arXiv subjects

Jitendra Bajpai

Publications and source records attributed to Jitendra Bajpai.

At least 19 recordsLinked to original sources

arXiv2026

On the classification of small cyclotomic integers

We give a general classification theorem for cyclotomic algebraic integers with all complex absolute values bounded by a fixed constant $c$, modeled on the theorem of Cassels which treats the case $c = \sqrt{5}$ up to finitely many exceptions. As a corollary, we establish that the range of the function taking a cyclotomic integer to its maximum complex absolute value is a well-ordered (but not closed) subset of the real numbers. We also formulate analogous statements for algebraic numbers in the maximal cyclotomic extension of a fixed number field. The proofs combine a result of Loxton, which bounds the number of roots of unity in the shortest additive representation of a cyclotomic integer in terms of the maximum complex absolute value, with an equidistribution theorem of Bilu et al. for Galois orbits of torsion points on algebraic tori.

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arXiv2026

Euler Characteristics of $\mathrm{SL}_4(\mathbb{Z})$ and $\mathrm{GL}_4(\mathbb{Z})$, and their cohomological consequences

We compute the homological Euler characteristics of $\mathrm{SL}_4(\mathbb{Z})$ and $\mathrm{GL}_4(\mathbb{Z})$ with coefficients in arbitrary irreducible rational highest-weight representations. Applying Wall's formula, we combine the orbifold Euler characteristics of centralizers of torsion elements with traces computed using the Jacobi-Trudi identity to derive explicit formulas and rational generating functions. Consequently, these Euler characteristics are quasi-polynomial functions of the highest-weight parameters, of total degree at most two. We further derive degreewise vanishing results and parity-sensitive lower bounds for the dimensions of cohomology groups. The two extensions of an $\mathrm{SL}_4(\mathbb{Z})$-coefficient system to $\mathrm{GL}_4(\mathbb{Z})$ yield sharper bounds, which grow linearly or quadratically in explicit infinite families. For symmetric powers, combining our formulas with Horozov's calculation of the determinant-twisted summand yields exact identities and lower bounds for the untwisted summand, together with a conjectural degreewise description of its cohomology.

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arXiv2025

Bloch-Beilinson conjectures for Hecke characters and Eisenstein cohomology of Picard surfaces

We consider certain families of Hecke characters $ϕ$ over a quadratic imaginary field $F$. According to the Bloch-Beilinson conjectures, the order of vanishing of the $L$-function $L(ϕ,s)$ at the central point $s=-1$ should be equal to the dimension of the space of extensions of the Tate motive $\mathbb{Q}(1)$ by the motive associated with $ϕ$. In this article, we construct candidates for the corresponding extensions of Hodge structures, assuming that the sign of the functional equation of $L(ϕ,s)$ is $-1$. This is accomplished through the cohomology of variations of Hodge structures over Picard modular surfaces associated with $F$ and Harder's theory of Eisenstein cohomology. Furthermore, we demonstrate that these extensions are naturally realized within certain biextensions. We outline a program to compute the biextension height and utilize it to establish the non-triviality of these extensions.

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arXiv2025

A CFSG-free explicit Jordan's theorem over arbitrary fields

We prove a version of Jordan's classification theorem for finite subgroups of $\mathrm{GL}_{n}(K)$ that is at the same time quantitatively explicit, CFSG-free, and valid for arbitrary $K$. This is the first proof to satisfy all three properties at once. Our overall strategy follows Larsen and Pink [24], with explicit computations based on techniques developed by the authors and Helfgott [2, 3], particularly in relation to dimensional estimates.

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arXiv2025

The exceptional set in Cassel's theorem on small cyclotomic integers

In a 1965 paper, R. Robinson made five conjectures about the classification of cyclotomic algebraic integers for which the maximum absolute value in any complex embedding (the house) is small, modulo the equivalence relation generated by Galois conjugation and multiplication by roots of unity. In response to one of these conjectures, Cassels showed in 1969 that when the house is at most $\sqrt{5}$, one obtains three parametric families plus an effectively computable finite set of equivalence classes of exceptions. Building on the work of Jones, Calegari-Morrison-Snyder, and Robinson-Wurtz, we determine this exceptional set. By specializing to the case where the house is strictly less than 2, we resolve the final outstanding conjecture from Robinson's 1965 paper.

