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Kirti Joshi

Publications and source records attributed to Kirti Joshi.

At least 37 records · Page 2Linked to original sources
arXiv2023

Construction of Arithmetic Teichmuller spaces II: Towards Diophantine Estimates

This paper deals with three consequences of the existence of Arithmetic Teichmuller spaces of arXiv:2106.11452. Let $\mathscr{X}_{F,\mathbb{Q}_p}$ (resp. $B=B_{\mathbb{Q}_p}$) be the complete Fargues-Fontaine curve (resp. the ring) constructed by Fargues-Fontaine with the datum $F={\mathbb{C}_p^\flat}$ (the tilt of $\mathbb{C}_p$), $E=\mathbb{Q}_p$. Fix an odd prime $\ell$, let $\ell^*=\frac{\ell-1}{2}$. The construction (§7) of an uncountable subset $Σ_{F}\subset \mathscr{X}_{F,\mathbb{Q}_p}^{\ell^*}$ with a simultaneous valuation scaling property (Theorem 7.8.1), Galois action and other symmetries. Now fix a Tate elliptic curve over a finite extension of $\mathbb{Q}_p$. The existence of $Σ_{F}$ leads to the construction (§9) of a set $\widetildeΘ\subset B^{\ell^*}$ consisting of lifts (to $B$), of values (lying in different untilts provided by $Σ_{F}$) of a chosen theta-function evaluated at $2\ell$-torsion points on the chosen elliptic curve. The construction of $\widetildeΘ$ can be easily adelized. Moreover I also prove a lower bound (Theorem 10.1.1) for the size of $\widetildeΘ$ (here size is defined in terms of the Fréchet structure of $B$). I also demonstrate (in §11) the existence of ``log-links'' in the theory of [Joshi 2021].

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arXiv2022

Untilts of fundamental groups: construction of labeled isomorphs of fundamental groups -- Arithmetic Holomorphic Structures

Let $p$ be a prime number. Let $X/E$ be a geometrically connected, smooth, quasi-projective variety over a finite extension $E/\mathbb{Q}_p$. In this paper I demonstrate the existence of isomorphs of the tempered (and hence also étale) fundamental group of $X/E$ which are labeled by distinct arithmetic holomorphic structures, just as isomorphs of the fundamental group of a Riemann surface $Σ$ may be labeled by Riemann surfaces (i.e. complex holomorphic structures) $Σ'$ in the Teichmuller space of $Σ$. This is the starting point of the theory elaborated in [Joshi, 2021a,b,c, 2022] for which this paper is intended as an brief sketch and announcement. Arithmetic holomorphic structures introduced here also provide distinct arithmetic holomorphic structures used by Shinichi Mochizuki in [Mochizuki,2021a,b,c,d]. Since the question of whether or not there exists distinct arith. hol. structures in [Mochizuki,2021a,b,c,d] was raised in [Scholze and Stix], I include a discussion of [Scholze and Stix]. See the introduction for additional details.

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arXiv2021

Corrigendum: On the construction of Weakly Ulrich bundles

This note corrects a mistake in Theorem~4.1 of https://doi.org/10.1016/j.aim.2021.107598 (arXiv:1901.03395). The error noted here does not affect any other results of loc. cit. To correct the error, here I prove a more general result (Theorem 3.1) and deduce Theorem~3.3 and also the correct version of the previously announced theorem in Theorem~3.4. Theorem~3.5 supplements Theorem~3.3. In it, I prove that if $k$ is an algebraically closed field of characteristic $p\geq 3$ and $X/k$ is any smooth, projective, minimal surface of general type and with $Pic(X)=\mathbb{Z}$, then for all integers $r\geq 5$, $X$ is embedded as a smooth surface by its pluricanonical linear system $X\hookrightarrow |ω_X^r|$, and $E=F_*(ω_X^{r+1})(1)$ is an almost Ulrich bundle for the pluricanonical embedding $X\hookrightarrow|ω_X^r|$ of $X$ and for the ample line bundle provided by this embedding. Corollary~3.6 generalizes Theorem 3.1 of https://doi.org/10.1016/j.aim.2021.107598 (arXiv:1901.03395). In particular this note corrects and extends the results of loc. cit.

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arXiv2021

On the construction of Weakly Ulrich bundles

I provide a construction of intrinsic weakly Ulrich bundles of large rank on any smooth complete surface in ${\bf P}^3$ over fields of characteristic $p>0$ and also for some classes of surfaces of general type in ${\bf P}^n$. I also construct intrinsic weakly Ulrich bundles on any Frobenius split variety of dimension at most three. The bundles constructed here are in fact ACM and weakly Ulrich bundles and so I call them almost Ulrich bundles. Frobenius split varieties in dimension three include as special cases: (1) smooth hypersurfaces in ${\bf P}^4$ of degree at most four, (2) Frobenius split, smooth quintics in ${\bf P}^4$ (3) Frobenius split Calabi-Yau varieties of dimension at most three (4) Frobenius split (i.e ordinary) abelian varieties of dimension at most three.

