Contributions to Algebraic Number Theory
from India
Dipendra Prasad
October 19, 2004
There was a conference organised at the Institite of Mathematical Sciences,
Madras in 1997 on the occasion of the 50th anniversary of India’s Independence
to review contributions by Indians to various branches of mathematics since Independence. This is a written account of the lecture I gave then on some of
the contributions by Indian mathematicians to Algebraic number theory for theperiod 1947-1997.
There has been a lot of work done in India during these years, and it will be
impossible to report on all of it in limited space. We have therefore chosen someof the representative works from the vast literature reflecting to some extent theauthor’s taste. Since this was a report about work done in the country, works ofseveral very prominent mathematicians from India during this period who worked
for most of their scientific career outside India has been omitted. In particular,
we do not speak about the work of S. Chowla who has contributed so much to somany areas of Algebraic number theory. We have also restricted ourselves to areas
very closely related to Algebraic number theory, and have therefore omitted topics
in transcendental and analytic number theory which was reported by others in
the conference. We have also omitted from our consideration the arithmetic
theory of algebraic groups. In particular, we do not speak on the work of K.G.
Ramanathan who initiated the study of arithmetic groups in the country which
has flourished into a very strong area of research at the Tata Institute, Bombay.
We begin the report by listing the two major schools in the country where
algebraic number theory has been pursued, and in each case list some of themathematicians from these schools. Limitation of knowledge on the part of theauthor has prevented him from a more complete list. In the list below, non-Indianmathematicians have been put in a bracket, and Y is on the immediate right of1

X, if Y is a student of X.

Panjab University, Chandigarh

I.S. Luthar

J.C. Parnami
Meenakshi Duggal
S.K. Khanduja(Cassels)

A.R. Rajwade

S.A. Katre
Dharam Bir Rishi(Coates)

M.K. Agarwal
S.K. Aggarwal

D.P. Gupta

Gurnam Kaur
(Mordell)

V.C. Dumir

Ranjit Sehmi
V.K. Grover

R.J. Hans-Gill

Madhu Raka

R.P. Bambah

Dharam Singh Khasa

2

Urmila Rani

Tata Institute of Fundamental Research, Bombay

S. Raghavan

Balakrishnan
Juliet Britto
Jyotsana Dani
Vasudevan

Manickam
Ramakrishnan

Neela S. Rege
S.S. Rangachari
C.P. Ramanujam
K.G. Ramanathan(1920–1992)(Studied with E. Artin
and C.L. Siegel)

V.C. Nanda

Goutami Bhowmik
Asha Narang

Sunder Lal

K. Ramachandra

Balasubramanian
Srinivasan
ShoreySankaranarayanan
K. Srinivas

We now take up some specific topics in Algebraic number theory to which
Indians have made a significant contribution.

1

Waring’s Problem

The Waring’s problem asks whether given any integer k ≥ 2, there exists an
integer g(k) ≥ 1 such that any integer n ≥ 1 is a sum of at most g(k) kth
powers of non-negative integers. The most classical result is for the exponent 2
in which case Lagrange proved that g(2) = 4.
Theorem 1 (Lagrange) Any integer n ≥ 1 can be written as n = n21 + n22 +n23 + n24 with integers ni ≥ 0.
Theorem 2 (Hilbert) Waring’s problem has an affirmative answer for any exponent k, i.e., for any integer k ≥ 2, there exists an integer g(k) such that any
integer n ≥ 1 is a sum of at most g(k) kth powers of non-negative integers.
3

Adhikari
Yogananda
Padma
Venkataraman
Amora
ManishaTheorem 3 (Pillai-Dickson as completed by Balasubramanian, Deshouillers andDress in the case k = 4 [BDD].)"  #
k3− 2 ifg(k) = 2 +2(  ) "  #
k
k
3
3
2
+
≤ 2k ,
22kk

Otherwise,h  i h  i
k
3 k
k

+ 43
− 2,

 2 + 2g(k) =h  i Or, h  i 2k + 3 k + + 4 k − 3,23

depending on whether"  # "  # "  # "  #
k
k
k
k
4
3
4
3
+
+
= 2k ,
3
2
3
2

or > 2k ,where {x} denotes the fractional part of x, and [x] the integral part of x.
Remark : It is expected but not known (amazing as such a lack of knowledge
might seem) that(  ) "  #
k
k
3
3
k
+
≤ 2k ,(∗)
22
2
and therefore only the first case in the statement of the theorem need be considered. Mahler has shown that (∗) holds for all but finitely many k.

