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Kirti Joshi

Publications and source records attributed to Kirti Joshi.

At least 19 recordsLinked to original sources
arXiv2026

The Categorical Local Langlands Correspondence and Anabelomorphy

Let $G/\mathbb{Q}_p$ be a connected, split, reductive group over $\mathbb{Q}_p$. In this paper I show that if $K$ and $L$ are anabelomorphic $p$-adic fields i.e. $K$ and $L$ have topologically isomorphic absolute Galois groups, then the stacks of Langlands parameters (for the fields $K$ and $L$) considered in [Fargues and Scholze, 2024], are also isomorphic (Theorem 2.2.1). This leads to Conjecture 3.3.1 which provides a precise relationship between the main conjecture of [Fargues and Scholze, 2024] and anabelomorphy of $p$-adic fields considered in [Joshi, 2020a]. I establish my conjecture for a split torus in Theorem 4.1.

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arXiv2025

On the equation $N_{p_1}(E)\cdot N_{p_2}(E)\cdots N_{p_k}(E)=n$

For a given elliptic curve $E/\mathbb{Q}$, let $N_p(E)$ be the number of points on $E$ modulo $p$ for a prime of good reduction for $E$. Given integer $n$, let $G_k(E,n)$ be the number of $k$-tuples of $p_1<p_2<\ldots <p_k$ primes of good reduction for $E$, for which the equation in the title holds, then on assuming the Generalized Riemann Hypothesis for elliptic curves without CM (and unconditionally if the curves have complex multiplication), I show that $\varlimsup_{n\to\infty} G_k(E,n)=\infty$ for any integer $k\geq 3$. I conjecture that this result also holds for $k=1,2$ i.e. this conjecture says that there are arbitrarily long ``elliptic progressions of primes'' i.e. sequences of primes $p_1<p_2<\cdots <p_m$ of arbitrary lengths $m$ such that $N_{p_1}(E)=N_{p_2}(E)=\cdots =N_{p_m}(E)$.

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arXiv2025

On surfaces satisfying $q=0,p_g=0,c_1^2=9$

I consider the class of surfaces $X$ over algebraically closed fields with numerical invariants given in the title. In characteristic zero, this class contains fake projective planes which were introduced by David Mumford. I prove that in characteristic $p>0$ such surfaces are Hodge-Witt and also ordinary under additional assumptions. In particular, fake projective planes are Hodge-Witt (Theorem 3.1). I show that if $X$ is Frobenius split then $X\simeq \mathbb{P}^2$ (Theorem 4.1). I also establish a characteristic free characterization of the projective plane using the Nori fundamental group scheme (Theorem 5.1). Finally, I show that any fake projective plane over a number field has good ordinary reduction at all but finitely many primes and in particular fake projective planes exist in positive characteristics (Theorem 6.1).

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arXiv2025

On the analytic rank of the twin prime elliptic curve $y^2=x(x-2)(x-p)$

Let $p\geq 7$ and suppose $(p,p-2)$ are twin prime numbers, in [Hatley, 2009], the elliptic curve $E_p:y^2=x(x-2)(x-p)$ was considered in the context of a conjecture by Jason Beers about the Mordell-Weil ranks of $E_p/\mathbb{Q}$. I show that for $p\equiv 3,5\bmod 8$, the analytic rank of $E_p$ is at least one (Theorem 1.1.2) in line with Beers' predictions. This is done by finding a formula (Theorem 4.1.1) for the global root number of $E_p$ for all twin prime pairs. I also show that Beers' conjecture, that for $p\equiv 1\bmod 8$ the rank of $E_p$ is two, is false as stated because $E_{73}$ has rank zero. In the light of Theorem 4.1.1, Beers' conjecture needs to be modified: if $p\equiv 1\bmod 8$ then the rank of $E_p$ is zero or two (Conjecture 5.3.1).

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arXiv2025

On determinantal equations for curves and Frobenius split hypersurfaces

I consider the problem of existence of intrinsic determinantal equations for plane projective curves and hypersurfaces in projective space and prove that in many cases of interest there exist intrinsic determinantal equations. In particular I prove (1) in characteristic two any ordinary, plane projective curve of genus at least one is given by an intrinsic determinantal equation (2) in characteristic three any plane projective curve is an intrinsic Pfaffian (3) in any positive characteristic any plane projective curve is set theoretically the determinant of an intrinsic matrix (4) in any positive characteristic, any Frobenius split hypersurface in ${\bf P}^n$ is given by set theoretically as the determinant of an intrinsic matrix with homogeneous entries of degree between $1$ and $n-1$. In particular this implies that any smooth, Fano hypersurface is set theoretically given by an intrinsic determinantal equation and the same is also true for any Frobenius split Calabi-Yau hypersurface.

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arXiv2025

On Belyi's Theorem in positive characteristic

The famous theorem of Belyi can be viewed as a characterization of compact Riemann surfaces which admit a non-empty open subset uniformized by a subgroup of $SL_2(\mathbb{Z})$ of finite index. I show that if $q\geq 5$, then ${\bf F}_q(T)$ is the one and only function field of positive characteristic for which such an analogous characterization of rigid analytic spaces of dimension one can exist and that Drinfel$'$d modular curves provide examples of rigid analytic spaces of this type.

