Kirti N Joshi
 Associate Professor, Mathematics
 Member of the Graduate Faculty
Contact
 (520) 6212743
 Mathematics, Rm. 609A
 Tucson, AZ 85721
 kirti@math.arizona.edu
Bio
No activities entered.
Interests
No activities entered.
Courses
202122 Courses

Independent Study
MATH 599 (Fall 2021) 
Intro to Linear Algebra
MATH 313 (Fall 2021)
202021 Courses

Calculus II
MATH 129 (Fall 2020)
201920 Courses

Calculus II
MATH 129 (Spring 2020) 
Calculus II
MATH 129 (Fall 2019)
201819 Courses

Complex Analysis
MATH 520B (Spring 2019) 
Independent Study
MATH 399 (Spring 2019) 
Calculus II
MATH 129 (Fall 2018)
201718 Courses

Calculus II
MATH 129 (Fall 2017)
201617 Courses

Calculus II
MATH 129 (Spring 2017) 
Calculus II
MATH 129 (Fall 2016)
201516 Courses

Commutative Algebra
MATH 516 (Spring 2016)
Scholarly Contributions
Journals/Publications
 Joshi, K. N. (2019). A remark on Ulrich and ACM bundles. journal of algebra. doi:https://doi.org/10.1016/j.jalgebra.2019.01.016More infoI show that on any smooth, projective ordinary curve of genus at least twoand a projective embedding, there is a natural example of a stable Ulrichbundle for this embedding: namely the sheaf $B^1_X$ of locally exactdifferentials twisted by $\O_X(1)$ given by this embedding and in particularthere exist ordinary varieties of any dimension which carry Ulrich bundles. Inhigher dimensions, assuming $X$ is Frobenius split variety I show that $B^1_X$is an ACM bundle and if $X$ is also a CalabiYau 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 anACM bundle on any ordinary CalabiYau variety. I also prove a characterizationof 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).[Journal_ref: ]
 Joshi, K. N. (2019). On determinantal equations for curves and Frobenius split hypersurfaces.More infoI consider the problem of existence of intrinsic determinantal equations forplane projective curves and hypersurfaces in projective space and prove that inmany cases of interest there exist intrinsic determinantal equations. Inparticular I prove (1) in characteristic two any ordinary, plane projectivecurve 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 theoreticallythe determinant of an intrinsic matrix (4) in any positive characteristic, anyFrobenius split hypersurface in ${\bf P}^n$ is given by set theoretically asthe determinant of an intrinsic matrix with homogeneous entries of degreebetween $1$ and $n1$. In particular this implies that any smooth, Fanohypersurface is set theoretically given by an intrinsic determinantal equationand the same is also true for any Frobenius split CalabiYau hypersurface.[Journal_ref: ]
 Joshi, K. N. (2019). On the Grothendieck ring of varieties in positive characteristic.More infoLet $k$ be an algebraically closed field of characteristic $p>0$. I provethat the ring of smooth, complete $k$varieties and Bittner relations containszero divisors if $p>13$ or $p=11$. In particular it follows, under the samehypothesis, that the isomorphism class of any supersingular elliptic curve is azero divisor in this ring.[Journal_ref: ]
 Joshi, K. N. (2019). On the construction of Weakly Ulrich bundles.More infoI provide a construction of intrinsic weakly Ulrich bundles of large rank onany 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 alsoconstruct intrinsic weakly Ulrich bundles on any Frobenius split variety ofdimension at most three. The bundles constructed here are in fact ACM andweakly Ulrich bundles and so I call them almost Ulrich bundles. Frobenius splitvarieties in dimension three include as special cases: (1) smooth hypersurfacesin ${\bf P}^4$ of degree at most four, (2) Frobenius split, smooth quintics in${\bf P}^4$ (3) Frobenius split CalabiYau varieties of dimension at most three(4) Frobenius split (i.e ordinary) abelian varieties of dimension at mostthree.[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/s002090182214y
 Joshi, K. N. (2018). A method for construction of rational points over elliptic curves II: Points over solvable extensions.More infoI provide a systematic construction of points, defined over finite radicalextensions, on any Legendre curve over any field of characteristic not equaltwo. This includes as special case Douglas Ulmer's construction of rationalpoints over a rational function field in characteristic $p>0$. In particular Ishow that if $n\geq 4$ is any even integer and not divisible by thecharacteristic of the field then any elliptic curve $E$ over this field has atleast $2n$ rational points over a finite solvable field extension. Underadditional hypothesis, when the ground field is a number field, I show thatthese are of infinite order. I also show that Ulmer's points lift tocharacteristic zero and in particular to the canonical lifting.[Journal_ref: ]
 Joshi, K. N. (2018). A question about Belyi's Theorem.More infoI discuss a natural version of Belyi's Theorem over $\fq(T)$ and prove thatthe situation I describe is unique and rigid for $q\geq5$ (in the sensedescribed below).[Journal_ref: ]
