Kirti N Joshi

Interests

Research

Algebraic Geometry and Number Theory

Courses

Scholarly Contributions

Journals/Publications

• Joshi, K. (2013). The t-motivic mixed Carlitz zeta category and Carlitz-Thakur multi-zeta values. arxiv.
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  We construct the t-motivic mixed Carlitz zeta category over $\F_q(t)$ and show that it contains all the (mixed) t-motives with Carlitz-Thakur multi-zeta values as periods constructed by Anderson and Thakur. Our construction is canonical and our category is Tannakian and neutral and every object is equipped with a weight filtration whose graded pieces are Carlitz motives over $\F_q(t)$. For any finite separable extension L/\F_q(t) we show that existence of a similar category over $L$ is a consequence of a version of a conjecture of L. Taelman. Along the way we also prove the existence of the category of mixed $t$-motives and the category of mixed Carlitz motives over any $L$ (these two existence results are independent of any conjectures). [Journal_ref: ]
• Joshi, K. (2014). Some remarks on Hodge symmetry. arxiv.
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  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 $X$. [Journal_ref: ]
• Joshi, K. (2018). A method for construction of rational points over elliptic curves II: Points over solvable extensions. arxiv.
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  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. [Journal_ref: ]
• Joshi, K. (2020). The Absolute Grothendieck Conjecture is false for Fargues-Fontaine Curves. arxiv.
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  I prove that the absolute Grothendieck Conjecture is false for Fargues-Fontaine curves. [Journal_ref: ]
• Joshi, K. (2020). Untilts of fundamental groups: construction of labeled isomorphs of fundamental groups. arxiv.
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  I show that one can explicitly construct topologically/geometrically distinguishable data which provide isomorphic copies (i.e. \emph{isomorphs}) of the tempered fundamental group of a geometrically connected, smooth, quasi-projective variety over $p$-adic fields. [Journal_ref: ]
• Joshi, K. (2022). Untilts of fundamental groups: construction of labeled isomorphs of fundamental groups -- Arithmetic Holomorphic Structures. arxiv.
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  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 \'etale) fundamental group of $X/E$ which are labeled by distinct arithmetic holomorphic structures, just as isomorphs of the fundamental group of a Riemann surface $\Sigma$ may be labeled by Riemann surfaces (i.e. complex holomorphic structures) $\Sigma'$ in the Teichmuller space of $\Sigma$. 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. [Journal_ref: ]
• Joshi, K. (2023). Construction of Arithmetic Teichmuller Spaces II(1/2): Deformations of Number Fields. arxiv.
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  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. 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. [Journal_ref: ]
• Joshi, K. (2024). Construction of Arithmetic Teichmuller Spaces III: A `Rosetta Stone' and a proof of Mochizuki's Corollary 3.12. arxiv.
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  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. [Journal_ref: ]
• Joshi, K. (2024). Construction of Arithmetic Teichmuller Spaces IV: Proof of the abc-conjecture. arxiv.
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  This is a continuation of my work on Arithmetic Teichmuller Spaces developed in the present series of papers. In this paper, I show that the Theory of Arithmetic Teichmuller Spaces leads, using Shinichi Mochizuki's rubric, to a proof of the $abc$-conjecture (as asserted by Mochizuki). [Journal_ref: ]
• Joshi, K., & Joshi, K. (2023). Construction of Arithmetic Teichmuller spaces II: Proof of a local prototype of Mochizuki's Corollary 3.12. arxiv.
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  This paper deals with consequences of the existence of Arithmetic Teichmuller spaces established in https://arxiv.org/abs/2106.11452 and https://arxiv.org/abs/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 $\Theta_{gau}$-links and the set of such links, called Mochizuki's Ansatz in {\S}6. Theorem~6.9.1 establishes valuation scaling property satisfied by points of Mochizuki's Ansatz (i.e. by my version of $\Theta_{gau}$-links). These results lead to the construction of a theta-values set ({\S}8) which is similar to Mochizuki's Theta-values set (differences between the two are in {\S}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. [Journal_ref: ]
• Joshi, K. (2022).

