Eknath Ghate's Papers


My papers are available below. A complete list of publications is here.
 

Special values of twisted tensor L-functions
 

• My thesis investigated the Critical values of the twisted tensor L-function in the imaginary quadratic case  (Duke Math. J. 96 (1999), no. 3, 595-638). The rationality result contained in this paper was motivated by Deligne's conjectures on the special values of motivic L-functions (translation available here).

• 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(1) (1999).

Congruences for modular forms
 

•  Congruences between base-change and non-base-change Hilbert modular forms  (Cohomology of arithmetic groups, Automorphic forms and L-functions, Narosa, 2001) is an expository  account of a conjecture of Doi, Hida and Ishii.

• Adjoint L-values and primes of congruence for Hilbert modular forms (Compositio Math. 132 (2002), no. 3, 243-281) shows that almost all primes that divide a certain adjoint L-value of a primitive Hilbert modular form are congruence primes for this form.

• On the freeness of the integral cohomology groups of Hilbert-Blumenthal varieties as Hecke modules (to appear in the Proceedings of the International Colloquium on Cycles, Motives and Shimura Varieties, TIFR, 2008) investigates the converse divisibility for ordinary congruence primes. The main technical tool used in this paper is a commutative algebra machine which forms the subject matter of the short expository note Taylor-Wiles systems (Ram. Math. Soc. Lect. Notes Ser., 1, 2005).

• 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.

• An introduction to congruences between modular forms  (Current trends in number theory, Hindustan Book Agency (2002), 39-58) introduces some of the basic concepts in the theory of congruences between modular forms.

Modular endomorphism algebras
 

• In Endomorphism algebras of motives attached to elliptic modular forms  (Ann. Inst. Fourier, Grenoble 53, no. 6 (2003), 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

• 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 (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  (Internat. Math. Res. Notices 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.

• The case of weight one forms is treated in Endomorphism algebras of Artin motives (joint with Debargha Banerjee).

• A description of the Brauer class of X, in terms of the slopes of the Gelbart-Jacquet adjoint lift of the underlying form, is given in Adjoint lifts and modular endomorphism algebras. This is also joint work with Debargha Banerjee. This paper completely answers Ribet's question on the Brauer class of X, under a finiteness hypothesis on the slopes. The main new ingredient is the use of the adjoint lift to decribe the local Brauer classes at the primes of bad reduction. This paper grew out of conjectures that we talked about in Roscoff in 2009 (slides).

Splitting of Galois representations
 

• 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.

• In a more recent paper On the local behaviour of ordinary $\Lambda$-adic representations (Ann. Inst. Fourier, Grenoble 54 (2004), 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.

• In Locally indecomposable Galois representations (to appear in the Canadian Journal of Mathematics) 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.

Filtered modules
 

• 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.

• In (p,p)-Galois representations attached to automorphic forms on GL_n  we use methods from p-adic Hodge theory to study the local irreducibility of Galois representations attached to automorphic forms on GL_n. This is joint work with Narasimha Kumar.

Counting weight one forms
 

• On the average number of octahedral forms of prime level  (Math. Ann. 344 (2009), 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.

Products of eigenforms
 

• In the note On monomial relations between Eisenstein series  (J. Ramanujan Math. Soc. 15 (2000), 71-79) we show that except for sixteen exceptions the product of two level one eigenforms is not an eigenform.

• The case of square-free level is investigated in On products of eigenforms  (Acta Arith. 102 (2002), 27-44).

Salem numbers
 

• Here is a joint article with Eriko Hironaka on the  The arithmetic and geometry of Salem numbers  (Bull. Amer. Math. Soc. 38 (2001), 293-314). It is an expository note on the minimization problem for Salem numbers.

Lecture notes, slides etc.
 

• 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.

• 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.

• Control theorems and applications are notes of lectures given by H. Hida during the special year (1998-1999) on automorphic forms at TIFR.  The phenomenon of trivial zeros are notes of 8 lectures given by R. Greenberg at TIFR in 2001.

• An overview of Lambda-adic forms is here. This is a compilation of notes of lectures given by the authors at the school on the Iwasawa Main Conjecture, at IIT Guwahati in 2008.

• Slides of my public talk The Tau of Ramanujan at Prithvi Theater, just before the ICM in 2010, are now available.