 Joshi, K. N. (2017). A method for construction of rational points over elliptic curves.More infoI provide a systematic construction of points (defined over number fields) onLegendre elliptic curves over $\mathbb{Q}$: for any odd integer $n\geq 3$ mymethod constructs $n$ points on the Legendre curve and I show that rank of thesubgroup of the MordellWeil group they generate is $n$ if $n\geq 7$. I alsoshow that every elliptic curve over any number field admits similar type ofpoints after a finite base extension.[Journal_ref: ]
 Joshi, K. N. (2017). Methods for constructing elliptic and hyperelliptic curves with rational points.More infoI provide methods of constructing elliptic and hyperelliptic curves overglobal fields with interesting rational points over the given fields or overlarge field extensions. I also provide a elliptic curves defined over any givennumber field equipped with a rational point, (resp. with two rational points)of infinite order over the given number field, and elliptic curves over therationals with two rational points over `simplest cubic fields.' I also providehyperelliptic curves of genus exceeding any given number over any given numberfields with points (over the given number field) which span a subgroup of rankat least $g$ in the group of rational points of the Jacobian of this curve. Ialso provide a method of constructing hyperelliptic curves over rationalfunction fields with rational points defined over field extensions with largefinite simple Galois groups, such as the Mathieu group $M_{24}$.[Journal_ref: ]
 Joshi, K. N. (2017). On the equation $N_{p_1}(E)\cdot N_{p_2}(E)\cdots N_{p_k}(E)=n$.More infoFor a given elliptic curve $E/\mathbb{Q}$, set $N_p(E)$ to be the number ofpoints 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 titleholds, then on assuming the Generalized Riemann Hypothesis for elliptic curveswithout CM (and unconditionally if the curves have complex multiplication), Ishow 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 $k=2$. Inparticular for $k=1$ this conjecture says that there are "elliptic progressionsof 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 HodgeWitt Reduction.More infoJeanPierre Serre has conjectured, in the context of abelian varieties, thatthere are infinitely primes of good ordinary reduction for a smooth, projectivevariety over a number field. We consider this conjecture and its naturalvariants. In particular we have conjectured (with C. S. Rajan) the existence ofinfinitely many primes of HodgeWitt reduction (any prime of ordinary reductionis also a prime of HodgeWitt reduction). The two conjectures are notequivalent but are related. We prove a precise relationship between the two; weprove several results which provide some evidence for these conjectures; weshow that primes of ordinary and HodgeWitt reduction can have differentdensities. We prove our conjecture on Hodge Witt and ordinary reduction forabelian varieties with complex multiplication. We include here an unpublishedjoint result with C. S. Rajan (also independently established by FedorBogomolov and Yuri Zarhin by a different method) on the existence of primes ofordinary reductions for K3 surfaces; our proof also shows that for an abelianthreefold over a number field there is a set of primes of positive density atwhich it has HodgeWitt reduction (this is also a joint result with C. S.Rajan). We give a number of examples including those of Fermat hypersurfacesfor which all the conjectures we make hold.[Journal_ref: ]
 Joshi, K. N. (2016). Ordinary reductions of abelian varieties.More infoI show that a conjecture of JoshiRajan on primes of HodgeWitt reduction andin particular a conjecture of JeanPierre Serre on primes of good, ordinaryreduction for an abelian variety over a number field follows from a certainconjecture on Galois rep resentations which may perhaps be easier to prove(and I prove this conjecture for abelian compatible systems of a suitabletype). This reduction (to a conjecture about certain sys tems of Galoisrepresentations) is based on a new slope estimate for non HodgeWitt abelianvarieties. In particular for any abelian variety over a number field with atleast one prime of good ordinary or split toric reduction, I show that theconjecture of JoshiRajan and the conjecture of Serre on ordinary reductionscan be reduced to proving that a certain rational trace of Frobenius is in factan integer. The assertion that this trace is an integer is proved for abeliansystems of Galois representations (of suitable type).[Journal_ref: ]
 Joshi, K. N., & Pauly, C. (2015). Hitchin–Mochizuki morphism, opers and Frobeniusdestabilized vector bundles over curves. Advances in Mathematics, 274, 3975 (38 pages).
 Joshi, K., & Petrov, A. (2014). On the action of Hecke operators on Drinfeld modular forms. Journal of Number Theory, 137, 186200.More infoAbstract: 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 nonzero semisimple part. In contrast, a wellknown 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), 13141336.More infoAbstract: We consider a variant of a question of N. Koblitz. For an elliptic curve E/Q which is not Qisogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes p such that Np(E)=#E(Fp)=p+1ap(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 k1+1a p(f) (here a p(f) is the pthFourier 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}sKac type theorem"). © 2012 Elsevier Inc..