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arXiv2024

Growth estimates and diameter bounds for untwisted classical groups

Babai's conjecture states that, for any finite simple non-abelian group $G$, the diameter of $G$ is bounded by $(\log|G|)^{C}$ for some absolute constant $C$. We prove that, for any untwisted classical group $G$ of rank $r$ defined over a field $\mathbb{F}_{q}$ with $q$ not too small with respect to $r$, \begin{equation*} \mathrm{diam}(G(\mathbb{F}_{q}))\leq(\log|G(\mathbb{F}_{q})|)^{408r^{4}}. \end{equation*} This bound improves on results by Breuillard, Green, and Tao [9], Pyber and Szabó [38], and, for $q$ large enough, also by Halasi, Maróti, Pyber, and Qiao [16]. Our approach is in several ways closer to that of preexistent work by Helfgott [20], in that we give dimensional estimates (that is, bounds of the form $|A\cap V(\mathbb{F}_{q})|\ll|A^{C}|^{\dim(V)/\dim(G)}$, where $A$ is any generating set) for varieties $V$ of specific types, and work in the Lie algebra whenever possible. One of our main tools is a new, more efficient form of escape from subvarieties.

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arXiv2024

Lifting of vector-valued automorphic forms

Recently, the first author [1] showed that the admissible vector-valued automorphic forms lift to the admissible ones. In this article, we study the lifts for the logarithmic vector-valued automorphic forms and explicitly compute the Fourier coefficients of the lifted vector-valued automorphic forms.

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arXiv2024

Relative Lie algebra cohomology of SU(2,1) and Eisenstein classes on Picard surfaces

We consider Picard surfaces, locally symmetric varieties $S_Γ$ attached to the Lie group SU(2,1), and we construct explicit differential forms on $S_Γ$ representing Eisenstein classes, i.e. cohomology classes restricting non-trivially to the boundary of the Borel-Serre compactification. This is needed for the computation of the class of the extensions of the Hodge structure that we have constructed in [2] according to the predictions of the Bloch-Beilinson conjectures. The tool for the construction of the differential forms is an analysis of relative Lie algebra cohomology of the principal series of SU(2,1) using recent methods of Buttcane and Miller.

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arXiv2024

New dimensional estimates for subvarieties of linear algebraic groups

For every connected, almost simple linear algebraic group $G\leq\mathrm{GL}_{n}$ over a large enough field $K$, every subvariety $V\subseteq G$, and every finite generating set $A\subseteq G(K)$, we prove a general dimensional bound, that is, a bound of the form \[|A\cap V(\overline{K})|\leq C_{1}|A^{C_{2}}|^{\frac{\dim(V)}{\dim(G)}}\] with $C_{1},C_{2}$ depending only on $n,\mathrm{deg}(V)$. The dependence of $C_1$ on $n$ (or rather on $\dim (V)$) is doubly exponential, whereas $C_2$ (which is independent of $\mathrm{deg}(V)$) depends simply exponentially on $n$. Bounds of this form have proved useful in the study of growth in linear algebraic groups since 2005 (Helfgott) and, before then, in the study of subgroup structure (Larsen-Pink: $A$ a subgroup). In bounds for general $V$ and $G$ available before our work, the dependence of $C_1$ and $C_2$ on $n$ was of exponential-tower type. We draw immediate consequences regarding diameter bounds for untwisted classical groups $G(\mathbb{F}_{q})$. (In a separate paper, we derive stronger diameter bounds from stronger dimensional bounds we prove for specific families of varieties $V$.)

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arXiv2023

Thin monodromy in O(5)

This article studies the orthogonal hypergeometric groups of degree five. We establish the thinness of 12 out of the 19 hypergeometric groups of type O(3,2) from [4, Table 6]. Some of these examples are associated with Calabi-Yau 4-folds. We also establish the thinness of 9 out of the 17 hypergeometric groups of type O(4,1) from [12], where the thinness of 7 other cases was already proven. The O(4,1) type groups were predicted to be all thin and our result leaves just one case open.

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arXiv2022

Thin monodromy in $\mathrm{Sp}(4)$ and $\mathrm{Sp}(6)$

We explore the thinness of hypergeometric groups of type $\mathrm{Sp}(4)$ and $\mathrm{Sp}(6)$ by applying a new approach of computer-assisted ping pong. We prove the thinness of $17$ hypergeometric groups with maximally unipotent monodromy in $\mathrm{Sp}(6)$, completing the classification of all $40$ such groups into arithmetic and thin cases. In addition, we establish the thinness of further $46$ hypergeometric groups in $\mathrm{Sp}(6)$, and of $3$ hypergeometric groups in $\mathrm{Sp}(4)$, completing the classification of all $\mathrm{Sp}(4)$ hypergeometric groups. To the best of our knowledge, this article produces the first $63$ examples in the cyclotomic family of Zariski dense non-arithmetic hypergeometric monodromy groups of real rank three.

