Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.35382

On a generating function of the basis of weakly holomorphic functions on an elliptic curve

Joshua S. Friedman·Jay Jorgenson·Lejla Smajlović
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $\Gz$ be a cofinite Fuchsian group whose quotient $\Gz\backslash\HH$ has genus one and a single cusp, normalized to be at $\infty$ with width one. Let $\Xg$ be its smooth compactification, which is an elliptic curve over $\CC$. Let $x,y$ be canonical generators of the function field on $\Xg$, which have poles of order $2,3$ respectively at the cusp and which {satisfy} a generalized Weierstrass relation $y^2+(a_1x+a_3)y=x^3+a_2x^2+a_4x+a_6$. Let $f$ be the unique (up to scale) weight two cusp form for $\Gz$, normalized so that the associated holomorphic differential is $ω=dx/(2y+a_1x+a_3)$. With all this, we prove that the generating function identity \[ \frac{\bigl(y(τ)+y(z)+a_1x(z)+a_3\bigr)f(z)}{x(z)-x(τ)}=\sum_{m\ge0}Φ_m(τ)q_z^m \] defines a family $\{Φ_m\}_{m\ge0}$ of weakly holomorphic modular functions on $\Xg$ such that the set $\{Φ_0\}\cup\{Φ_m\}_{m\ge2}$ is a canonical basis of the space $M_{0,\Gz}^{!,\infty}$ of weakly holomorphic modular functions on $\Xg$ with poles supported only at the cusp. As an application, by comparing the generating function of the family $\{Φ_m\}_{m\ge0}$ with the generating function of the Niebur--Poincaré series, we show that the generating function of the special values of the Kloosterman zeta function at $1$ is the holomorphic part of a certain weight two harmonic Maass form up to an additive constant.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Joshua S. Friedman, Jay Jorgenson, Lejla Smajlović. 2026-09-28. On a generating function of the basis of weakly holomorphic functions on an elliptic curve. https://arxiv.org/abs/2609.35382

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On a uniform bound for exponential sums modulo $p^m$ for Deligne polynomials

Let $f$ be a polynomial of degree $d>1$ in $n$ variables over $\mathbb{Z}$. Let $f_d$ be the homogeneous part of degree $d$ of $f$ and $s$ be the dimension of the critical locus of $f_d$. In this paper, we prove Igusa's conjecture for exponential sums with the exponent $(n-s)/(2(d-1))$. This implies a weak solution for a recent conjecture raised by Cluckers and the author (2020) about an analogue of the results of Deligne (1974) and Katz (1999) for exponential sums over finite fields in the finite ring setting. Moreover, this also improves the result of Cluckers, Mustaţă and the author (2019) in case $n-s>2(d-1)$. In particular, this result improves the conditions $n-s>2^d(d-1)$ of Birch (1962) and $n-s>3(d-1)2^{d-2}$ of Browning-Prendiville (2017) on the validity of the estimation for the major arcs to $(n-s)>4(d-1)$. Therefore this result may have further applications on subjects related to the Hardy-Littlewood circle method such as the Hasse principle or distribution of rational points in algebraic varieties. On the other hand, we also improve the recent work of Cluckers, Kollár and Mustaţă (2019) on the strong monodromy conjecture in the range $(-{\rm lct}((f)+J_f^2),0]$ in case of bad reduction and bad Schwartz-Bruhat function. Namely, in the range $(-{\rm lct}((f)+J_f^2),0]$, the real part of any pole of the Igusa local zeta functions associated with $f$ and any Schwartz-Bruhat function over any $p$-adic field is a root of the Bernstein-Sato polynomial of $f$.

math.NT↗
arXiv2026

Local points on twists of $X(p)$ with applications

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parametrizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to~$E[p]$. We give a complete classification of when $X_E^-(p)(\mathbb{Q}_\ell)$ is non-empty, for all primes $\ell$. We give two different applications. First, we classify CM curves $E/\mathbb{Q}$ for which the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for infinitely many~$p$. Assuming the Frey--Mazur conjecture, we prove that for at least $60\%$ of rational elliptic curves $E$, the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for at least $50\%$ of primes~$p$. Secondly, we introduce a new technique to the elimination stage of the modular method and apply it to show that $x^3+y^3=5^αz^p$ has no non-trivial primitive solutions for various primes $p$ satisfying $(α/p)=-1$. Moreover, as a by-product of our work, we simplify the assumptions of several local symplectic criteria due to the first author and Alain Kraus.

math.NT↗
arXiv2026

Involution on a quotient space of multiple zeta values in positive characteristic

In this paper, we introduce $*$-inverse multiple zeta values and $*$-inverse Carlitz multiple polylogarithms in positive characteristic and study their algebraic structures and relations. Using special values of Carlitz multiple polylogarithms, we show that a natural quotient of the space of multiple zeta values in positive characteristic admits a Hopf algebra structure and that the original space admits a compatible comodule structure over this quotient. These structures may be regarded as function field analogues of the Hopf algebra and comodule structures arising from motivic multiple zeta values in characteristic zero. In particular, the antipode on this quotient gives a non-trivial involution corresponding to the $*$-inverse operation. We also establish, in certain cases, the $q$-shuffle product formula and linear relations for $*$-inverse multiple zeta values, providing evidence for our conjectures on their relations. Finally, we consider another characteristic-zero analogue of our construction, based on the Hopf algebra structure given by the harmonic product and the deconcatenation coproduct, and formulate a related problem.

math.NT↗