arXiv subjects

Ian Kiming

Publications and source records attributed to Ian Kiming.

14 recordsLinked to original sources

arXiv2023

Eisenstein series, $p$-adic modular functions, and overconvergence, II

Let $p$ be a prime number. Continuing and extending our previous paper with the same title, we prove explicit rates of overconvergence for modular functions of the form $\frac{E_k^{\ast}}{V(E_k^{\ast})}$ where $E_k^{\ast}$ is a classical, normalized Eisenstein series on $\Gamma_0(p)$ and $V$ the $p$-adic Frobenius operator. In particular, we extend our previous paper to the primes $2$ and $3$. For these primes our main theorem improves somewhat upon earlier results by Emerton, Buzzard and Kilford, and Roe. We include a detailed discussion of those earlier results as seen from our perspective. We also give some improvements to our earlier paper for primes $p\ge 5$. Apart from establishing these improvements, our main purpose here is also to show that all of these results can be obtained by a uniform method, i.e., a method where the main points in the argumentation is the same for all primes. We illustrate the results by some numerical examples.

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arXiv2022

A Conjecture of Coleman on the Eisenstein Family

We prove for primes $p\ge 5$ a conjecture of Coleman on the analytic continuation of the family of modular functions $\frac{E^\ast_\kappa}{V(E^\ast_\kappa)}$ derived from the family of Eisenstein series $E^\ast_\kappa$. The precise, quantitative formulation of the conjecture involved a certain on $p$ depending constant. We show by an example that the conjecture with the constant that Coleman conjectured cannot hold in general for all primes. On the other hand, the constant that we give is also shown not to be optimal in all cases. The conjecture is motivated by its connection to certain central statements in works by Buzzard and Kilford, and by Roe, concerning the "halo" conjecture for the primes $2$ and $3$, respectively. We show how our results generalize those statements and comment on possible future developments.

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arXiv2020

Eisenstein series, p-adic modular functions, and overconvergence

Let $p$ be a prime $\ge 5$. We establish explicit rates of overconvergence for members of the "Eisenstein family", notably for the $p$-adic modular function $V(E_{(1,0)}^{\ast})/E_{(1,0)}^{\ast}$ ($V$ the $p$-adic Frobenius operator) that plays a pi\-votal role in Coleman's theory of $p$-adic families of modular forms. The proof goes via an in-depth analysis of rates of overconvergence of $p$-adic modular functions of form $V(E_k)/E_k$ where $E_k$ is the classical Eisenstein series of level $1$ and weight $k$ divisible by $p-1$. Under certain conditions, we extend the latter result to a vast generalization of a theorem of Coleman--Wan regarding the rate of overconvergence of $V(E_{p-1})/E_{p-1}$. We also comment on previous results in the literature. These include applications of our results for the primes $5$ and $7$.

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arXiv2018

Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4

We work out a non-trivial example of lifting a so-called weak eigenform to a true, characteristic 0 eigenform. The weak eigenform is closely related to Ramanujan's tau function whereas the characteristic 0 eigenform is attached to an elliptic curve defined over ${\mathbb Q}$. We produce the lift by showing that the coefficients of the initial, weak eigenform (almost all) occur as traces of Frobenii in the Galois representation on the 4-torsion of the elliptic curve. The example is remarkable as the initial form is known not to be liftable to any characteristic 0 eigenform of level 1. We use this example as illustrating certain questions that have arisen lately in the theory of modular forms modulo prime powers. We give a brief survey of those questions.

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arXiv2016

On certain finiteness questions in the arithmetic of modular forms

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms. Specifically, we conjecture that for fixed N, m, and prime p with p not dividing N, there is only a finite number of reductions modulo p^m of normalized eigenforms on Γ_1(N). We consider various variants of our basic finiteness conjecture, prove a weak version of it, and give some numerical evidence.

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arXiv2015

On the theta operator for modular forms modulo prime powers

We consider the classical theta operator $θ$ on modular forms modulo $p^m$ and level $N$ prime to $p$ where $p$ is a prime greater than 3. Our main result is that $θ$ mod $p^m$ will map forms of weight $k$ to forms of weight $k+2+2p^{m-1}(p-1)$ and that this weight is optimal in certain cases when $m$ is at least 2. Thus, the natural expectation that $θ$ mod $p^m$ should map to weight $k+2+p^{m-1}(p-1)$ is shown to be false. The primary motivation for this study is that application of the $θ$ operator on eigenforms mod $p^m$ corresponds to twisting the attached Galois representations with the cyclotomic character. Our construction of the $θ$-operator mod $p^m$ gives an explicit weight bound on the twist of a modular mod $p^m$ Galois representation by the cyclotomic character.

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arXiv2012

Lifts of projective congruence groups, II

We continue and complete our previous paper `Lifts of projective congruence groups' [2] concerning the question of whether there exist noncongruence subgroups of $\SL_2(\Z)$ that are projectively equivalent to one of the groups $Γ_0(N)$ or $Γ_1(N)$. A complete answer to this question is obtained: In case of $Γ_0(N)$ such noncongruence subgroups exist precisely if $N\not\in {3,4,8}$ and we additionally have either that $4\mid N$ or that $N$ is divisible by an odd prime congruent to 3 modulo 4. In case of $Γ_1(N)$ these noncongruence subgroups exist precisely if $N>4$. As in our previous paper the main motivation for this question is the fact that the above noncongruence subgroups represent a fairly accessible and explicitly constructible reservoir of examples of noncongruence subgroups of $\SL_2(\Z)$ that can serve as basis for experimentation with modular forms on noncongruence subgroups.