  Corrigendum to “On the construction of weakly Ulrich bundles” [Adv. Math. 381 (2021) 107598]

  . Advances in Mathematics, 394, 108025. doi:10.1016/j.aim.2021.108025
• Joshi, K. (2022). Comments on Arithmetic Teichmuller Spaces. arxiv, 12.
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  I provide some comments on Arithmetic Teichmuller Spaces constructed in my paper arXiv:2106.11452. [Journal_ref: ]
• Joshi, K. (2021).

  On varieties with trivial tangent bundles in characteristic p>0.

  . Nagoya Mathematical Journal, 242, 35-51. doi:10.1017/nmj.2019.19
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  In this article, I give a crystalline characterization of abelian varieties amongst the class of smooth projective varieties with trivial tangent bundles in characteristic $p>0$. Using my characterization, I show 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 KeZheng Li about smooth projective varieties with trivial tangent bundles in characteristic $p>0$ is true for smooth projective surfaces. I give a new proof of a result by 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.
• Joshi, K. (2021). On the construction of weakly Ulrich bundles. Advances in Mathematics, 381, 107598. doi:https://doi.org/10.1016/j.aim.2021.107598
• Joshi, K. (2020). Crystalline aspects of geography of low dimensional varieties I: numerology. European Journal of Mathematics, 6(4), 1111-1175.
• Joshi, K., & Pauly, C. (2020). Opers of higher types, Quot-schemes and Frobenius instability loci. Épijournal de Géométrie Algébrique, Volume 4. doi:https://doi.org/10.46298/epiga.2020.volume4.5721
• Joshi, K. (2019). A remark on Ulrich and ACM bundles. Journal of Algebra, 527, 20-29.
• Joshi, K. N. (2018). On the Grothendieck ring of varieties in positive characteristic. arxiv.
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  Let $k$ be an algebraically closed field of characteristic $p>0$. I prove that the ring of smooth, complete $k$-varieties and Bittner relations contains zero divisors if $p>13$ or $p=11$. In particular it follows, under the same hypothesis, that the isomorphism class of any supersingular elliptic curve is a zero divisor in this ring. [Journal_ref: ]
• Joshi, K. N. (2019). On determinantal equations for curves and Frobenius split hypersurfaces. arxiv.
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  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. [Journal_ref: ]
• Joshi, K. N. (2019). On varieties with trivial tangent bundle. Nagoya Math. Journal. doi:DOI: https://doi.org/10.1017/nmj.2019.19
• Joshi, K. N., & Zhang, Y. (2018). Eigenform product identities for Hilbert modular forms. Mathematische Zeitschrift, 18. doi:10.1007/s00209-018-2214-y
• Joshi, K. N. (2018). A question about Belyi's Theorem. arxiv, 6.
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  I discuss a natural version of Belyi's Theorem over $\fq(T)$ and prove that the situation I describe is unique and rigid for $q\geq5$ (in the sense described below). [Journal_ref: ]
• Joshi, K. N. (2017). A method for construction of rational points over elliptic curves. arxiv.
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  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. [Journal_ref: ]
• Joshi, K. N. (2017). Methods for constructing elliptic and hyperelliptic curves with rational points. arxiv.
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  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 $M_{24}$. [Journal_ref: ]
• Joshi, K. N. (2017). On the equation $N_{p_1}(E)\cdot N_{p_2}(E)\cdots N_{p_k}(E)=n$. arxiv.
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  For a given elliptic curve $E/\mathbb{Q}$, set $N_p(E)$ to 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 pairwise distinct primes $p_1,\ldots,p_k$ of good reduction for $E$, for which 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 also conjecture that this result also holds for $k=1$ and $k=2$. In particular for $k=1$ this conjecture says that there are "elliptic progressions of primes" i.e. sequences of primes $p_1