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arXiv2020

Opers of higher types, Quot-schemes and Frobenius instability loci

In this paper we continue our study of the Frobenius instability locus in the coarse moduli space of semi-stable vector bundles of rank $r$ and degree $0$ over a smooth projective curve defined over an algebraically closed field of characteristic $p>0$. In a previous paper we identified the "maximal" Frobenius instability strata with opers (more precisely as opers of type $1$ in the terminology of the present paper) and related them to certain Quot-schemes of Frobenius direct images of line bundles. The main aim of this paper is to describe for any integer $q \geq 1$ a conjectural generalization of this correspondence between opers of type $q$ (which we introduce here) and Quot-schemes of Frobenius direct images of vector bundles of rank $q$. We also give a conjectural formula for the dimension of the Frobenius instability locus.

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arXiv2019

A method for construction of rational points over elliptic curves II: Points over solvable extensions

I provide a systematic construction of points, defined over finite radical extensions, on any Legendre curve over any field of characteristic not equal two. This includes as special case Douglas Ulmer's construction of rational points over a rational function field in characteristic $p>0$. In particular I show that if $n\geq 4$ is any even integer and not divisible by the characteristic of the field then any elliptic curve $E$ over this field has at least $2n$ rational points over a finite solvable field extension. Under additional hypothesis, when the ground field is a number field, I show that these are of infinite order. I also show that Ulmer's points lift to characteristic zero and in particular to the canonical lifting.

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arXiv2019

A remark on Ulrich and ACM bundles

I show that on any smooth, projective ordinary curve of genus at least two and a projective embedding, there is a natural example of a stable Ulrich bundle for this embedding: namely the sheaf $B^1_X$ of locally exact differentials twisted by $Ø_X(1)$ given by this embedding and in particular there exist ordinary varieties of any dimension which carry Ulrich bundles. In higher dimensions, assuming $X$ is Frobenius split variety I show that $B^1_X$ is an ACM bundle and if $X$ is also a Calabi-Yau variety and $p>2$ then $B^1_X$ is not a direct sum of line bundles. In particular I show that $B^1_X$ is an ACM bundle on any ordinary Calabi-Yau variety. I also prove a characterization of projective varieties with trivial canonical bundle such that $B^1_X$ is ACM (for some projective embedding datum): all such varieties are Frobenius split (with trivial canonical bundle).

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arXiv2018

Crystalline aspects of geography of low dimensional varieties I: Numerology

This is a modest attempt to study, in a systematic manner, the structure of low dimensional varieties in positive characteristics using $p$-adic invariants. The main objects of interest in this paper are surfaces and threefolds. It is known that there are many (counter) examples of `pathological' or unexpected behavior in surfaces and even of threefolds. Our focus has been on, obtaining systematically, general positive results instead of counter examples. There are many results we prove in this paper and not all can be listed in this abstract. Here are some of the results we prove: We show that $c_1^2\leq 5c_2+6b_1$ holds for a large class of surfaces (of general type). We prove that for a smooth, projective, Hodge-Witt, minimal surface of general type (with additional assumptions such as slopes of Frobenius on $H^2_{cris}(X)$ are $\geq\frac{1}{2}$) that $$c_1^2\leq 5c_2.$$ Novelty of our method lies in our use of slopes of Frobenius (and Hodge-Witt assumption which allows us to control the slope spectral sequence). We also construct new birational invariants of surfaces. Applying our methods to threefolds, we characterise Calabi-Yau threefolds with $b_3=0$.

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arXiv2018

A method for construction of rational points over elliptic curves

I provide a systematic construction of points (defined over number fields) on Legendre elliptic curves over $\mathbb{Q}$: for any odd integer $n\geq 3$ my method constructs $n$ points on the Legendre curve and I show that rank of the subgroup of the Mordell-Weil group they generate is $n$ if $n\geq 7$. I also show that every elliptic curve over any number field admits similar type of points after a finite base extension.

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arXiv2018

Methods for constructing elliptic and hyperelliptic curves with rational points

I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number field equipped with a rational point, (resp. with two rational points) of infinite order over the given number field, and elliptic curves over the rationals with two rational points over `simplest cubic fields.' I also provide hyperelliptic curves of genus exceeding any given number over any given number fields with points (over the given number field) which span a subgroup of rank at least $g$ in the group of rational points of the Jacobian of this curve. I also provide a method of constructing hyperelliptic curves over rational function fields with rational points defined over field extensions with large finite simple Galois groups, such as the Mathieu group $M_{24}$.

