Eknath Ghate's Papers
My papers are available below. A complete list of publications is
Special values of twisted tensor L-functions
A generalization of the main result of the above paper to CM fields
This article appeared in the Shimura honorary volume:
Proc. Symp. Pure Math., vol 66, part 1, Amer. Math. Soc. (1999), 87-109.
Congruences for modular forms
congruence primes and class fields of real quadratic fields
Number Theory 95 (2002), no. 1, 14-37), Alex Brown and I show
dihedral congruence primes for elliptic cusp forms of quadratic
can be characterized in terms of simple expressions involving
units of real quadratic fields. We apply our results to explicitly
(ray) class fields of real quadratic fields by torsion points on
Modular endomorphism algebras
algebras of motives attached to elliptic modular forms (Ann.
Inst. Fourier, Grenoble 53 (2003), no. 6, 1615-1676) Alex Brown and I
the endomorphism algebra X_f of the motive attached to a non-CM
modular form f, of weight k at least two. We show that if k > 2 then
contains a certain crossed product algebra X defined over a number
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
There is a program that we wrote which illustrates some of
the theorems proved in the paper above. The code builds on the
C++ version of the modular symbols engine HECKE written by W. Stein.
program can be run locally by
logging on to the machine homotopy, and,
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
nebentypus is specified by typing in a string of 1's and 2's). As
the program prints the (norm to Q) of the Brauer class of X by
a list of finite primes where it is ramified. As for the infinite
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
k > 2, and sharpen some of our results concerning the Brauer class
contained in the paper above. We show that in many cases that the
class is locally at p determined by the p-adic valuations of the p-th
coefficients (slopes) of the form. This is joint work with Enrique
and Jordi Quer.
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).
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
In the paper On
the local behaviour of ordinary modular Galois representations
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.
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
the weight 1 specializations of families of p-ordinary forms. As a
of this study, we show that for all but finitely many specializations
weight 2 or larger, the local Galois representation is split if and
if the form has complex multiplication (we assume p is odd and work
some technical conditions on the mod p representation). We also treat
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
conditions, the local splitting of an ordinary modular characteristic 0
Galois representation is related to whether the corresponding form is
the image of a suitable power of the theta derivation. We deduce some
towards a question of Coleman regarding the existence of non-CM forms
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.
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).
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
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.
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
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.
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
The missing and very interesting case of slope 1 is treated in the joint work
Reductions of Galois representations of slope 1 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.
Weight one forms
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.)
On classical weight one
forms in Hida families
(J. Théorie 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
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
In a recent joint article with T. N. Venkataramana Sums of Fractions and Finiteness of Monodromy (to appear in Indag. Math.), 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.
Lecture notes, slides etc.
are some slides of a talk I gave at the graduate student seminar at
on the nature of the values of zeta functions.
are notes of lectures given at the Pune summer school on cyclotomic
1999, on work of Soule' and Kurihara on Vandiver's Conjecture. I
also gave a more introductory set of lectures on class field theory
can be found here.
Notes of lectures on complex multiplication given at HRI, Allahabad
the winter school on elliptic curves, 2000, are here.
An overview of
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).
Here are slides of my outreach talk
The Tau of Ramanujan at Prithvi Theater, given just before the ICM in 2010.