1.1

Waring’s problem for number fields

One can ask if Waring’s problem has an affirmative answer in a number field too.
However, one sees immediately that the problem may not have an affirmative
answer because of local conditions. If for instance q is a power of prime and notitself a prime, then for any element x of F∗q , xq−1 = 1, and therefore sum of(q − 1)-th powers of elements in F∗q is not all of F∗q . The following theorem of
C.P. Ramanujam [RCP1] is therefore the best (except for the explicit number ofmth powers needed) one can hope for.

4

Theorem 4 (C.P. Ramanujam) Any totally positive algebraic integer in any algebraic number field belonging to the order generated by the mth powers of algebraic integers in that number field is actually a sum of atmost max(2m + 1, 8m5 )mth powers of totally positive integers.
Remark on the proof : The proof depends on a theorem of Birch that local to
global principle holds in sufficiently large number of variables. Therefore it suffices
to prove the above theorem for p-adic fields which is what C.P. Ramanujam does.
Generalising Waring’s problem, one can ask about the set of values taken by
a general form, or about the zero set of a general form: whether local to global
principle holds, or whether the weak approximation holds. Both these questions
are an active area of research. Here is a theorem due to C.P. Ramanujam [RCP2],
together with refinements due to Pleasant [Pl] and Hooley [Ho].
Theorem 5 (C.P. Ramanujam) Any cubic form over any algebraic number fieldin ≥ 54 variables has a non-trivial zero.
The strongest theorem in this direction is due to Pleasant obtained by refining
the method of C.P. Ramanujam.
Theorem 6 (Pleasant) Any cubic form over any algebraic number field in ≥ 16
variables has a non-trivial zero.
Assuming the non-singularity of the cubic form, one has a much better theorem due to Hooley.
Theorem 7 (Hooley) If f (X) is a non-singular cubic form in 9 variables over Q
which has a non-trivial zero in Qp for all p, then f (X) has a non-trivial zero.
Remarks(a) The above theorem is due to Heath-Brown for 10 variables ormore. In 10 variables, there are no local conditions: every cubic form in ≥ 10
variables over a p-adic field has a non-trivial zero.
(b) The theorem of Hooley is not yet available for number fields.
(c) It is expected that the theorem of Hooley is true without the nonsingularity assumption.

5

2

Geometry of Numbers

This area was inaugurated by Minkowski in his work on discriminants and classnumbers of number fields. R. P. Bambah and his school at Chandigarh havemade several contributions to this area. We state below some of the questionsin this subject in which this school has contributed. We begin by recalling thefollowing basic and elementary theorem of Minkowski.
Theorem 8 Let S be a convex, symmetrical domain in Rn containing the origin.
Then S contains a point other than the origin from any lattice Λ whenevervol(S) ≥ 2n vol(Rn /Λ).
Before proceeding further, we need to introduce the following definition.
Definition : A lattice Λ ⊂ Rn is called a covering lattice for a subset S in Rn
if Λ + S = Rn . The covering constant c(S) of S is defined to bec(S) = sup vol(Rn /Λ),where the supremum is taken over all latices Λ which form a covering lattice forS.
One also defines θ(S), the density of best covering lattice to beθ(S) =