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arXiv2022

Arithmetic Monodromy in Sp(2n)

Based on a result of Singh--Venkataramana, Bajpai--Dona--Singh--Singh gave a criterion for a discrete Zariski-dense subgroup of Sp(2n,Z) to be a lattice. We adapt this criterion so that it can be used in some situations that were previously excluded. We apply the adapted method to subgroups of Sp(6,Z) and Sp(4,Z) that arise as the monodromy groups of hypergeometric differential equations. In particular, we show that out of the 40 maximally unipotent Sp(6) hypergeometric groups more than half are arithmetic, answering a question of Katz in the negative.

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arXiv2022

Commentary on Sp(6) hypergeometric groups

We study the examples mentioned in [2,Tables A & C] and establish the arithmeticity of four examples of symplectic hypergeometric groups of degree six. Following [2] we know that there are 458 inequivalent symplectic hypergeometric groups of degree six, and combining the results of this article with the work of [1,2,11], we now know that at least 384 are arithmetic and at least 63 are thin whereas the arithmeticity and thinness of remaining 11 examples are still unknown.

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arXiv2022

Bilateral series and Ramanujan's radial limits

Ramanujan's last letter to Hardy explored the asymptotic properties of modular forms, as well as those of certain interesting $q$-series which he called \emph{mock theta functions}. For his mock theta function $f(q)$, he claimed that as $q$ approaches an even order $2k$ root of unity $ζ$, \[\lim_{q\to ζ} \big(f(q) - (-1)^k (1-q)(1-q^3)(1-q^5)\cdots (1-2q + 2q^4 - \cdots)\big) = O(1),\] and hinted at the existence of similar statements for his other mock theta functions. Recent work of Folsom-Ono-Rhoades provides a closed formula for the implied constant in this radial limit of $f(q)$. Here, by different methods, we prove similar results for all of Ramanujan's 5th order mock theta functions. Namely, we show that each 5th order mock theta function may be related to a modular bilateral series, and exploit this connection to obtain our results. We further explore other mock theta functions to which this method can be applied.

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arXiv2022

Arithmeticity of Some Hypergeometric Groups

We show that the hypergeometric groups associated to the pairs of the parameters $\left(0,0,\frac{1}{3}, \frac{2}{3}\right)$, $\left(\frac{1}{2},\frac{1}{2},\frac{1}{4},\frac{3}{4}\right)$; and $\left(0,\frac{1}{12}, \frac{5}{12},\frac{7}{12},\frac{11}{12}\right), \left(\frac{1}{2},\frac{1}{3},\frac{1}{3},\frac{2}{3},\frac{2}{3}\right)$ are arithmetic.

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arXiv2021

Growth of Fourier coefficients of vector-valued automorphic forms

In this article, we establish polynomial-growth bound for the sequence of Fourier coefficients associated to even integer weight vector-valued automorphic forms of Fuchsian groups of the first kind. At the end, their $L$-functions and exponential sums have been discussed.

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arXiv2021

Euler characteristic and cohomology of $\mathrm{Sp}_4(\mathbb{Z})$ with nontrivial coefficients

In this article, the cohomology of the arithmetic group $\mathrm{Sp}_4(\mathbb{Z})$ with coefficients in any finite dimensional highest weight representation $\mathcal{M}_λ$ have been studied. Euler characteristic with coefficients in $\mathcal{M}_λ$ have been carried out in detail. Combining the results obtained on Euler characteristic and the work of Harder on Eisenstein cohomology, the description of the cuspidal cohomology has been achieved. At the end, we employ our study to compute the dimensions for the cohomology spaces $H^{\bullet}(\mathrm{Sp}_4(\mathbb{Z}), \mathcal{M}_λ)$.

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arXiv2021

Ghost classes in $\mathbb{Q}$-rank two orthogonal Shimura varieties

In this article, the existence of ghost classes for the Shimura varieties associated to algebraic groups of orthogonal similitudes of signature (2, n) is investigated. We make use of the study of the weights in the mixed Hodge structures associated to the corresponding cohomology spaces and results on the Eisenstein cohomology of the algebraic group of orthogonal similitudes of signature (1, n-1). For the values of n = 4, 5 we prove the non-existence of ghost classes for most of the irreducible representations (including most of those with an irregular highest weight). For the rest of the cases, we prove strong restrictions on the possible weights in the space of ghost classes and, in particular, we show that they satisfy the weak middle weight property.

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