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Olivia Beckwith

Publications and source records attributed to Olivia Beckwith.

At least 19 recordsLinked to original sources
arXiv2026

Selmer groups of CM-twists of elliptic curves

Let $F$ be a totally real Galois number field of degree $g\leq5$ and let $\ell$ be an odd prime. Let $E/F$ be an elliptic curve with an $F$-rational point of order $\ell$. Under explicit arithmetic and local hypotheses, together with an explicit non-vanishing condition modulo $\ell$, we prove that there are $\gg_{F,E,\ell}\frac{X^{1/(2g)}}{\log X}$ totally negative square classes $d\in F^\times/(F^\times)^2$ with $\left|\mathrm{N}_{F/\mathbb{Q}}\bigl(D(F(\sqrt{d})/F)\bigr)\right|<X $ for which $\operatorname{Sel}_\ell(E^d,F)$ is trivial. Such twists have rank zero and trivial $\ell$-part of the Shafarevich--Tate group. The condition is verifiable by a finite computation, which we carry out in an example. We lift the method of James and Ono from $\mathbb{Q}$ to the totally real setting, combining a theorem of Morrow, which relates the Selmer group of such a twist to the class group of the CM extension $F(\sqrt{d})$, with an indivisibility theorem of Takai for relative class numbers. Takai twists only by quadratic Hecke characters, whereas Morrow's conditions are local; we extend the twisting argument to primitive quadratic residue-class characters to make the two results compatible.

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arXiv2026

Riesz Means of Quadratic Class Numbers

We prove an asymptotic formula for a weighted Riesz mean of Hurwitz class numbers and real quadratic class numbers. To do this, we introduce L-functions for weight $\frac {1}{2} $ sesquiharmonic Maass forms of moderate growth and prove a formula for the Riesz means of the corresponding generalized mock modular forms, generalizing a recent result of the first author with Diamantis, Gupta, Rolen, and Thalagoda for mock modular forms. We then apply this formula to a sesquiharmonic Maass form that was first introduced by Duke, Imamoglu, and Tóth.

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arXiv2026

A modular framework for generalized Hurwitz class numbers I

We discover a non-trivial relation between the mock modular generating functions of the level $1$ and level $N$ Hurwitz class numbers. This relation yields a holomorphic modular form of weight $\frac{3}{2}$ and level $4N$, where $N > 1$ is stipulated to be odd and square-free. We extend this observation to a non-holomorphic framework and obtain a higher level non-holomorphic Zagier Eisenstein series as well as a preimage $\mathcal{G}$ of it under the differential operator $ξ_{\frac{1}{2}}$. All of these observations are deduced from a more general inspection of a certain weight $\frac{1}{2}$ Maass--Eisenstein series of level $4N$ at its spectral point $s=\frac{3}{4}$. This idea goes back to Duke, Imamoglu and Tóth in level $4$ and relies on the theory of so-called sesquiharmonic Maass forms. We calculate the Fourier expansion of $\mathcal{G}$ and $ξ_{\frac{1}{2}}\mathcal{G}$. We conclude by offering examples if $N=5$ or $N=7$ as well as some questions for future work.

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arXiv2025

Scarcity of partition congruences on semiprime progressions

In recent work with Raum the authors considered congruences for the ordinary partition function $p(n)$ of the form $p(\ell Q^r n+β)\equiv 0\pmod\ell$ where $\ell, Q\geq 5$ are prime and $r\in \{1,2\}$, and proved a number of results which show that such congruences are scarce in a precise sense. Here we improve one of our results when $r=1$; in particular we prove (outside of trivial cases) that the set of primes $Q$ such that there exists $β\in \mathbb{Z}$ with $p(\ell Q n+β)\equiv 0\pmod \ell$ for all $n$ has density zero. The proof involves a modification of part of our previous argument and an application of a recent theorem of Dicks regarding modular forms of half-integral weight and level one modulo $\ell$.

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arXiv2025

Summation formulas for Hurwitz class numbers and other mock modular coefficients

We prove a formula for weighted sums of the first $n$ coefficients of mock modular forms of moderate growth and apply it to Hurwitz class numbers and coefficients of negative half integral weight Eisenstein series, which take the form of certain quadratic Dirichlet $L$-values. Our formula is a mock modular version of a Bessel-sum identity proved by Chandrasekharan and Narasimhan for Dirichlet series satisfying a functional equation. Our proof utilizes $L$-functions for mock modular Eisenstein series defined by Shankadhar and Singh.

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arXiv2024

A modular framework for generalized Hurwitz class numbers II

In a recent preprint, we constructed a sesquiharmonic Maass form $\mathcal{G}$ of weight $\frac{1}{2}$ and level $4N$ with $N$ odd and squarefree. Extending seminal work by Duke, Imamoglu, and Tóth, $\mathcal{G}$ maps to Zagier's non-holomorphic Eisenstein series and a linear combination of Pei and Wang's generalized Cohen--Eisenstein series under the Bruinier--Funke operator $ξ_{\frac{1}{2}}$. In this paper, we realize $\mathcal{G}$ as the output of a regularized Siegel theta lift of $1$ whenever $N=p$ is an odd prime building on more general work by Bruinier, Funke and Imamoglu. In addition, we supply the computation of the square-indexed Fourier coefficients of $\mathcal{G}$. This yields explicit identities between the Fourier coefficients of $\mathcal{G}$ and all quadratic traces of $1$. Furthermore, we evaluate the Millson theta lift of $1$ and consider spectral deformations of $1$.