vol(S).
c(S)Clearly, θ(S) ≥ 1. The following theorem is due to Bambah and Roth [BR].
Theorem 9 For a closed convex symmetrical region S in Rn containing theorigin,π nnθ(S) ≤ √.
3 3 n!
Remark: θ(Dn ) even for the n-disc Dn = {x ∈ Rn ||x| ≤ 1} in Rn is notknown for n ≥ 6!
Minkowski’s Conjecture : Let L1 , · · · , Ln be linearly independent linear
forms on Rn with ∆ = det(L1 , · · · , Ln ). Let α1 , · · · , αn ∈ R. Then there exists
x ∈ Zn such thatnY∆
|Li (x) + αi | ≤ n .
2
i=1
6Bambah has proved this conjecture for n = 4, 5 in a joint paper with AC
Woods, cf. [BW]. The conjecture is unknown for n ≥ 6.
Oppenheim Conjecture : Given an indefinite quadratic form Q on Rn (n ≥3) which is not a multiple of a rational quadratic form, and given  > 0, there
exists v ∈ Zn , such that
0 < |Q(v)| < .
Much work on this conjecture was done by Bambah, Raghavan and Ramanathan, cf.[RR2], besides Oppenheim, Davenport and Birch, and was finallyproved by Margulis in 1986-87 who proved a conjecture of M.S. Raghunathanon the closure of orbits of unipotent flows on Γ\G where G is a Lie group, andΓ is an arithmetic subgroup. S.G. Dani and Gopal Prasad have also contributed
to the Oppenheim Conjecture, see the papers [DM] and [BP]. Oppenheim conjecture is open to generalisation for forms of higher degree. Here we state thequestion in a rather crude form, cf. [RR1] for one case of such a conjecture.
Generalised Oppenheim Conjecture : Given a form Q of degree d on Rn(n ≥ n(d)) which takes both positive and negative values and is not a multipleof a rational form, and given  > 0, there exists v ∈ Zn , such that
0 < |Q(v)| < .
Here is another theorem relating to values of quadratic forms due to Blaney,
cf. [Bl].
Theorem 10 (Blaney, 1948) Let Q be an indefinite quadratic form in n variables
and of discriminant D 6= 0. Then there exists a constant Γr,s depending only on
the signature (r, s) of Q at infinity, and not on Q itself, such that for any real
numbers c1 , · · · , cn , there exists integers x1 , · · · , xn such that
0 < Q(x1 + c1 , · · · , xn + cn ) < (Γr,s |D|)1/n .
Much work has been done by Bambah, Hans-Gill, Dumir, Madhu Raka, Urmila
Rani all from Chandigarh, to find the optimum Γr,s . The values of Γr,s is now
known for all r, s except (r, s) = (1, 4). We refer to the survey paper [BDH] by
Bambah, Dumir and Hans-Gill for a detailed account.
Examples :(a) Γ1,1 = 4, due to Davenport and Heilbronn.
(b) Γ2,1 = 4, due to Blaney.
, Γ2,2 = 16, due to Dumir.
(c) Γ1,2 = 8, Γ3,1 = 163(d) Γ1,3 = 16, due to Dumir and Hans-Gill.
7

3Work on Elliptic curves and curves of higher
genus

There has been a large number of papers by A.R. Rajwade, M.K. Agrawal, J.C.
Parnami, D. B. Rishi, S.A. Katre, all from Chandigarh, cf. the bibliography
at the end, and also by Padma and Venkataraman, cf. [PV], students of R.
Balasubramanian from Mat. Science, Madras, on the explicit determination of
the number of points on an elliptic curve with complex multiplication. Results
have also been obtained for the number of points on curves of the form
y 2 = xl + a.
Instead of stating various theorems obtained by the above mentioned mathematicians, we state one classical result of this kind due to Davenport and Hasse togive a flavour for the kind of the result that is obtained in this direction. The
reader will easily notice that the problem one is really solving is one of signs.
Theorem 11 (Davenport-Hasse) For the curve y 2 = x3 − dx,
p + 1 if p ≡ 3 mod 4
Np =if p ≡ 1 mod 4.
p + 1 − π π̄d 4 − π̄ πd 4Here p = ππ̄ in Q(i) such that π, π̄ are congruent to 1 modulo (2 + 2i);d
is the 4th power residue symbol, and is defined to be the unique element
π 4x ∈ {±1, ±i} such thatp−1

d 4 ≡ x mod π.

4Siegel-Ramachandra-Robert Units

These are explicit units in abelian extensions of quadratic imaginary fields constructed using elliptic functions. After the initial work by Siegel, they wereconstructed by Ramachandra in [Ram], and then streamlined by Robert. These
were called Siegel-Ramachandra-Robert units for some time, but are now called
elliptic units and play a fundamental role in many works dealing with arithmeticof elliptic curves with complex multiplication as in the fundamental works of
Coates-Wiles and Rubin.
Before we describe elliptic units, we describe the simpler case of cyclotomic
units.
8

