arXiv subjects
Publications and source records attributed to Isabella Negrini.
We show that arithmetic definability of henselian valuations is preserved by the tilting correspondence. Moreover, we show that if a perfectoid valuation is arithmetically definable, then no parameters are needed. We also investigate whether these definitions can be chosen uniformly, and discuss the required quantifier complexity.
We construct a Shintani lift for rigid analytic cocycles of higher weight, attaching modular forms of half-integral weight to such cocycles. The expression for the Fourier coefficients of the modular form $\mathcal{RS}(J)$ attached to a cocycle $J$ is given in terms of the residues of $J$, and shares a striking similarity with the expression for the coefficients of the classical Shintani lift $\mathcal{S}(f)$ of an integral weight modular form $f$. This work aligns with the ideas of the nascent $p$-adic Kudla program and strengthens the analogy between rigid cocycles and modular forms.
The classical Maass Spezialschar is a Hecke-stable subspace of the space of holomorphic Siegel modular forms of genus two and level one cut out by certain linear relations among Fourier coefficients. We define an analogous quaternionic Maass Spezialschar, which consists of the quaternionic modular forms of level one on split $\mathrm{SO}(8)$ whose Fourier coefficients satisfy certain linear relations. We characterize this space in terms of a theta lift from the space of holomorphic Siegel modular forms on $\mathrm{Sp}(4)$, and in terms of periods. We also give a conjecture for the Dirichlet series of the standard $L$-function of quaternionic modular eigenforms on $\mathrm{SO}(8)$ and verify our conjecture on the quaternionic Maass Spezialschar.
In this paper we add to the literature on the combinatorial nature of the mock theta functions, a collection of curious $q$-hypergeometric series introduced by Ramanujan in his last letter to Hardy in 1920, which we now know to be important examples of mock modular forms. Our work is inspired by Beck's conjecture, now a theorem of Andrews, related to Euler's identity: the excess of the number of parts in all partitions of $n$ into odd parts over the number of partitions of $n$ into distinct parts is equal to the number of partitions with only one (possibly repeated) even part and all other parts odd. We establish Beck-type identities associated to partition identities due to Andrews, Dixit, and Yee for the third order mock theta functions $ω(q), ν(q)$, and $ϕ(q)$. Our proofs are both analytic and combinatorial in nature, and involve mock theta generating functions and combinatorial bijections.
This paper takes the first steps towards a systematic study of additive rigid meromorphic cocycles of higher weight. These were introduced by Darmon and Vonk, who focused on multiplicative and weight two cocycles. After classifying certain rigid meromorphic cocycles of weight $2k$, we construct an explicit holomorphic kernel function realising a Shimura-Shintani style correspondence from modular forms of weight $k+1/2$ and level $4p^2$ to rigid analytic cocycles of weight $2k$ on SL$_2(\mathbb{Z}[1/p])$.
Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called "Beck-type" companions to other identities. In this paper, we establish a collection of Beck-type companion identities to the following result mentioned by Lehmer at the 1974 International Congress of Mathematicians: the excess of the number of partitions of n with an even number of even parts over the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct, odd parts. We also establish various generalizations of Lehmer's identity, and prove related Beck-type companion identities. We use both analytic and combinatorial methods in our proofs.