Eknath Ghate's Papers
My papers are available below. A complete list of publications is
here.
Special values of twisted tensor L-functions
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A generalization of the main result of the above paper to CM fields
is here.
This article appeared in the Shimura honorary volume:
Proc. Symp. Pure Math., vol 66, part 1, Amer. Math. Soc. (1999), 87-109.
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A mostly expository article with Ravitheja Vangala on Panchishkin's construction
of the p-adic
Rankin product L-function using the method of abstract Kummer congruences, in which we correct a sign
error. To appear in the Ramanujan Math Society Lecture Notes Series (2019).
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In the paper with Baskar Balasubramanyam and
Ravitheja Vangala, we prove an algebraicity result for the twisted Asai L-values, using a variation
of the method used in the Duke paper above. We then outline the main steps needed to construct
the corresponding p-adic Asai L-function, using the method of abstract Kummer congruences discussed
in the previous paper. To appear in a Springer Proceedings (2019).
Congruences for modular forms
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In Dihedral
congruence primes and class fields of real quadratic fields
(J.
Number Theory 95 (2002), no. 1, 14-37), Alex Brown and I show
that
dihedral congruence primes for elliptic cusp forms of quadratic
nebentypus
can be characterized in terms of simple expressions involving
fundamental
units of real quadratic fields. We apply our results to explicitly
generate
(ray) class fields of real quadratic fields by torsion points on
modular
abelian varieties.
Modular endomorphism algebras
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In Endomorphism
algebras of motives attached to elliptic modular forms (Ann.
Inst. Fourier, Grenoble 53 (2003), no. 6, 1615-1676) Alex Brown and I
study
the endomorphism algebra X_f of the motive attached to a non-CM
elliptic
modular form f, of weight k at least two. We show that if k > 2 then
X_f
contains a certain crossed product algebra X defined over a number
field.
If k = 2 then Ribet and Momose have shown that X_f is isomorphic to X.
We also investigate the Brauer class of X for all weights k at least 2.
Here are some slides
of a talk given at the AMS-India meeting in Bangalore in 2003 about
this
work.
There is a program that we wrote which illustrates some of
the theorems proved in the paper above. The code builds on the
1999
C++ version of the modular symbols engine HECKE written by W. Stein.
Our
program can be run locally by
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logging on to the machine homotopy, and,
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typing the command: ~eghate/public_html/endohecke
The program asks for input a Galois conjugacy class of an elliptic
modular cusp form of arbitrary weight and level and real nebentypus
(the
nebentypus is specified by typing in a string of 1's and 2's). As
output
the program prints the (norm to Q) of the Brauer class of X by
specifying
a list of finite primes where it is ramified. As for the infinite
places
recall that a result of Momose says that the algebra is ramified at an
infinite place if and only if the weight of the form is odd.
- In the paper On
the Brauer class of modular endomorphism algebras (Int.
Math. Res. Not. 12 (2005), 701-723.) we show X_f is isomorphic to X
in weights
k > 2, and sharpen some of our results concerning the Brauer class
of X
contained in the paper above. We show that in many cases that the
Brauer
class is locally at p determined by the p-adic valuations of the p-th
Fourier
coefficients (slopes) of the form. This is joint work with Enrique
Gonzalez-Jimenez
and Jordi Quer.
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In
Adjoint lifts
and modular endomorphism algebras (joint with Debargha
Banerjee, Israel J. Math. 195 (2013), 507-543) we use the Gelbart-Jacquet adjoint lift to specify the Brauer class of X, especially at the primes of bad reduction.
This paper completely determines the Brauer class in weights k > 1, under a finiteness
hypothesis on the slopes of the adjoint lift. This work grew out of conjectures
discussed in Roscoff in 2009 (slides).
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The case of supercuspidal primes is treated in
Supercuspidal ramfication of modular endomorphism algebras. This is
joint work with Shalini Bhattacharya (Proc. Amer. Math. Soc. 143 (2015), no. 11, 4669-4684). Since the slope is not finite at supercuspidal
primes, we describe the ramification in terms of an auxiliary Fourier coefficient.
Splitting of Galois representations
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In the paper On
the local behaviour of ordinary modular Galois representations
(Progress
in Mathematics 224, Birkhauser-Verlag (2004), 105-124) we investigate
a question of Greenberg concerning the spitting behaviour of the
restriction to a decomposition group at a prime p of the p-adic Galois
representation attached to a p-ordinary elliptic modular cusp form.