For x ∈ Q/Z, x 6= 0, let g(x) = e2πix − 1. Define the group of cyclotomic
units to be the intersection of all units with the group generated by g(x).
Theorem 12 (a) If the denominator of x is composite, then g(x) is a unit in
the ring of algebraic integers. If the denominator of x is a power of p then g(x)is a p-unit.
(b) The mapping x → g(x) satisfies the distribution relation:Yg(y) = g(x),N y=x

for all integers N ≥ 1.
(c) For x ∈ N1 Z/Z, g(x) ∈ Q(ζN ). For a ∈ (Z/N )∗ , let σa be the naturalelement in Gal(Q(ζN )/Q). Theng(x)σa = g(ax).
(d) The index of the cyclotomic units in all units is closely related to the classnumber of the real subfield. The index is equal to the class number in the caseof prime power level.
Siegel Units: We will define Siegel units here which do not have as niceproperties as the case of cyclotomic units described above, but are easier todefine than the elliptic units which have properties much more analogous tocyclotomic units which we will be taking up next.
Let K be a quadratic imaginary field with HK the Hilbert class field of K.
We recall that HK is the maximal unramified abelian extension of K and that
its Galois group over K is canonically isomorphic to the class group of K. For
an ideal I of K, we let σI denote the corresponding element of the Galois groupof the Hilbert class field of K.
For an ideal I of K, defineu(I) =∆(OK ),∆(I −1 )where OK is the ring of integers in K, and ∆ is the Ramanujan delta function.
Theorem 13 (Siegel)(a) u(I) ∈ HK .
(b) u(IJ ) = u(I)σJ u(J ).
(c) (u(I)) = I −12 .
9If h is the class number of K, then I h is a principal ideal, say I h = (α), α ∈K . Clearly, α is unique only up to a root of unity in K, and therefore α12 iswell-defined. It follows that δ(I) = u(I)h α12 is a well defined unit of HK . The
group generated by such units is called the group of Siegel units.
∗

Elliptic Units: Let L ⊂ C be a lattice which is invariant under multiplication
by elements of OK . For an integral ideal I of K, defineΘ(z, L, I) ==θ(z,L)N I
θ(z,I −1 L)∆(L)∆(L) Q0u∈I −1 L/L (℘(z,L)−℘(u,L))6 .
∆(I −1 L)

The prime over the product notation signifies that the product is to be takenover non-zero elements. (Since we do not define θ(z, L) here, one could take thesecond equality for the definition of Θ(z, L, I).) The numbers Θ(z, L, I) satisfy
a ‘distribution relationship’ as in theorem 12(b):YΘ(z + v, L, I) = Θ(z, J −1 L, I),v∈J −1 L/L

which holds for any coprime ideals I, J in OK .
Theorem 14 Let m be a non-trivial integral ideal of K and v a primitive mdivision point of L (i.e., v ∈ m−1 L but v 6∈ n−1 L for any proper divisor n of m).
Then if (I, m) = 1,
(1) Θ(v, L, I) belongs to the m-ray class field of K.
(2) σc (Θ(v, L, I)) = Θ(v, c−1 L, I).
(3) Θ(v, L, I) is a unit if m is not a power of a prime ideal. If m = ℘n , it isa unit outside ℘.

5

Theta function associated to quadratic forms

Let q : L → Z be a positive definite quadratic form on a lattice L in Rk . The
simplest theta function associated to q isP
θq (z) = Pv∈L eπiq(v)zπinz=
n≥0 rq (n)ewhere rq (n) = ]{v ∈ L|q(v) = n}.
10

It is known that θq (z) is a modular form of weight k/2 on the upper-halfplane.
The Fourier coefficients of a theta function are all positive integers with 1 asits constant term. For many weights for the full modular group, Manickam, Ramakrishnan, Kalyan Chakraborty, Arbind Lal have determined all modular forms
with these properties for their Fourier coefficients, cf. [CLR].
The space of modular forms is spanned by cusp forms and Eisenstein series.
Eisenstein series have a simple Fourier expansion while cusp forms have niceestimates for their Fourier coefficients. Combining the two, one gets the following
theorem.
kkk
π 2 n 2 −1
Theorem 15 (Hardy-Ramanujan) rq (n) =σk (n) + O(n 4 ), where k ≥k!
24, and σk (n) is the ‘singular series’.

