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Niranjan Ramachandran

Publications and source records attributed to Niranjan Ramachandran.

At least 19 recordsLinked to original sources
arXiv2023

Derived categories of curves of genus one and torsors over abelian varieties

Suppose $C$ is a smooth projective curve of genus 1 over a perfect field $F$, and $E$ is its Jacobian. In the case that $C$ has no $F$-rational points, so that $C$ and $E$ are not isomorphic, $C$ is an $E$-torsor with a class $δ(C)\in H^1(\text{Gal}(\bar F/F), E(\bar F))$. Then $δ(C)$ determines a class $β\in \text{Br}(E)/\text{Br}(F)$ and there is a Fourier-Mukai equivalence of derived categories of (twisted) coherent sheaves $\mathcal D(C) \xrightarrow{\cong} \mathcal D(E, β^{-1})$. We generalize this result to higher dimensions; namely, we prove it also for torsors over abelian varieties.

math.AG↗
arXiv2021

Fiber integration of gerbes and Deligne line bundles

Let $π: X \to S$ be a family of smooth projective curves, and let $L$ and $M$ be a pair of line bundles on $X$. We show that Deligne's line bundle $\langle{L,M}\rangle$ can be obtained from the $\mathcal{K}_2$-gerbe $G_{L,M}$ constructed in a previous work by the authors via an integration along the fiber map for gerbes that categorifies the well known one arising from the Leray spectral sequence of $π$. Our construction provides a full account of the biadditivity properties of $\langle {L,M}\rangle$. The functorial description of the low degree maps in the Leray spectral sequence for $π$ we develop are of independent interest, and along the course we provide an example of their application to the Brauer group.

math.AG↗
arXiv2015

Cup products, the Heisenberg group, and codimension two algebraic cycles

We define higher categorical invariants (gerbes) of codimension two algebraic cycles and provide a categorical interpretation of the intersection of divisors on a smooth proper algebraic variety. This generalization of the classical relation between divisors and line bundles furnishes a new perspective on the Bloch-Quillen formula.

math.AG↗
arXiv2015

Higher Euler characteristics

We provide a natural interpretation of the secondary Euler characteristic and introduce higher Euler characteristics. For a compact oriented manifold of odd dimension, the secondary Euler characteristic recovers the Kervaire semi-characteristic. We prove basic properties of the higher invariants and illustrate their use. We also introduce motivic variants.

math.KT↗
arXiv2015

A note on the Bloch-Tamagawa space and Selmer groups

For any abelian variety $A$ over a number field, we construct an extension of the Tate-Shafarevich group by the Bloch-Tamagawa space using the recent work of Lichtenbaum and Flach. This gives a new example of a Zagier sequence for the Selmer group of $A$.

math.NT↗
arXiv2014

Exponentiation of motivic measures

In this short note we establish some properties of all those motivic measures which can be exponentiated. As a first application, we show that the rationality of Kapranov's zeta function is stable under products. As a second application, we give an elementary proof of a result of Totaro.

math.AG↗
arXiv2014

Zeta functions, Grothendieck groups, and the Witt ring

We prove some results connecting the zeta functions of varieties over finite fields with the big Witt ring over $\mathbb Z$. We explore relations with motivic measures and a classical formula of Macdonald on invariants of symmetric products of a variety.

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arXiv2013

Motivic complexes and special values of zeta functions

Beginning with the conjecture of Artin and Tate in 1966, there has been a series of successively more general conjectures expressing the special values of the zeta function of an algebraic variety over a finite field in terms of other invariants of the variety. In this article, we present the ultimate such conjecture, and provide evidence for it. In particular, we enhance Voevodsky's Z[1/p]-category of etale motivic complexes with a p-integral structure, and show that, for this category, our conjecture follows from the Tate and Beilinson conjectures. As the conjecture is stated in terms of motivic complexes, it (potentially) applies also to algebraic stacks, log varieties, simplicial varieties, etc..

math.AG↗
arXiv2013

The p-cohomology of algebraic varieties and special values of zeta functions

The p-cohomology of an algebraic variety in characteristic p lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl-Illusie-Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc. and the special values of their zeta functions. These results provide compelling evidence that $D_{c}^{b}(R)$ is the correct target for p-cohomology in characteristic p.

math.NT↗
arXiv2008

Motivic complexes over finite fields and the ring of correspondences at the generic point

Already in the 1960s Grothendieck understood that one could obtain an almost entirely satisfactory theory of motives over a finite field when one assumes the full Tate conjecture. In this note we prove a similar result for motivic complexes. In particular Beilinson's Q-algebra of "correspondences at the generic point" is then defined for all connected varieties. We compute this for all smooth projective varieties (hence also for varieties birational to such a variety).

math.AG↗
arXiv2005

Values of zeta functions at s=1/2

We study the behaviour near s=1/2 of zeta functions of varieties over finite fields F_q with q a square. The main result is an Euler-characteristic formula for the square of the special value at s=1/2. The Euler-characteristic is constructed from the Weil-etale cohomology of a certain supersingular elliptic curve.

math.NT↗
arXiv2005

KMS states and complex multiplication

We construct a quantum statistical mechanical system which generalizes the Bost-Connes system to imaginary quadratic fields K of arbitrary class number and fully incorporates the explict class field theory for such fields. This system admits the Dedekind zeta function as partition function and the Idele class group as group of symmetries. The extremal KMS states at zero temperature intertwine this symmetry with the Galois action on the values of the states on the arithmetic subalgebra. We also give an interpretation of the original BC system and of the GL(2) system in terms of Shimura varieties, which motivates the construction for imaginary quadratic fields. The geometric notion underlying the construction is that of commensurability of K-lattices.

math.OA↗
arXiv2003

One-motives and a conjecture of Deligne

We introduce new motivic invariants of arbitrary varieties over a perfect field. These cohomological invariants take values in the category of one-motives (considered up to isogeny in positive characteristic). The algebraic definition of these invariants presented here proves a conjecture of Deligne. Applications include some cases of conjectures of Serre, Katz and Jannsen on the independence of $\ell$ of parts of the étale cohomology of arbitrary varieties over number fields and finite fields.

math.AG↗
arXiv2002

A Weil-Barsotti formula for Drinfeld modules

We study the group of extensions in the category of Drinfeld modules and Anderson's t-modules, and we show in certain cases that this group can itself be given the structure of a t-module. Our main result is a Drinfeld module analogue of the Weil-Barsotti formula for abelian varieties. Extensions of general t-modules are also considered, in particular extensions of tensor powers of the Carlitz module. We motivate these results from various directions and compare to the situation of elliptic curves.

math.AG↗
arXiv2002

Integral Motives and Special Values of Zeta Functions

For each field k, we define an abelian category of rationally decomposed mixed motives with integer coefficients. When k is finite, we show that the category is Tannakian, and we prove formulas relating the behaviour of zeta functions near integers to certain Ext groups. This is the submitted version, with minor corrections and additions from the first version.

math.NT↗
arXiv2001

From Jacobians to one-motives: exposition of a conjecture of Deligne

Deligne has conjectured that certain mixed Hodge theoretic invariants of complex algebraic invariants are motivic. This conjecture specializes to an algebraic construction of the Jacobian for smooth projective curves, which was done by A. Weil. The conjecture (and one-motives) are motivated by means of Jacobians, generalized Jacobians of Rosenlicht, and Serre's generalized Albanese varieties. We discuss the connections with the Hodge and the generalized Hodge conjecture. We end with some applications to number theory by providing partial answers to questions of Serre, Katz and Jannsen.

math.AG↗