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In a more recent paper On
the local behaviour of ordinary $\Lambda$-adic representations
(Ann. Inst. Fourier, Grenoble 54 (2004), no. 7, 2143-2162) Vinayak Vatsal and
I study
the weight 1 specializations of families of p-ordinary forms. As a
result
of this study, we show that for all but finitely many specializations
of
weight 2 or larger, the local Galois representation is split if and
only
if the form has complex multiplication (we assume p is odd and work
under
some technical conditions on the mod p representation). We also treat
the
corresponding question for ordinary Lambda-adic forms.
- Ordinary forms and their local Galois
representations is an exposition of some
related issues (this paper appeared in the proceedings of a conference held in Hyderabad:
Algebra and number theory,
Hindustan Book Agency (2005), 226-242;
some slides of the talk are here).
Among other things we show that, for p > 3 and under similar
technical
conditions, the local splitting of an ordinary modular characteristic 0
Galois representation is related to whether the corresponding form is
in
the image of a suitable power of the theta derivation. We deduce some
information
towards a question of Coleman regarding the existence of non-CM forms
in
this image.
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In Locally indecomposable Galois representations (Canad.
J. Math. 63 (2011), no. 2, 277-297) Vinayak Vatsal and I give
examples of non-CM families for which every arithmetic member has
a locally non-split Galois representation. The residual representations
in the examples we can treat fully have solvable image. The proofs
use the deformation theory of Galois representations.
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The case p = 2 is investigated in the paper Control
theorems for ordinary $2$-adic families of modular forms (joint work with
Narasimha Kumar). Along the way, we develop Hida theory for the prime p = 2 and
prove a control theorem for the ordinary Lambda-adic Hecke algebra in the 2-adic setting.
This paper appeared in the Proceedings of the International Colloquium on
Automorphic Representations and L-functions held at TIFR in 2012
(Automorphic representations and L-functions, 231-261, Tata Inst. Fundam. Res. Stud. Math., 22,
Tata Inst. Fund. Res., Mumbai, 2013).
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The case of totally real fields is treated in On local
Galois representations attached to ordinary Hilbert modular forms
(joint with B. Balasubramanyam and V. Vatsal). We extend the main result of the
AIF (2004) paper above, proving that a p-ordinary Lambda-adic Hilbert modular Galois
representation is locally split at all primes above p if and only if the underlying
primitive family is of CM type. We work under the same technical conditions as in that paper,
assuming in addition that the prime p splits completely in the totally real field.
The corresponding result for classical Hilbert cuspforms follows for a
Zariski dense subset of arithmetic specializations. (Manuscripta Math. 142 (2013), no. 3-4, 513-524).
Filtered modules and Galois representations
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In Filtered
modules with coefficients (Trans. Amer. Math. Soc. 361 (2009), 2243-2261)
Ariane Mezard and I write down
some rank two admissible filtered modules with coefficents, concentrating
on the new features that arise when the coefficients are not
necessarily Q_p. In particular we write down explict Galois stable lines
which are candidates for the filtration.
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In (p,p)-Galois
representations attached to automorphic forms on GL_n (see Pacific J. Math. 252 (2011), no. 2, 379-406 for an abridged version)
Narasimha Kumar and I
use methods from p-adic Hodge theory to study the local irreducibility of Galois
representations attached to automorphic forms on GL_n. The case where the underlying Weil-Deligne
representation is indecomposable is described along with other things in these slides of a talk given at Hida 60th birthday
conference at UCLA in 2012.
Reductions of Galois representations
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In Reductions of Galois representations via the mod p Local Langlands Correspondence (J. Number Theory 147 (2015), 250-286), Abhik Ganguli and I describe the reductions of certain crystalline two-dimensional Galois
representations of weights roughly less than p^2 and slopes in (1,2). We make key use of the compatibility between the mod p and p-adic Local Langlands Correspondences with respect to the process of reduction.
We also describe the submodules of the symmetric power representations of GL(2,F_p) generated by the top two monomials in
this range of weights. The version here includes an extra appendix containing proofs of some of the combinatorial identities we use.