• Joshi, K. N. (2017). The Degree of the Dormant Operatic Locus. International Mathematics Research Notices, 2017(9), 2599–2613. doi:https://doi.org/10.1093/imrn/rnw066
• Joshi, K. N. (2016). On Primes of Ordinary and Hodge-Witt Reduction. arxiv.
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  Jean-Pierre Serre has conjectured, in the context of abelian varieties, that there are infinitely primes of good ordinary reduction for a smooth, projective variety over a number field. We consider this conjecture and its natural variants. In particular we have conjectured (with C. S. Rajan) the existence of infinitely many primes of Hodge-Witt reduction (any prime of ordinary reduction is also a prime of Hodge-Witt reduction). The two conjectures are not equivalent but are related. We prove a precise relationship between the two; we prove several results which provide some evidence for these conjectures; we show that primes of ordinary and Hodge-Witt reduction can have different densities. We prove our conjecture on Hodge- Witt and ordinary reduction for abelian varieties with complex multiplication. We include here an unpublished joint result with C. S. Rajan (also independently established by Fedor Bogomolov and Yuri Zarhin by a different method) on the existence of primes of ordinary reductions for K3 surfaces; our proof also shows that for an abelian threefold over a number field there is a set of primes of positive density at which it has Hodge-Witt reduction (this is also a joint result with C. S. Rajan). We give a number of examples including those of Fermat hypersurfaces for which all the conjectures we make hold. [Journal_ref: ]
• Joshi, K. N. (2016). Ordinary reductions of abelian varieties. arxiv, 13.
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  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). [Journal_ref: ]
• Joshi, K. N., & Pauly, C. (2015). Hitchin–Mochizuki morphism, opers and Frobenius-destabilized vector bundles over curves. Advances in Mathematics, 274, 39--75 (38 pages).
• Joshi, K., & Petrov, A. (2014). On the action of Hecke operators on Drinfeld modular forms. Journal of Number Theory, 137, 186-200.
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  Abstract:In this paper we determine the explicit structure of the semisimple part of the Hecke algebra that acts on Drinfeld modular forms of full level modulo T. We show that modulo T the Hecke algebra has a non-zero semisimple part. In contrast, a well-known theorem of Serre asserts that for classical modular forms the action of Tℓ for any odd prime ℓ is nilpotent modulo 2. After proving the result for Drinfeld modular forms modulo T, we use computations of the Hecke action modulo T to show that certain powers of the Drinfeld modular form h cannot be eigenforms. Finally, we pose a question a positive answer to which will mean that the Hecke algebra that acts on Drinfeld modular forms of full level is not smooth for large weights, which again contrasts the classical situation. © 2014 Elsevier Inc.
• Joshi, K. (2012). Remarks on the Fourier coefficients of modular forms. Journal of Number Theory, 132(6), 1314-1336. doi:https://doi.org/10.1016/j.jnt.2011.10.004
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  Abstract:We consider a variant of a question of N. Koblitz. For an elliptic curve E/Q which is not Q-isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes p such that Np(E)=#E(Fp)=p+1-ap(E) is also a prime. We consider a variant of this question. For a newform f, without CM, of weight k≥4, on Γ 0(M) with trivial Nebentypus χ 0 and with integer Fourier coefficients, let N p(f)=χ 0(p)p k-1+1-a p(f) (here a p(f) is the p-th-Fourier coefficient of f). We show under GRH and Artin's Holomorphy Conjecture that there are infinitely many p such that N p(f) has at most [5k+1+log(k)] distinct prime factors. We give examples of about hundred forms to which our theorem applies. We also show, on GRH, that the number of distinct prime factors of N p(f) is of normal order log(log(p)) and that the distribution of these values is asymptotically a Gaussian distribution ("Erdo{double acute}s-Kac type theorem"). © 2012 Elsevier Inc..
• Joshi, K. (2011).