Such a theorem has been generalised by Raghavan in [RS1] who obtained
an asymptotic formula for the number of integral representations of an n rowed
positive semi-definite integral matrix T by an m rowed positive definite integral
matrix S, i.e., the cardinality of the set of integral matrices X of size (n, m)
with
XS t X = T.
We will denote the cardinality of this set of X’s by rS (T ).
The estimate on rS (T ) is via an association of Siegel modular form to S. We
therefore fix some notation regarding Siegel modular forms.
Let
Hn = {Z = X + iY ∈ Mn (C)|t Z = Z, Y > 0}
be the Siegel upper half space. The symplectic group Sp(2n, R) operates on
Hn and one has the notion of a Siegel modular form of genus n and weight kanalogous to modular forms in one variable. A Siegel modular form has Fourier
expansionX
F (Z) =
f (A)eπitr(AZ) .
t A=A≥0

We now state Raghavan’s theorem which estimates the Fourier coefficients ofa Siegel modular form in terms of the Fourier coefficients of a linear combinationof Eisenstein series. The difficulty in the proof of this theorem arises because
unlike the one variable case, it is not true that a Siegel modular form is up to a
cusp form, a linear combination of Eisenstein series. Raghavan has to use Siegel’s
11

generalisation of the ‘Farey dissection’ to matrix spaces to obtain an estimate on
the Fourier coefficient of the difference.
Theorem 16 (Raghavan) Given a Siegel modular form
X
F (Z) =a(T )eπitr(T Z) ,T ≥0

one can associate to F (Z) a linear combination of Eisenstein seriesX
φ(Z) =b(T )eπitr(T Z) ,T ≥0such thata(T ) − b(T ) = 0([minT −1 ]n(n+1−2k)2[minT ]

n+1−k
2

)

for |T | ≥ ν, for some constant ν, and where minT denotes the minimum value
taken by T at a non-zero vector.
Singularmodular forms: A Siegel modular form F given by F (Z) =P
πitr(T Z)
is called a cusp form if f (A) = 0 whenever A is singuT ≥0 f (T )elar. At the other extreme we have the notion of a singular modular form which
are Siegel modular forms which have the property that f (A) = 0 whenever A
is non-singular. Certain theta functions constructed below provide examples of
singular modular forms.
To a positive definite quadratic form q : L → Z, define θq (Z) to be thefunction on the Siegel upper-half plane Hn defined byX
rq (A)eπitr(AZ) .
θq (Z) =t A=A≥0

The function θq (Z) is called the theta function of genus n associated to thequadratic form q generalising the theta function in one variable defined at the
beginning of this section. This Siegel modular form is singular if k < n. The
first of the following two theorems, due to Resnikoff, states that the weights
of singular modular forms is exactly like this, and the second due to Raghavan,
cf. [RS2], states that singular modular forms are linear combinations of theta
functions.
Theorem 17 (Resnikoff): Singular modular forms of genus n exist only forweights < n/2.
12

Theorem 18 (Raghavan) A singular modular form is a linear combination oftheta series.
Remark : Raghavan proved the above theorem for the full modular group. The
general case is due to Resnikoff and Freitag.
Raghavan also has several papers, some jointly with others, in which hestudies estimates on the Fourier coefficients of Siegel modular forms, cf. [BR],for one such work.

6

Modular forms of half-integral weight

There has been an extensive study by Vasudevan of Vivekananda College, Chennai
together with his former students Manickam and Ramakrishnan on the theoryof new forms for half-integral weight modular forms and its relation to Jacobi
forms. They have also made an explicit construction of the so called Shintani
lifting which gives a connection between modular forms of integral and halfintegral weights and have thereby obtained an explicit version of Waldspurger’s
theorem. We mention only one of their papers [MRV] out of several that they
have written.

7Henselian Rings

There has been several contributions from mathematicians at Chandigarh on
Henselian rings.
The classical result of Hasse and Arf on higher ramification groups in a cyclic
Galois extension of a complete discrete rank 1 valued field states that the jumps
in the Herbrand upper indexing occur at integers. In a paper by Ram Avtar
and N. Sankaran, cf. [AS], an analogous result has been proved for higher rank
valued fields which are Henselian. They extend a result of S. Sen conjectured by
A. Grothendieck, on wildly ramified automorphisms of a complete discrete rank
1 valued field.
Sudesh K. Khanduja has several papers around Hensel’s lemma. We only
mention [KS] here.

13

8

Brauer-Siegel theorem

There is the famous theorem in Algebraic number theory which tells how thevarious basic invariants of a number field (class number, regulator and the discriminant) vary as we vary the number field. In an interesting work, Sudesh
K. Gogia (now Sudesh K. Khanduja after marriage!) and IS Luther prove theanalogue for function fields.