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arXiv2025

Construction of Arithmetic Teichmuller Spaces I

In this paper after proving (in Section 2) the Berkovich analytic space analog of the familiar fact that there exist many non-isomorphic Riemann surfaces of the fixed topological type, I introduce the precise notion of Arithmetic Holomorphic Structures. This leads, for a fixed geometrically connected, smooth quasi-projective variety $X/E$ over a $p$-adic field, to the construction of a category which can be called Arithmetic Teichmuller Space of $X/E$. After establishing the properties of this local i.e. $p$-adic Arithmetic Teichmuller Space, I proceed to the global (adelic) construction, for a geometrically connected, smooth quasi-projective variety $X/L$ over a number field $L$, of the Adelic Arithmetic Teichmuller Space of $X/L$. A fixed number field itself has an Arithmetic Teichmuller Space--this is detailed in Constructions II(1/2) paper in this series of papers. All of these constructions extend the analogy between Number fields and Riemann surfaces and are inspired by (and directly related to) Shinichi Mochizuki's ideas on Inter-Universal Teichmuller Theory and his work on the abc-conjecture. But my approach is based on a completely different set of ideas.

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arXiv2025

Construction of Arithmetic Teichmuller spaces II: Proof of a local prototype of Mochizuki's Corollary 3.12

This paper deals with consequences of the existence of Arithmetic Teichmuller spaces established in arXiv:2106.11452 and arXiv:2210.11635. Theorem~9.2.1 provides a proof of a local version of Mochizuki's Corollary~3.12. Local means for a fixed $p$-adic field. There are several new innovations in this paper. Some of the main results are as follows. Theorem~3.5.1 shows that one can view the Tate parameter of Tate elliptic curve as a function on the arithmetic Teichmuller space of [Joshi, 2021a], [Joshi, 2022b]. The next important point is the construction of Mochizuki's $Θ_{gau}$-links and the set of such links, called Mochizuki's Ansatz in §6. Theorem~6.9.1 establishes valuation scaling property satisfied by points of Mochizuki's Ansatz (i.e. by my version of $Θ_{gau}$-links). These results lead to the construction of a theta-values set (§8) which is similar to Mochizuki's Theta-values set (differences between the two are in §8.7.1). Finally Theorem~9.2.1 is established. For completeness, I provide an intrinsic proof of the existence of Mochizuki's $\log$-links (Theorem 10.9.1), $\mathfrak{log}$-links (Theorem~10.15.1) and Mochizuki's log-Kummer Indeterminacy (Theorem~10.20.1) in my theory.

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arXiv2025

Construction of Arithmetic Teichmuller Spaces II$\frac{1}{2}$: Deformations of Number Fields

This paper lays the foundation of the Theory of Arithmetic Teichmuller Spaces of Number Fields by explicitly constructing many arithmetically inequivalent avatars of a fixed number field. This paper also constructs a topological space of such avatars and describes its symmetries. Notably amongst these symmetries is a global Frobenius morphism which changes the avatar of the number field! The existence of such avatars has been suggested and used by Shinichi Mochizuki in his work on the arithmetic Vojta and Szpiro conjectures. Important to the global aspect of this theory is the fact that the product formula for a number field defines an arithmetic period mapping (Section 5.9). The key advantage of my approach is that one can quantify the difference between two inequivalent avatars and this renders my theory fundamentally and quantitatively more precise than Mochizuki's approach. In the appendix, I provide a discussion of the proofs of the geometric Szpiro Conjectures due to [Bogomolov et. al. 2000] and [Zhang 2001] from the point of view of this paper. I also discuss applications of my theory to the theory of arithmetic loops and arithmetic knots.

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arXiv2025

Construction of Arithmetic Teichmuller Spaces III: A `Rosetta Stone' and a proof of Mochizuki's Corollary 3.12

This is a continuation of my work on Arithmetic Teichmuller Spaces (arXiv:2106.11452, arXiv:2210.11635, arXiv:2303.01662, arXiv:2305.10398). This paper establishes a number of important results including (1) a proof Mochizuki's Corollary 3.12 (2) establishes a `Rosetta Stone' for a parallel reading of Mochizuki's Inter-Universal Teichmuller Theory and my Theory of Arithmetic Teichmuller Spaces, and (3) a proof that Mochizuki's gluing of Hodge-Theaters, Frobenioids along prime-strips as described in his theory is naturally provided by the existence of Arithmetic Teichmuller Spaces. (4) Includes the geometric case of Mochizuki's Corollary 3.12 in §12.

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arXiv2024

Algebraization of Mochizuki's anabelian variation of ring structures, perfectoid geometry and formal groups

Let $M$ be a multiplicative monoid with identity. Then I show that there is a universal one dimensional formal group law equipped with an action of $M$. If $M$ is $p$-perfect (i.e. $m\mapsto m^p$ is an isomorphism for some prime number $p$) then the universal $M$-formal group law comes equipped with a natural Frobenius endomorphism. There are a number of concrete applications of this result. If $K$ is a $p$-adic field and $\mathcal{O}=\mathcal{O}_K$ is the multiplicative monoid of the ring of integers of $K$, then there is a universal formal group (over a suitable (non-zero) ring) which is equipped with an action of the multiplicative monoid $\mathcal{O}$. Lubin-Tate formal groups arise from this universal monoid formal group law. This has applications to Mochizuki's anabelian ideas: if two p-adic fields have isomorphic absolute Galois groups then they have isomorphic multiplicative monoids $\mathcal{O}$ (but possibly non-isomorphic ring structures). The existence of the universal monoid formal group law for the monoid $\mathcal{O}$ implies that the additive structures of a ring can be interpolated into a universal algebraic family (while keeping the multiplicative structure of the ring fixed). Here is another important example covered by my result: let $R$ be a perfectoid ring and let $R^\flat$ be its tilt and the multiplicative monoid $R^\flat$ of $R^\flat$. Then there exists a universal monoid formal group law for this monoid which interpolates the additive structures of untilts with tilt $R^\flat$. Thus in some sense one has a unified approach to various phenomenon which are well-known in anabelian geometry and in perfectoid geometry. These results also provide a natural number field version of Fontaine's fundamental ring $A_{inf}$ of $p$-adic Hodge Theory (Section 4.3).

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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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