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arXiv2012

On modular Galois representations modulo prime powers

We study modular Galois representations mod $p^m$. We show that there are three progressively weaker notions of modularity for a Galois representation mod $p^m$: we have named these `strongly', `weakly', and `dc-weakly' modular. Here, `dc' stands for `divided congruence' in the sense of Katz and Hida. These notions of modularity are relative to a fixed level $M$. Using results of Hida we display a `stripping-of-powers of $p$ away from the level' type of result: A mod $p^m$ strongly modular representation of some level $Np^r$ is always dc-weakly modular of level $N$ (here, $N$ is a natural number not divisible by $p$). We also study eigenforms mod $p^m$ corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod $p^m$ to any `dc-weak' eigenform, and hence to any eigenform mod $p^m$ in any of the three senses. We show that the three notions of modularity coincide when $m=1$ (as well as in other, particular cases), but not in general.

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arXiv2012

Quadratic twists of rigid Calabi-Yau threefolds over $\QQ$

We consider rigid Calabi--Yau threefolds defined over $\QQ$ and the question of whether they admit quadratic twists. We give a precise geometric definition of the notion of a quadratic twists in this setting. Every rigid Calabi--Yau threefold over $\QQ$ is modular so there is attached to it a certain newform of weight 4 on some $Γ_0(N)$. We show that quadratic twisting of a threefold corresponds to twisting the attached newform by quadratic characters and illustrate with a number of obvious and not so obvious examples. The question is motivated by the deeper question of which newforms of weight 4 on some $Γ_0(N)$ and integral Fourier coefficients arise from rigid Calabi--Yau threefolds defined over $\QQ$.

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arXiv2011

On some new invariants for strong shift equivalence for shifts of finite type

We introduce a new computable invariant for strong shift equivalence of shifts of finite type. The invariant is based on an invariant introduced by Trow, Boyle, and Marcus, but has the advantage of being readily computable. We summarize briefly a large-scale numerical experiment aimed at deciding strong shift equivalence for shifts of finite type given by irreducible $2\times 2$-matrices with entry sum less than 25, and give examples illustrating to power of the new invariant, i.e., examples where the new invariant can disprove strong shift equivalence whereas the other invariants that we use can not.

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arXiv2010

Lifts of projective congruence groups

We show that noncongruence subgroups of SL_2(Z) projectively equivalent to congruence subgroups are ubiquitous. More precisely, they always exist if the congruence subgroup in question is a principal congruence subgroup Gamma(N) of level N>2, and they exist in many cases also for Gamma_0(N). The motivation for asking this question is related to modular forms: projectively equivalent groups have the same spaces of cusp forms for all even weights whereas the spaces of cusp forms of odd weights are distinct in general. We make some initial observations on this phenomenon for weight 3 via geometric considerations of the attached elliptic modular surfaces. We also develop algorithms that construct all subgroups projectively equivalent to a given congruence subgroup and decides which of them are congruence. A crucial tool in this is the generalized level concept of Wohlfahrt.

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arXiv2007

New models for the action of Hecke operators in spaces of Maass wave forms

Utilizing the theory of the Poisson transform, we develop some new concrete models for the Hecke theory in a space $M_λ(N)$ of Maass forms with eigenvalue $1/4-λ^2$ on a congruence subgroup $Γ_1(N)$. We introduce the field $F_λ = {\mathbb Q} (λ,\sqrt{n}, n^{λ/2} \mid ñ\in {\mathbb N})$ so that $F_λ$ consists entirely of algebraic numbers if $λ= 0$. The main result of the paper is the following. For a packet $Φ= (ν_p \mid p\nmid N)$ of Hecke eigenvalues occurring in $M_λ(N)$ we then have that either every $ν_p$ is algebraic over $F_λ$, or else $Φ$ will - for some $m\in {\mathbb N}$ - occur in the first cohomology of a certain space $W_{λ,m}$ which is a space of continuous functions on the unit circle with an action of $\mathrm{SL}_2({\mathbb R})$ well-known from the theory of (non-unitary) principal representations of $\mathrm{SL}_2({\mathbb R})$.

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arXiv2003

On modular mod $\ell$ Galois representations with exceptional images

We give a parametrization of the possible Serre invariants $(N,k,ν)$ of modular mod $\ell$ Galois representations of the exceptional types $A_4$, $S_4$, $A_5$, in terms of local data attached to the fields cut out by the associated projective representations. We show how this result combined with certain global considerations leads to an effective procedure that will determine for a given eigenform $f$ and prime $\ell$ whether a mod $\ell$ representation attached to $f$ is exceptional. We illustrate with numerical examples.

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arXiv2002

Mod pq Galois representations and Serre's conjecture

Motives and automorphic forms of arithmetic type give rise to Galois representations that occur in {\it compatible families}. These compatible families are of p-adic representations with p varying. By reducing such a family mod p one obtains compatible families of mod p representations. While the representations that occur in such a p-adic or mod p family are strongly correlated, in a sense each member of the family reveals a new face of the motive. In recent celebrated work of Wiles playing off a pair of Galois representations in different characteristics has been crucial. In this paper we investigate when a pair of mod p and mod q representations of the absolute Galois group of a number field K simultaneously arises from an {\it automorphic motive}: we do this in the 1-dimensional (Section 2) and 2-dimensional (Section 3: this time assuming $K={\mathbb Q}$) cases. In Section 3 we formulate a mod pq version of Serre's conjecture refining in part a question of Barry Mazur and Ken Ribet.

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