  Ordinarity of configuration spaces and of wonderful compactifications

  . Documenta Mathematica, 16, 669-676. doi:10.4171/dm/347
• Joshi, K., & Mcleman, C. (2011).

  Infinite Hilbert Class Field Towers from Galois Representations

  . International Journal of Number Theory, 07(01), 1-8. doi:10.1142/s1793042111003879
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  We investigate class field towers of number fields obtained as fixed fields of modular representations of the absolute Galois group of the rational numbers. First, for each k ∈ {12, 16, 18, 20, 22, 26}, we give explicit rational primes l such that the fixed field of the mod-l representation attached to the unique normalized cusp eigenform of weight k on SL2(ℤ) has an infinite class field tower. Further, under a conjecture of Hardy and Littlewood, we prove the existence of infinitely many cyclotomic fields of prime conductor, providing infinitely many such primes l for each k in the list. Finally, given a non-CM curve E/ℚ, we show that there exists an integer ME such that the fixed field of the representation attached to the n-division points of E has an infinite class field tower for a set of integers n of density one among integers coprime to ME.
• Joshi, K. (2010).

  Musings on $\Q(1/4)$: Arithmetic spin structures on elliptic curves

  . Mathematical Research Letters, 17(6), 1013-1028. doi:10.4310/mrl.2010.v17.n6.a1
• Joshi, K., & Mehta, V. B. (2009). Vectors bundles with theta divisors I-bundles on castelnuovo curves. Archiv der Mathematik, 92(6), 574-584.
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  Abstract:In this paper we show that semistable vector bundles on a Castelnuovo curve of genus g ≥ 2 have theta divisors. As a corollary, we deduce that semistable vector bundles on a smooth, general curve of genus g ≥ 2 which extend to semistable vector bundles on any flat Castelnuovo degeneration of the general curve admit a theta divisor. © 2009 Birkhäuser Verlag Basel/Switzerland.
• Joshi, K. (2008).