 Joshi, K., & McLeman, C. (2011). Infinite hilbert class field towers from galois representations. International Journal of Number Theory, 7(1), 18.More infoAbstract: 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 κ ∈ {12, 16, 18, 20, 22, 26}, we give explicit rational primes ℓ such that the fixed field of the modℓ representation attached to the unique normalized cusp eigenform of weight κ 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 κ in the list. Finally, given a nonCM curve E/ℚ, we show that there exists an integer ME such that the fixed field of the representation attached to the ndivision points of E has an infinite class field tower for a set of integers n of density one among integers coprime to ME. © 2011 World Scientific Publishing Company.
 Joshi, K. (2010). Musings on Q(1=4): Arithmetic spin structures on elliptic curves. Mathematical Research Letters, 17(6), 10131028.More infoAbstract: We introduce and study arithmetic spin structures on elliptic curves. We show that there is a unique isogeny class of elliptic curves over F p2 which carries a unique arithmetic spin structure and provides a geometric object of weight 1=2 in the sense of Deligne and Grothendieck. This object is thus a candidate for ℚ(1=4). © International Press 2010.
 Joshi, K., & Mehta, V. B. (2009). Vectors bundles with theta divisors Ibundles on castelnuovo curves. Archiv der Mathematik, 92(6), 574584.More infoAbstract: 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(2), 237243.More infoAbstract: We study two natural questions about subvarieties of moduli spaces. In the first section, we study the locus of curves equipped with Fnilpotent bundles and its relationship to the prank 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. © 2008 World Scientific Publishing Company.
 Joshi, K. (2007). Exotic torsion, frobenius splitting and the slope spectral sequence. Canadian Mathematical Bulletin, 50(4), 567578.More infoAbstract: In this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic p > 0 is HodgeWitt. This is proved by generalizing to the case of threefolds a wellknown criterion due to N. Nygaard for surfaces to be HodgeWitt. 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 pullback. Compositio Mathematica, 142(3), 616630.More infoAbstract: 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 semistable bundles of rank 2 which are destabilized by Frobenius pullback. 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 3g4. 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 preopers 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 lfunctions. Illinois Journal of Mathematics, 49(3), 885891.More infoAbstract: 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 padic proof of Hodge symmetry for threefolds. Comptes Rendus Mathematique, 338(10), 781786.More infoAbstract: We give a padic 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), 869872.More infoAbstract: 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, 109121.
 Gordon, B. B., & Joshi, K. (2002). Griffiths groups of supersingular abelian varieties. Canadian Mathematical Bulletin, 45(2), 213219.More infoAbstract: 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 pprimary 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 nontrivial; 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 semistable representations of HodgeTate type (0,1). Mathematische Zeitschrift, 241(3), 479483.
 Joshi, K. (2000). Kodairaakizukinakano vanishing: A variant. Bulletin of the London Mathematical Society, 32(2), 171176.
 Joshi, K., & Xia, E. Z. (2000). Moduli of Vector Bundles on Curves in Positive Characteristics. Compositio Mathematica, 122(3), 315321.More infoAbstract: 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 rank2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semistable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pullback is not semistable. 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), X11209.
 Joshi, K., & Tzermias, P. (1999). On the ColemanChabauty bound. Comptes Rendus de l'Academie des Sciences  Series I: Mathematics, 329(6), 459463.More infoAbstract: The ColemanChabauty bound is an upper bound for the number of rational points on a curve of genus g ≥ 2 whose Jacobian has MordellWeil rank r less than g. The bound is given in terms of the genus of the curve and the number of Fppoints 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 MordellWeil 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 Lfunctions. Acta Arithmetica, 91(4), 325327.
 Joshi, K. (1996). A family of Étale coverings of the affine line. Journal of Number Theory, 59(2), 414418.More infoAbstract: 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 nonzero, proper ideals of A. This result is proved by using the moduli of Drinfel'd Amodules of rank two over C with Ilevel 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. Math Department, University of Vermont: Math Department, University of Vermont.