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Building on the paper above, Shalini Bhattacharya and I give an essentially complete description of the
reductions of crystalline two-dimensional Galois representations of all weights and slopes in (1,2) in our paper Reductions of Galois representations for slopes in (1,2) (Doc. Math. 20 (2015), 943-987). We work
under a mild hypothesis, which applies only for weights congruent to 5 mod p-1 and for slope 3/2. We again make key use of the compatibility between the p-adic and mod p Local Langlands Correspondences with respect to the process of reduction.
We also describe the submodules of the symmetric power representations of GL(2,F_p) generated by the top two monomials for
all weights.
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The missing and very interesting case of slope 1 is treated in the joint work
Reductions of Galois representations of slope 1
(J. Algebra 508 (2018), 98-156) with
Shalini Bhattacharya and Sandra Rozensztajn. We compute the semisimplifiction of the reduction completely
for all weights, discovering an interesting trichotomy in the most difficult case of weights congruent to 4
mod p-1. Unlike the fractional slopes cases treated above, we show that the reduction is
often reducible. We therefore also investigate whether the reduction is peu or tr\`es ramifi\'ee,
in the relevant reducible non-semisimple cases. This involves studying the reductions of
both the standard and non-standard lattices in certain p-adic Banach spaces.
A video of a talk given at
the Fields Symposium, Toronto, 2016, summarzing many of the results
obtained in the papers above (among other things) is here, and the
slides are here.
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Vivek Rai and I treated the tricky missing cases for slopes in (1,2) in the Doc. Math. paper above. In
Reductions of Galois representations of slopes 3/2 we establish that the
reductions of two-dimensional crystalline representations of weights congruent to
5 mod p-1 and of slope 3/2 satisfy a tetrachotomy that alternates betwen various irreducible and reducible cases.
Again the proof uses the compatibility with respect to reduction between the p-adic and mod p Local Langlands Correspondences.
This paper will appear in the Kyoto Journal of Mathematics.
This and the previous paper provide evidence for a more general zig-zag conjecture that we make in the paper below.
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We make a general zig-zag conjecture on the behaviour of the reductions of two-dimensional crystalline representations
of small half-integral slopes and exceptional weights in the paper A zig-zag conjecture and local constancy for Galois representations (RIMS Kokyuroku Bessatsu B86 (2021), 249-268). These are weights which are two more than twice the slope mod p-1.
We discuss known cases
of this conjecture, and show that these results force local constancy in the weight for the reductions to not hold at some small weights.
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Let \bar{\rho}_f be the mod p Galois representation attached to a cuspidal eigenform f of level relatively prime to p and finite slope \alpha,
and let \omega be the mod p cyclotomic character. In
Reductions of Galois representations and the theta operator (Int. J. Number Theory 18 (2022), no. 10, 2217-2240),
Arvind Kumar and I prove, under an assumption on the weight of f, that there exists a cuspidal eigenform g of level coprime to p of slope
\alpha+1 such that
\bar{\rho}_f twisted by \omega is isomorphic to \bar{\rho}_g.
See here for an example.
The proof uses Hida-Coleman families and the theta operator acting on overconvergent forms. This shows that
the irreducibility / reducibility of the reductions of two-dimensional crystalline representations tends to propogate as
the slope increases by 1.
We also check that all known cases of the reductions for slopes less than 2, in spite of their somewhat complicated behavior,
are compatible with the displayed formula above.
Moreover, the displayed formula allows us to make some predictions about
the shape of the reductions of crystalline representations attached to
eigenforms of slope larger than 2. Finally, the methods of this paper allow us
to obtain upper bounds on the radii of certain Coleman families.
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In the paper The Monomial Lattice in Modular Symmetric Power Representations with Ravitheja
Vangala, we study the structure of the lattice generated by the monomial submodules in the symmetric
power representations of the standard representation of GL_2(F_p). We give the complete structure for the first p submodules.
We also determine the structure of certain related quotients of the symmetric power representations which arise when studying the
reductions of local Galois representations of slope at most p. Many of our results are stated in terms
of the sizes of various sums of digits in base p-expansions and in terms of the vanishing or non-vanishing
of certain binomial coefficients modulo p. An abridged version of this paper appeared in Algebras and Representation Theory 25 (2022), no. 1, 121-185.