9

Some more recent works

In this section we briefly mention about some of the recent works done in Algebraic number theory. All these mathematicians have been associated with theTata Institute of Fundamental Research, Bombay.
Kirti Joshi : He has constructed coverings of the affine line using Drinfeld
modules, cf. [J].
C. Khare : He has proved several results about congruences of modular forms,
cf. [Kh1]. He also has proved that Artin’s conjecture is a consequence of Serre’s
conjecture on modularity of certain Galois representations, cf. [Kh2].
Madhav Nori : He has a very beautiful construction for unramified coveringsof the affine line in positive characteristic by finite Chevalley groups in thatcharacteristic, cf. [N1]. He also has a new proof of the algebraicity of values of
L-functions associated to Grössencharacters on totally real number fields whichis due originally to Siegel and Klingen, cf. [N2].
Dipendra Prasad : According to conjectures of Langlands which have been
proved in many cases, representation theory of p-adic algebraic groups is closely
related to the representation theory of the Galois group of the p-adic field. Severalof my works, some done in collaboration with B.H.Gross, cf. [GP], use thisparametrization to give a description of “branching laws”: explicit descriptionof the decomposition of representations of a group restricted to a subgroup. In
particular, there is a p-adic analogue of a theorem of Clebsch and Gordan about
the tensor product of representations of GL(2), cf. [P1],[P2].
C S Rajan : He has studied estimates on the orders of the Tate-Shafarevichgroup for elliptic curves over function field of curves over finite fields, cf. [Raj].
14

He also has proved multiplicity one theorem for `-adic representations.
Nimish Shah : Counting integral points on Algebraic varieties of boundedheight is a basic and important problem in number theory. Nimish Shah together
with his collaborators have an impressive paper dealing with this problem whenthe variety in question is a closed subvariety of the affine n-space which is acted
upon transitively by a reductive algebraic group, and for more general such homogeneous varieties. The proofs depend on the ‘unipotent flows’ on measures,see [EMS], for a paper in this direction.
Acknowledgement:
The author would like to thank A. Raghuram and
C.S.Yogananda, and Prof. R.J.Hans-Gill for their help.

10

Bibliography[AS] Ram Avtar and N. Sankaran, Higher ramification in Hensel fields. J. Reine
Angew. Math. 306 (1979), 185–191.
[BDD] R. Balasubramanian, J-M Deshouillers, F. Dress, Probleme de Waring
pour les bicarres, I and II, CR Acad. Sci. Paris Series I, Math. 303(1986) 85-88
and 161-163.
[BDH] R.P. Bambah, V.C. Dumir, and R.J. Hans-Gill, Non-homogeneous problems: conjectures of Minkowski and Watson. Number theory, 15–41, Trends
Math., Birkhuser, Basel, 2000.
[BR] R.P. Bambah and K.F. Roth, A note on lattice coverings, Journal of Indian
Mathematical Society, 16(1952), 7-12.
[BW] R.P. Bambah, A.C. Woods, Minkowski’s conjecture for n = 5; a theoremof Skubenko, J. of Number theory 12 (1980) 27-48.
[Bl] H. Blaney, Indefinite quadratic forms in n variables, Journal of London Mathematical Society, 23(1948), 153-160.
[BR] S. Bocherer, S. Raghavan, On Fourier coefficients of Siegel modular forms,
J. Reine Angew. Math 384 (1988) 8-101.
[BP] A. Borel and G. Prasad, Values of isotropic quadratic forms at S-integral
points,Compositio Math. 83 (1992), no. 3, 347–372.
[CLR]K. Chakraborty, A.K. Lal, B. Ramakrishnan, Modular forms which behave
like theta functions, Math. Computations 66(1997) 1169-1183.