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arXiv2024

Holomorphic projection for sesquiharmonic Maass forms

We study the holomorphic projection of mixed mock modular forms involving sesquiharmonic Maass forms. As a special case, we numerically express the holomorphic projection of a function involving real quadratic class numbers multiplied by a certain theta function in terms of eta quotients. We also analyze certain shifted convolution $L$-series involving mock modular forms and bound certain shifted convolution sums.

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arXiv2023

Imaginary quadratic fields with $\ell$-torsion-free class groups and specified split primes

Given an odd prime $\ell$ and finite set of odd primes $S_+$, we prove the existence of an imaginary quadratic field whose class number is indivisible by $\ell$ and which splits at every prime in $S_+$. Notably, we do not require that $p \not\equiv -1 \pmod{\ell}$ for any of the split primes $p$ that we impose. Our theorem is in the spirit of a result by Wiles, but we introduce a new method. It relies on a significant improvement of our earlier work on the classification of non-holomorphic Ramanujan-type congruences for Hurwitz class numbers.

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arXiv2022

Scarcity of congruences for the partition function

The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+β)\equiv 0\pmod\ell$ for the primes $\ell=5, 7, 11$, and it is known that there are no others of this form. On the other hand, for every prime $\ell\geq 5$ there are infinitely many examples of congruences of the form $p(\ell Q^m n+β)\equiv 0\pmod\ell$ where $Q\geq 5$ is prime and $m\geq 3$. This leaves open the question of the existence of such congruences when $m=1$ or $m=2$ (no examples in these cases are known). We prove in a precise sense that such congruences, if they exist, are exceedingly scarce. Our methods involve a careful study of modular forms of half integral weight on the full modular group which are related to the partition function. Among many other tools, we use work of Radu which describes expansions of such modular forms along square classes at cusps of the modular curve $X(\ell Q)$, Galois representations and the arithmetic large sieve.

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arXiv2022

Integer partitions with large Dyson rank

The Dyson rank of an integer partition is the difference between its largest part and the number of parts it contains. Using Fine-Dyson symmetry, we give formulas for the number of partitions of n with rank larger than n/2, and we prove identities for counts of partitions with large rank in fixed residue classes.

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arXiv2022

Theta-type congruences for colored partitions

We investigate congruence relations of the form $p_r(\ell m n + t) \equiv 0 \pmod{\ell}$ for all $n$, where $p_r(n)$ is the number of $r$-colored partitions of $n$ and $m,\ell$ are distinct primes.

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arXiv2022

Congruences of Hurwitz class numbers on square classes

We extend a holomorphic projection argument of our earlier work to prove a novel divisibility result for non-holomorphic congruences of Hurwitz class numbers. This result allows us to establish Ramanujan-type congruences for Hurwitz class numbers on square classes, where the holomorphic case parallels previous work by Radu on partition congruences. We offer two applications. The first application demonstrates common divisibility features of Ramanujan-type congruences for Hurwitz class numbers. The second application provides a dichotomy between congruences for class numbers of imaginary quadratic fields and Ramanujan-type congruences for Hurwitz class numbers.

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arXiv2020

Non-Holomorphic Ramanujan-type Congruences for Hurwitz Class Numbers

In contrast to all other known Ramanujan-type congruences, we discover that Ramanujan-type congruences for Hurwitz class numbers can be supported on non-holomorphic generating series. We establish a divisibility result for such non-holomorphic congruences of Hurwitz class numbers. The two keys tools in our proof are the holomorphic projection of products of theta series with a Hurwitz class number generating series and a theorem by Serre, which allows us to rule out certain congruences.

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arXiv2018

Class number divisibility for imaginary quadratic fields

In this note we revisit classic work of Soundararajan on class groups of imaginary quadratic fields. Let $A,B,g \ge 3$ be positive integers such that $\gcd(A,B)$ is square-free. We refine Soundararajan's result to show that if $4 \nmid g$ or if $A$ and $B$ satisfy certain conditions, then the number of negative square-free $D \equiv A \pmod{B}$ down to $-X$ such that the ideal class group of $\mathbb{Q} (\sqrt{D})$ contains an element of order $g$ is bounded below by $X^{\frac{1}{2} + ε(g) - ε}$, where the exponent is the same as in Soundararajan's theorem. Combining this with a theorem of Frey, we give a lower bound for the number of quadratic twists of certain elliptic curves with $p$-Selmer group of rank at least $2$, where $p \in \{3,5,7\}$.

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arXiv2017

Indivisibility of class numbers of imaginary quadratic fields

We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to -X whose class numbers are indivisible by a given prime and whose imaginary quadratic fields satisfy any given set of local conditions. This estimate matches the best results in the direction of the Cohen-Lenstra heuristics for the number of imaginary quadratic fields with class number indivisible by a given prime. This general result is applied to study rank 0 twists of certain elliptic curves.

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arXiv2016

Asymptotic bounds for special values of shifted convolution Dirichlet series

Hoffstein and Hulse defined the shifted convolution series of two cusp forms by "shifting" the usual Rankin-Selberg convolution L-series by a parameter h. We use the theory of harmonic Maass forms to study the behavior in h-aspect of certain values of these series and prove a polynomial bound as h approaches infinity. Our method relies on a result of Mertens and Ono, who showed that these values are Fourier coefficients of mixed mock modular forms.

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