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In the recent joint work Semi-stable representations as limits of crystalline representations with Anand Chitrao and Seidai Yasuda, we construct an explicit sequence of crystalline representations converging to a given irreducible two-dimensional semi-stable representation of the Galois group of Q_p. The convergence takes place in the blow-up space of two-dimensional trianguline representations studied by Colmez and Chenevier. The process of blow-up is described in detail in the rigid analytic setting and may be of independent interest. Also,
we recover a variant of a formula of Stevens expressing the L-invariant as a logarithmic derivative.
Our convergence result can be used to compute the reductions of any irreducible two-dimensional semi-stable representation in terms of the reductions of certain nearby crystalline representations of exceptional weight. In particular, this provides an alternative approach to computing the reductions of irreducible two-dimensional semi-stable representations that circumvents the somewhat technical machinery of integral p-adic Hodge theory. For instance, using the zig-zag conjecture made in the paper a few papers above this one ([Gha21]) on the reductions of crystalline representations of exceptional weights, we recover (resp. extend) completely the work of Breuil-Mezard and Guerberoff-Park on the reductions of irreducible semi-stable representations of weights at most p-1 (resp. p+1), at least on the inertia subgroup. In the cases where the zig-zag conjecture is known, we are further able to obtain some new information about the reductions of semi-stable representations of small odd weights. Finally, we use the above ideas to explain away some apparent violations to local constancy in the weight of the reductions of crystalline representations of small weight that were noted in [Gha21] and which
provided the initial impetus for this work. To appear in Alegbra & Number Theory.
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In the very recent paper Zig-zag for Galois representations,
we prove the zig-zag conjecture ([Gha21])] on the reductions of two-dimensional crystalline p-adic Galois representations for all
half-integral slopes 1/2 \leq v \leq (p-1)/2 and for weights which are p-adically close enough to the weight
2v+2 (and congruent to it modulo (p-1))
in a smooth family of such representations, at least if p > 3.
In version 1, we were able to prove this conjecture on inertia
and for slopes up to (p-3)/2.
The key idea was to reverse the limiting arguments in the paper of Chitrao-Ghate-Yasuda above ([CGY21]): we used
the work of Breuil-Mezard and Guerberoff-Park on the reduction of two-dimensional semi-stable p-adic
Galois representations of weights 3 \leq 2v+2 \leq p-1 to show that zig-zag is true for the reductions of
nearby crystalline representations, on inertia.
In version 2, we extend the proof to the full Galois group of Q_p and to the top two
missing slopes (p-2)/2 and (p-1)/2, by using as extra input the main result with Anand Chitrao (two papers below this one)
on the reductions of semi-stable representations, which also covers weights p and p+1.
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In the paper Modular induced representations of GL_2(F_q) as cokernels of symmetric power representations,
we prove that certain induced representations of GL_2(F_q) can be written as cokernels of operators acting on symmetric power representations. When the induction is from the Borel subgroup, these operators generalize multiplication by the
classical Dickson polynomial, and when the induction is from the anisotropic torus, these operators generalize Serre's operator. All our isomorphisms are explicitly defined using certain differential operators. The proofs are elementary and just use facts from calculus. As a corollary we extend some periodicity results in the ART paper with Ravitheja Vangala above. This is joint work with Arindam Jana. An abridged version of this paper (with a few details omitted and
a slighly different title) is here: Modular representations of GL_2(F_q) using calculus. This will appear in Forum Math.
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In the paper Reductions of semi-stable representations using the Iwahori
mod p Local Langlands Correspondence, Anand Chitrao and I describe the reductions of all two-dimensional semi-stable
representations of the Galois group of Q_p of weights between 3 and p+1 inclusive
(with any L-invariant and for all primes p at least 5). Qualitatively speaking, the main theorem
says that the reduction varies through
an alternating sequence of irreducible and reducible representations depending on the size of a parameter.
This recovers work of Breuil-Mezard and Guerberoff-Park
for even, respectively odd, weights up to p-1 (the latter on inertia, but see work of Lee-Park).
One feature is that we describe the constants in the
unramified characters appearing in the reduction completely. In particular, we describe the
tricky `self-dual' constants which arise when the parameter is in the
`last' interval, where the reduction is scalar on inertia.