15

[DM] S.G. Dani and G.A. Margulis, Values of quadratic forms at integral points:
an elementary approach, Enseign. Math. (2) 36 (1990), no. 1-2, 143–174.
[EMS] A. Eskin, S. Mozes, and N. Shah, Unipotent flows and counting lattice
points on homogeneous varieties, Ann. of Math. (2) 143 (1996), no. 2, 253–299.
[GL]Sudesh K. Gogia and I.S. Luthar, The Brauer-Siegel theorem for algebraic
function fields. J. Reine Angew. Math. 299/300 (1978), 28–37.
[GP1] B.H. Gross and D. Prasad, On the decomposition of a representation ofSO(n) when restricted to SO(n-1), Canadian J. of Maths 44, 974-1002 (1992).
[H] C. Hooley, On nonary cubic forms, J. Reine Angew. Math. 386(1988) 32-98.
[J] K. Joshi, A family of etale coverings of the Affine Line, Journal of Number
Theory, 59(1996), 414-418.
[KR] S.A. Katre and A.R. Rajwade, Resolution of sign ambiguity in the determination of the cyclotomic numbers of order 4 and corresponding Jacobsthal sum,
Math Scand. 60 (1987) 52-62.
[KS] Sudesh K. Khanduja and Jayanti Saha, Generalized Hensel’s lemma. Proc.
Edinburgh Math. Soc. (2) 42 (1999), no. 3, 469–480.
[Kh1] C. Khare, Congruences between cusp forms, Duke Mathematical Journal,
80(1995) 631-667.
[Kh2]C. Khare, Remarks on mod p forms of weight one, International Mathematics Research Notices, 3(1997), 127-133.
[MRV] M. Manickam, B. Ramakrishnan, TC Vasudevan, The theory of new formsof half integral weight, J. of Number theory 34 (1990) 210-224.
[N1] M.V.Nori, Unramified coverings of the affine line in positive characteristic,
Algebraic Geometry and its applications (West Lafayette, IN, 1990), SpringerVerlag (1994) 209-212.
[N2] M.V.Nori, Some Eisenstein cohomolgy classes, Proceedings of the International Congress of Mathematicians, Zurich 1994, Birkhauser (1995) 690-696.
[PAR] J.C. Parnami, M.K. Agrawal, A.R. Rajwade, The number of points on the
curve y 2 = x5 + a in Fq and applications to local zeta functions, Math. Student
48 (1980) 205-211.
[Pl] P.A.B. Pleasant, Cubic polynomials over algebraic number fields, J. of Number Theory 7(1975), 310-344.
[PV] R. Padma and S. Venkataraman, Elliptic curves with complex multiplication
and a character sum, Journal of Number Theory 61 (1996) 274-282.

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[P1] D. Prasad, Trilinear forms for representations of GL(2) and local epsilon
factors, Compositio Math 75, 1-46 (1990).
[P2] D. Prasad, Invariant linear forms for representations of GL(2) over a local
field, American J. of Maths 114, 1317-1363 (1992).
[RS1] S. Raghavan, Modular forms of degree n and representation by quadratic
forms, Annals of Mathematics, 70 (1959) 446-470.
[RS2] S. Raghavan, Singular modular forms of degree s, in C.P. Ramanujam: A
tribute, published by Springer Verlag for Tata Institute of Fundamental Research(1978) 263-272.
[RR1] S. Raghavan and K.G. Ramanathan, Solvability of Diophantine inequality
in Algebraic number fields, Acta Arithmetica (1972) 299-315.
[RR2] S. Raghavan and K.G. Ramanathan, Values of quadratic forms, J. of IndianMath. Society 34(1970), 253-257.
[Raj] C.S. Rajan, On the size of the Shafarevich-Tate group of elliptic curves
over function fields, Compositio Math. 105(1997) 29-41.
[R1] A.R.Rajwade, The number of solutions of the congruence y 2 ≡ x6 − a modp, Indian J. of pure and applied Math. 4(1973) 325-332.
[R2] A.R.Rajwade, On the congruence y 2 ≡ x5 − a mod p, Proc. Cambridge
Philosophical society, 74(1973) 473-475.
[R3] A. R.Rajwade, Notes on the congruence y 2 ≡ x5 −a mod p, l’Enseignement
Math 21 (1975) 49-56.
[RR] Madhu Raka and Urmila Rani, Positive values of inhomogeneous indefinite
ternary quadratic forms of type (2, 1), Hokkaido Math. J. 25(1996) 215-230.
[RCP1]C.P. Ramanujam, Sums of mth powers in p-adic rings, Mathematika,10(1963), 137-146.
[RCP2] C.P. Ramanujam, Cubic forms over algebraic number fields, Proceedingsof Cambridge Philosophical society, 59(1963), 683-705.
[Ram] K. Ramachandra, Some applications of Kronecker limit formula, Annalsof Mathematics, vol. 80, 1964, 104-148.
[Re] H.L.Resnikoff, Automorphic forms of singular weight are singular forms,
Math. Annalen 215(1975) 173-193.
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