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Rajneesh Kumar Singh

Publications and source records attributed to Rajneesh Kumar Singh.

7 recordsLinked to original sources
arXiv2021

The large sieve for square moduli in function fields

We prove a lower and an upper bound for the large sieve with square moduli for function fields. These bounds correspond to bounds for the classical large sieve with square moduli established in arXiv:1812.05844 by Baier, Lynch and Zhao and arXiv:math/0512271 by Baier and Zhao. Our lower bound in the function field setting contradicts an upper bound obtained in arXiv:1802.03131 by Baier and Singh. Indeed, we point out an error in arXiv:1802.03131.

math.NT↗
arXiv2021

Periods of Drinfeld modules and local shtukas with complex multiplication

Colmez conjectured a product formula for periods of abelian varieties over number fields with complex multiplication and proved it in some cases. His conjecture is equivalent to a formula for the Faltings height of CM abelian varieties in terms of the logarithmic derivatives at $s=0$ of certain Artin $L$-functions. In a series of articles we investigate the analog of Colmez's theory in the arithmetic of function fields. There abelian varieties are replaced by Drinfeld modules and their higher dimensional generalizations, so-called $A$-motives. In the present article we prove the product formula for the Carlitz module and we compute the valuations of the periods of a CM $A$-motive at all finite places in terms of Artin $L$-series. The latter is achieved by investigating the local shtukas associated with the $A$-motive.

math.NT↗
arXiv2020

Product Formulas for Periods of CM Abelian Varieties and the Function Field Analog

We survey Colmez's theory and conjecture about the Faltings height and a product formula for the periods of abelian varieties with complex multiplication, along with the function field analog developed by the authors. In this analog, abelian varieties are replaced by Drinfeld modules and $A$-motives. We also explain the necessary background on abelian varieties, Drinfeld modules and $A$-motives, including their cohomology theories and comparison isomorphisms and their theory of complex multiplication.

math.AG↗
arXiv2019

Large sieve inequality with power moduli for function fields

In this paper, we establish a general version of the large sieve with additive characters for restricted sets of moduli in arbitrary dimension for function fields. From this, we derive function field versions for the large sieve in high dimensions and for power moduli.

math.NT↗
arXiv2019

Divisibility problems for function fields

We investigate three combinatorial problems considered by Erdös, Rivat, Sarközy and Schön regarding divisibility properties of sum sets and sets of shifted products of integers in the context of function fields. Our results in this function field setting are better than those previously obtained for subsets of the integers. These improvements depend on a version of the large sieve for sparse sets of moduli developed recently by the first and third-named authors.

math.NT↗
arXiv2019

Local Shtukas and Divisible Local Anderson Modules

We develop the analog of crystalline Dieudonné theory for p-divisible groups in the arithmetic of function fields. In our theory p-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson's abelian t-modules and t-motives, and Drinfeld shtukas. Moreover, we discuss the existence of a Verschiebung map and apply it to deformations of local shtukas and divisible local Anderson modules. As a tool we use Faltings's and Abrashkin's theory of strict modules, which we review to some extent.

math.AG↗