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arXiv2016

The degree of the dormant operatic locus

Let $X$ be a smooth, projective curve of genus $g\geq 2$ over an algebraically closed field of characteristic $p>0$. I provide a conjectural formula for the degree of the scheme of dormant ${\rm PGL}(r)$-opers on $X$ where $r\geq 2$ (I assume that $p$ is greater than an explicit constant depending on $g,r$). For $r=2$ a dormant ${\rm PGL}(2)$-oper is a dormant indigenous bundle on $X$ in the sense of Shinichi Mochuzki (and his work provides a formula only for $g=2,r=2,p\geq 5$, from a different point of view). Recently Yasuhiro Wakabayashi has shown that my conjectural formula holds for $X$ generic, $r\geq 2$ and $p$ large.

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arXiv2016

Ordinary reductions of abelian varieties

I show that a conjecture of Joshi-Rajan on primes of Hodge-Witt reduction and in particular a conjecture of Jean-Pierre Serre on primes of good, ordinary reduction for an abelian variety over a number field follows from a certain conjecture on Galois rep- resentations which may perhaps be easier to prove (and I prove this conjecture for abelian compatible systems of a suitable type). This reduction (to a conjecture about certain sys- tems of Galois representations) is based on a new slope estimate for non Hodge-Witt abelian varieties. In particular for any abelian variety over a number field with at least one prime of good ordinary or split toric reduction, I show that the conjecture of Joshi-Rajan and the conjecture of Serre on ordinary reductions can be reduced to proving that a certain rational trace of Frobenius is in fact an integer. The assertion that this trace is an integer is proved for abelian systems of Galois representations (of suitable type).

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arXiv2016

On varieties with trivial tangent bundle

I prove a crystalline characterization of abelian varieties in characteristic $p>0$ amongst the class of varieties with trivial tangent bundle. I show using my characterization that a smooth, projective, ordinary variety with trivial tangent bundle is an abelian variety if and only if its second crystalline cohomology is torsion free. I also show that a conjecture of Ke-Zheng Li about characteristic $p>0$ varieties with trivial tangent bundles is true for surfaces. I give a new proof of a result of Li and prove a refinement of it. Based on my characterization of abelian varieties I propose modifications of Li's conjecture which I expect to be true.

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arXiv2015

Asymptotics of conductors of elliptic curves over Q

In this note we study numbers which occur as conductors of elliptic curves over Q. We show, by constructing families of elliptic curves with quadratic discriminant and invoking a theorem of Iwaniec, that this set contains infinitely many almost primes. We show, assuming a strong version of the Cohen-Lenstra heuristics, that the set of prime conductors has an explicitly bounded density in the set of primes. Studying the Cremona and Stein-Watkins databases of elliptic curves we conjecture that the set of conductors should be of density zero in the set of natural numbers and that the set of prime conductors should be of density zero in the set of prime numbers.

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arXiv2015

Eigenform Product Identities For Hilbert Modular Forms

We prove that amongst all real quadratic fields and all spaces of Hilbert modular forms of full level and of weight $2$ or greater, the product of two Hecke eigenforms is not a Hecke eigenform except for finitely many real quadratic fields and finitely many weights. We show that for $\mathbb Q(\sqrt 5)$ there are exactly two such identities.

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arXiv2015

Hitchin-Mochizuki morphism, Opers and Frobenius-destabilized vector bundles over curves

Let X be a smooth projective curve of genus g \textgreater{}1 defined over an algebraically closed field k of characteristic p \textgreater{}0. For p sufficiently large (explicitly given in terms of r,g) we construct an atlas for the locus of all Frobenius-destabilized bundles (i.e. we construct all Frobenius-destabilized bundles of degree zero up to isomorphism). This is done by exhibiting a surjective morphism from a certain Quot-scheme onto the locus of stable Frobenius-destabilized bundles. Further we show that there is a bijective correspondence between the set of stable vector bundles E over X such that the pull-back F^*(E) under the Frobenius morphism of X has maximal Harder-Narasimhan polygon and the set of opers having zero p-curvature. We also show that, after fixing the determinant, these sets are finite, which enables us to derive the dimension of certain Quot-schemes and certain loci of stable Frobenius-destabilized vector bundles over X. The finiteness is proved by studying the properties of the Hitchin-Mochizuki morphism; an alternative approach to finiteness has been realized in a recent preprint by Chen and Zhu. In particular we prove a generalization of a result of Mochizuki to higher ranks.

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arXiv2014

Some remarks on Hodge symmetry

I make some remarks on Hodge symmetry, and prove for instance that if $k$ is a perfect field of characteristic $p>0$ and $X/k$ smooth, proper and Hodge-Witt scheme, and Hodge de Rham sequence of $X$ degenerates at $E_1$ and $X$ has torsion-free crystalline cohomology, then Hodge symmetry holds for $X$.

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