The proof is quite different from previous approaches: it
studies the reduction of Breuil's GL_2(Q_p)-Banach space by analyzing certain logarithmic functions
using background material due to Colmez, and then uses an
Iwahori theoretic reformulation of the mod p LLC
worked out by Anand Chitrao to conclude.
In principle, the method works for all weights and allows one to go beyond the previous
glass ceiling of weight p-1 which arises naturally when using strongly divisible modules
from integral p-adic Hodge theory. In particular, we treat
the weights p and p+1 completely (in the latter case, a non-commutative Iwahori-Hecke algebra reappears
when the parameter is at the `first' point - it had already appeared in our argument for even
weights at most p+1 when the parameter is at the `last' point). The main theorem for these
two weights will allow us to put the final
nails in the coffin in our proof of the
zig-zag conjecture (announced a few papers above) by allowing us to treat the last two slopes
(p-2)/2 and (p-1)/2.
Appendix: this is the main addition to version 2. It provides theoretical proofs of some
pending claims regarding solutions to certain matrix equations (see footnotes 10,11,12,16,17).
The proofs hinge on the evaluation of certain finite binomial sums involving Harmonic
numbers which were proved using the help of programs developed by mathematicians
at RISC, Austria (we thank Peter Paule for his help with this).
Non-admissible representations
Weight one forms
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On the
average number of octahedral forms of prime level
(Math. Ann. 344 (2009), no. 4, 749-768) is concerned with counting exotic weight
one forms. Using results on the asymptotic enumeration of quartic fields,
Manjul Bhargava and I show that, on average,
the number of octahedral forms of prime level is bounded by a constant.
(Errata: Prop. 5.1 and Cor. 5.2 only hold for square-free levels. When the level
is cube-free, it is possible for the \psi_i in Prop. 5.1 to be characters of inertia which do not
extend to the Galois group of Q so one cannot twist globally by them, and even
if one could, the fact that \psi_1 \neq \psi_2 does not imply the local
representation on inertia injects into the projective representation. This
does not affect the main result, Theorem 1.1, for prime levels, but Theorem 1.2
and the estimates in Section 6 for good levels will change, and will be reworked elsewhere.)
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On classical weight one
forms in Hida families
(J. Theorie Nombres Bordeaux 24 (2012), no. 3, 669-690)
is concerned with
a) giving explicit bounds on the number of weight 1 forms in non-CM Hida
families, and b) investigating uniqueness and etaleness of weight 1 points
in Hida families. In particular, we give the first explicit examples
of two non-Galois conjugate Hida families passing through the same weight
1 form. This is joint work with Mladen Dimitrov.
Images of modular Galois representations
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In the paper On uniform
large Galois images for modular abelian varieties (Bull.
London Math. Soc. 44 (2012), no. 6, 1169-1181) Pierre
Parent and I investigate the existence of uniform bounds for the
images of residual Galois representations attached to abelian varieties
of GL_2 type. We show that uniform bounds depending on the dimesion
exist in the exceptional image case, and also investigate the other cases.
Products of Eigenforms
Sums of Fractions and Monodromy
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In a joint article with T. N. Venkataramana, Sums of Fractions and Finiteness of Monodromy (Indag. Math. (N.S.) 28 (2017), no. 6, 1183-1199), we solve an elementary number theory problem on sums of fractions using methods from group theory and some direct calculations.
The case of three fractions is equivalent to Schwarz's classification
of algebraic Euler-Gauss hypergeometric functions. As an application we
deduce the finiteness of certain monodromy representations.
Salem numbers
Lecture notes, slides etc.
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Here ([dvi],
[ps])
are some slides of a talk I gave at the graduate student seminar at
UCLA
on the nature of the values of zeta functions.
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These
are notes of lectures given at the Pune summer school on cyclotomic
fields,
1999, on work of Soule' and Kurihara on Vandiver's Conjecture. I
also gave a more introductory set of lectures on class field theory
which
can be found here.
Notes of lectures on complex multiplication given at HRI, Allahabad
during
the winter school on elliptic curves, 2000, are here.
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An overview of
Lambda-adic
forms is here. This is a compilation of notes of lectures given by
the authors at the workshop on the Iwasawa Main Conjecture, at IIT
Guwahati in 2008 (Ram. Math. Soc. Lect. Notes Ser., 12, 2010).