situation, showing that virtually rectangular lattices in the plane correspond to elliptic curves isogenous to those with real $j$-invariant. We also identify planar virtually rectangular lattices in terms of a natural rationality condition of the geodesics on the modular curve carrying the corresponding points.

math.NT↗
arXiv2019

Central $L$-values of elliptic curves and local polynomials

Here we study the recently introduced notion of a locally harmonic Maass form and its applications to the theory of $L$-functions. In particular, we find finite formulas for certain twisted central $L$-values of a family of elliptic curves in terms of finite sums over canonical binary quadratic forms. This yields vastly simpler formulas related to work of Birch and Swinnerton-Dyer for such $L$-values, and extends beyond their framework to special non-CM elliptic curves.

math.NT↗
arXiv2019

Periodicities for Taylor coefficients of half-integral weight modular forms

Congruences of Fourier coefficients of modular forms have long been an object of central study. By comparison, the arithmetic of other expansions of modular forms, in particular Taylor expansions around points in the upper-half plane, has been much less studied. Recently, Romik made a conjecture about the periodicity of coefficients around $τ=i$ of the classical Jacobi theta function. Here, we prove this conjecture and generalize the phenomenon observed by Romik to a general class of modular forms of half-integral weight.

math.NT↗
arXiv2019

Farkas' identities with quartic characters

Farkas in \cite{Farkas} introduced an arithmetic function $δ$ and found an identity involving $δ$ and a sum of divisor function $σ'$. The first-named author and Raji in \cite{Guerzhoy} discussed a natural generalization of the identity by introducing a quadratic character $χ$ modulo a prime $p \equiv 3 \pmod 4$. In particular, it turns out that, besides the original case $p=3$ considered by Farkas, an exact analog (in a certain precise sense) of Farkas' identity happens only for $p=7$. Recently, for quadratic characters of small composite moduli, Williams in \cite{Williams} found a finite list of identities of similar flavor using different methods. Clearly, if $p \not \equiv 3 \pmod 4$, the character $χ$ is either not quadratic or even. In this paper, we prove that, under certain conditions, no analogs of Farkas' identity exist for even characters. Assuming $χ$ to be odd quartic, we produce something surprisingly similar to the results from \cite{Guerzhoy}: exact analogs of Farkas' identity happen exactly for $p=5$ and $13$.

math.NT↗
arXiv2016

On arithmetic lattices in the plane

We investigate similarity classes of arithmetic lattices in the plane. We introduce a natural height function on the set of such similarity classes, and give asymptotic estimates on the number of all arithmetic similarity classes, semi-stable arithmetic similarity classes, and well-rounded arithmetic similarity classes of bounded height as the bound tends to infinity. We also briefly discuss some properties of the $j$-invariant corresponding to similarity classes of planar lattices.

math.NT↗
arXiv2015

Congruences for Taylor expansions of quantum modular forms

Recently, a beautiful paper of Andrews and Sellers has established linear congruences for the Fishburn numbers modulo an infinite set of primes. Since then, a number of authors have proven refined results, for example, extending all of these congruences to arbitrary powers of the primes involved. Here, we take a different perspective and explain the general theory of such congruences in the context of an important class of quantum modular forms. As one example, we obtain an infinite series of combinatorial sequences connected to the "half-derivatives" of the Andrews-Gordon functions and with Kashaev's invariant on $(2m+1,2)$ torus knots, and we prove conditions under which the sequences satisfy linear congruences modulo at least $50\%$ of primes of primes.

math.NT↗
arXiv2014

Shintani lifts and fractional derivatives for harmonic weak Maass forms

In this paper, we construct Shintani lifts from integral weight weakly holomorphic modular forms to half-integral weight weakly holomorphic modular forms. Although defined by different methods, these coincide with the classical Shintani lifts when restricted to the space of cusp forms. As a side effect, this gives the coefficients of the classical Shintani lifts as new cycle integrals. This yields new formulas for the $L$-values of Hecke eigenforms. When restricted to the space of weakly holomorphic modular forms orthogonal to cusp forms, the Shintani lifts introduce a definition of weakly holomorphic Hecke eigenforms. Along the way, auxiliary lifts are constructed from the space of harmonic weak Maass forms which yield a "fractional derivative" from the space of half-integral weight harmonic weak Maass forms to half-integral weight weakly holomorphic modular forms. This fractional derivative complements the usual $ξ$-operator introduced by Bruinier and Funke.

math.NT↗
arXiv2014

On cycle integrals of weakly holomorphic modular forms

In this paper, we investigate cycle integrals of weakly holomorphic modular forms. We show that these integrals coincide with the cycle integrals of classical cusp forms. We use these results to define a Shintani lift from integral weight weakly holomorphic modular forms to half-integral weight holomorphic modular forms.

math.NT↗
arXiv2010

Mock modular forms as $p$-adic modular forms

In this paper, we consider the question of correcting mock modular forms in order to obtain $p$-adic modular forms. In certain cases we show that a mock modular form $M^+$ is a $p$-adic modular form. Furthermore, we prove that otherwise the unique correction of $M^+$ is intimately related to the shadow of $M^+$.

math.NT↗
arXiv2005

Some congruences for traces of singular moduli

We address a question posed by Ono, prove a general result for powers of an arbitrary prime, and provide an explanation for the appearance of higher congruence moduli for certain small primes. One of our results coincides with a recent result of Edixhoven, and we hope that the comparison of the methods, which are entirely different, may reveal a connection between the p-adic geometry and the arithmetic of half-integral weight Hecke operators.

math.NT↗