  Two Remarks On Subvarieties Of Moduli Spaces

  . International Journal of Mathematics, 19(02), 237-243. doi:10.1142/s0129167x08004649
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  We study two natural questions about subvarieties of moduli spaces. In the first section, we study the locus of curves equipped with F-nilpotent bundles and its relationship to the p-rank zero locus of the moduli space of curves of genus g. In the second section, we study subvarieties of moduli spaces of vector bundles on curves. We prove an analogue of a result of F. Oort about proper subvarieties of moduli of abelian varieties.
• Joshi, K. (2007). Exotic torsion, frobenius splitting and the slope spectral sequence. Canadian Mathematical Bulletin, 50(4), 567-578.
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  Abstract: In this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic p > 0 is Hodge-Witt. This is proved by generalizing to the case of threefolds a well-known criterion due to N. Nygaard for surfaces to be Hodge-Witt. We also show that the second crystalline cohomology of any smooth, projective Frobenius split variety does not have any exotic torsion. In the last two sections we include some applications. ©Canadian Mathematical Society 2007.
• Joshi, K., Ramanan, S., Xia, E. Z., & Yu, J. (2006). On vector bundles destabilized by Frobenius pull-back. Compositio Mathematica, 142(3), 616-630.
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  Abstract: Let X be a smooth projective curve of genus g > 1 over an algebraically closed field of positive characteristic. This paper is a study of a natural stratification, defined by the absolute Frobenius morphism of X, on the moduli space of vector bundles. In characteristic two, there is a complete classification of semi-stable bundles of rank 2 which are destabilized by Frobenius pull-back. We also show that these strata are irreducible and obtain their respective dimensions. In particular, the dimension of the locus of bundles of rank two which are destabilized by Frobenius is 3g-4. These Frobenius destabilized bundles also exist in characteristics two, three and five with ranks 4, 3 and 5, respectively. Finally, there is a connection between (pre)-opers and Frobenius destabilized bundles. This allows an interpretation of some of the above results in terms of pre-opers and provides a mechanism for constructing Frobenius destabilized bundles in large characteristics. © Foundation Compositio Mathematica 2006.
• Joshi, K., & Raghunathan, R. (2005). Infinite product identities for l-functions. Illinois Journal of Mathematics, 49(3), 885-891.
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  Abstract: We establish certain infinite product identities for Dirichlet series twisted by Dirichlet characters and give examples where the products have meromorphic continuation to the whole complex plane. ©2005 University of Illinois.
• Joshi, K. (2004). A p-adic proof of Hodge symmetry for threefolds. Comptes Rendus Mathematique, 338(10), 781-786.
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  Abstract: We give a p-adic proof of Hodge symmetry for smooth and projective varieties of dimension three over the field of complex numbers. © 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.
• Joshi, K. (2004). Stability and locally exact differentials on a curve. Comptes Rendus Mathematique, 338(11), 869-872.
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  Abstract: We show that the locally free sheaf B1⊂F*(Ω1X) of locally exact differentials on a smooth projective curve of genus g≥2 over an algebraically closed field k of characteristic p is a stable bundle. This answers a question of Raynaud. © 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.
• Joshi, K., & Rajan, C. S. (2003). Frobenius splitting and ordinarity. International Mathematics Research Notices, 109-121.
• Gordon, B. B., & Joshi, K. (2002). Griffiths groups of supersingular abelian varieties. Canadian Mathematical Bulletin, 45(2), 213-219.
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  Abstract: The Griffiths group Grr(X) of a smooth projective variety X over an algebraically dosed field is defined to be the group of homologically trivial algebraic cycles of codimension r on X modulo the subgroup of algebraically trivial algebraic cycles. The main result of this paper is that the Griffiths group Gr2 (Ak̄) of a supersingular abelian variety Ak̄ over the algebraic closure of a finite field of characteristic p is at most a p-primary torsion group. As a corollary the same conclusion holds for supersingular Fermat threefolds. In contrast, using methods of C. Schoen it is also shown that if the Tare conjecture is valid for all smooth projective surfaces and all finite extensions of the finite ground field k of characteristic p > 2, then the Griffiths group of any ordinary abelian threefold Ak̄ over the algebraic closure of k is non-trivial; in fact, for all but a finite number of primes ℓ ≠ p it is the case that Gr2 (Ak̄) ⊗ ℤℓ ≠ 0.
• Joshi, K., & Kim, M. (2002). A remark on potentially semi-stable representations of Hodge-Tate type (0,1). Mathematische Zeitschrift, 241(3), 479-483.
• Joshi, K. (2000). Kodaira-akizuki-nakano vanishing: A variant. Bulletin of the London Mathematical Society, 32(2), 171-176.
• Joshi, K., & Xia, E. Z. (2000). Moduli of Vector Bundles on Curves in Positive Characteristics. Compositio Mathematica, 122(3), 315-321.
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  Abstract: Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.
• Joshi, K. (1999). Remarks on Methods of Fontaine and Faltings. International Mathematics Research Notices, 1999(22), X1-1209.
• Joshi, K., & Tzermias, P. (1999). On the Coleman-Chabauty bound. Comptes Rendus de l'Academie des Sciences - Series I: Mathematics, 329(6), 459-463.
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  Abstract: The Coleman-Chabauty bound is an upper bound for the number of rational points on a curve of genus g ≥ 2 whose Jacobian has Mordell-Weil rank r less than g. The bound is given in terms of the genus of the curve and the number of Fp-points on the reduced curve, for all primes p of good reduction such that p > 2g. In this Note we show that the hypothesis on the Mordell-Weil rank is essential. We do so by exhibiting, for each prime p ≥ 5, an explicit family of curves of genus (p - 1) /2 (and rank at least (p - 1) /2) for which the bound in question does not hold. Our examples show that the difference between the number of rational points and the bound in question can in fact be linear in the genus. Under mild assumptions, our curves have rank at least twice their genus. © 1999 Académie des Sciences/Éditions scientifiques et médicales Elsevier SAS.
• Joshi, K., & Yogananda, C. S. (1999). A remark on product of Dirichlet L-functions. Acta Arithmetica, 91(4), 325-327.
• Joshi, K. (1996). A family of Étale coverings of the affine line. Journal of Number Theory, 59(2), 414-418.
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  Abstract: It this note we prove the following theorem. Let Πalg1(A1C) be the algebraic fundamental group of the affine line over C, where C is the completion of the algebraic closure of Fq((1/T)), and Fq is a field with q elements. If Fq has at least four elements, then we show that there is a continuous surjection Πalg1(A1C) → ←lim SL2(A/I)/{+ ± 1}, where A = Fq[T] and the inverse limit is over the family of non-zero, proper ideals of A. This result is proved by using the moduli of Drinfel'd A-modules of rank two over C with I-level structures; these curves give (tamely) ramified covers of the line and the tame ramification is removed using a variant of Abhyankar's lemma. © 1996 Academic Press, Inc.

Presentations

• Joshi, K. N. (2019, November). Galois Theory of Fields and Addition and Multiplication Laws. Colloquium Talk U Vermont. Math Department, University of Vermont: Math Department, University of Vermont.