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Let f be a primitive Hilbert modular form over F of weight k with coefficient field E-f, generated by the Fourier coefficients C(p, f) for p is an element of Spec(O-F). Under certain assumptions on the image of the residual Galois representations attached to f, we calculate the Dirichlet density of {p is an element of Spec(O-F)vertical bar E-f = Q(C(p, f))}. For k = 2, we show that those assumptions are satisfied when [E-f : Q] = [ F : Q] is an odd prime. We also study analogous results for F-f, the fixed field of E-f by the set of all inner twists of f. Then, we provide some examples of f to support our results. Finally, we compute the density of {p is an element of Spec(O-F)vertical bar C(p, f) is an element of K} for fields K with F-f subset of K subset of E-f.

Let f be a primitive Hilbert modular form over F of weight k with coefficient field E-f, generated by the Fourier coefficients C(p, f) for p is an element of Spec(O-F). Under certain assumptions on the image of the residual Galois representations attached to f, we calculate the Dirichlet density of {p is an element of Spec(O-F)vertical bar E-f = Q(C(p, f))}. For k = 2, we show that those assumptions are satisfied when [E-f : Q] = [ F : Q] is an odd prime. We also study analogous results for F-f, the fixed field of E-f by the set of all inner twists of f. Then, we provide some examples of f to support our results. Finally, we compute the density of {p is an element of Spec(O-F)vertical bar C(p, f) is an element of K} for fields K with F-f subset of K subset of E-f.

Let f be a primitive Hilbert modular form over F of weight k with coefficient field E-f, generated by the Fourier coefficients C(p, f) for p is an element of Spec(O-F). Under certain assumptions on the image of the residual Galois representations attached to f, we calculate the Dirichlet density of {p is an element of Spec(O-F)vertical bar E-f = Q(C(p, f))}. For k = 2, we show that those assumptions are satisfied when [E-f : Q] = [ F : Q] is an odd prime. We also study analogous results for F-f, the fixed field of E-f by the set of all inner twists of f. Then, we provide some examples of f to support our results. Finally, we compute the density of {p is an element of Spec(O-F)vertical bar C(p, f) is an element of K} for fields K with F-f subset of K subset of E-f.

In this lecture notes, we give an introduction to the p-adic analogues of the Birch and Swinnerton-Dyer conjecture for elliptic curves over Q , when p is a prime of split multiplicative reduction for the elliptic curve. We quickly go through the p-adic methods and the tools from Hida theory, state the exceptional zero conjecture, and give a sketch of the proof of a conjecture of Mazur, Tate and Teitelbaum on the first derivative of p-adic L-functions due to Greenberg and Stevens.

In this lecture notes, we give an introduction to the p-adic analogues of the Birch and Swinnerton-Dyer conjecture for elliptic curves over Q , when p is a prime of split multiplicative reduction for the elliptic curve. We quickly go through the p-adic methods and the tools from Hida theory, state the exceptional zero conjecture, and give a sketch of the proof of a conjecture of Mazur, Tate and Teitelbaum on the first derivative of p-adic L-functions due to Greenberg and Stevens.

In this lecture notes, we give an introduction to the p-adic analogues of the Birch and Swinnerton-Dyer conjecture for elliptic curves over Q , when p is a prime of split multiplicative reduction for the elliptic curve. We quickly go through the p-adic methods and the tools from Hida theory, state the exceptional zero conjecture, and give a sketch of the proof of a conjecture of Mazur, Tate and Teitelbaum on the first derivative of p-adic L-functions due to Greenberg and Stevens.

We show that signs of Fourier coefficients, on certain sub-families, determine the half-integral weight cuspidal eigenform uniquely, up to a positive constant. We also study sign change results for the product of the Fourier coefficients of two distinct half-integral weight eigenforms.

The Manin-Drinfeld theorem is an important result of the theory of modular forms on the existing of an isomorphism between the space of Eisenstein series E2(T0(N)) and a first homology group H1(X0(N)3 Z) of a modular curve X0(N). Explicitly such an isomorphism was first constructed in [1] in the case when N = p is prime. In the thesis we review the structure of an isomorphism in the case when N = p2 and p ≥ 3 is prime.

We show that signs of Fourier coefficients, on certain sub-families, determine the half-integral weight cuspidal eigenform uniquely, up to a positive constant. We also study sign change results for the product of the Fourier coefficients of two distinct half-integral weight eigenforms.

We show that signs of Fourier coefficients, on certain sub-families, determine the half-integral weight cuspidal eigenform uniquely, up to a positive constant. We also study sign change results for the product of the Fourier coefficients of two distinct half-integral weight eigenforms.

The Manin-Drinfeld theorem is an important result of the theory of modular forms on the existing of an isomorphism between the space of Eisenstein series E2(T0(N)) and a first homology group H1(X0(N)3 Z) of a modular curve X0(N). Explicitly such an isomorphism was first constructed in [1] in the case when N = p is prime. In the thesis we review the structure of an isomorphism in the case when N = p2 and p ≥ 3 is prime.

In this article, we study simultaneous sign changes of the Fourier coefficients of two Hilbert cusp forms of different non-parallel weights. We also study simultaneous non-vanishing of Fourier coefficients of two distinct non-zero primitive Hilbert cuspidal non-CM eigenforms of integral weights at powers of prime ideals.

In this article, we study simultaneous sign changes of the Fourier coefficients of two Hilbert cusp forms of different non-parallel weights. We also study simultaneous non-vanishing of Fourier coefficients of two distinct non-zero primitive Hilbert cuspidal non-CM eigenforms of integral weights at powers of prime ideals.

In this article, we study simultaneous sign changes of the Fourier coefficients of two Hilbert cusp forms of different non-parallel weights. We also study simultaneous non-vanishing of Fourier coefficients of two distinct non-zero primitive Hilbert cuspidal non-CM eigenforms of integral weights at powers of prime ideals.

The Manin-Drinfeld theorem is an important result of the theory of modular forms on the existing of an isomorphism between the space of Eisenstein series E2(T0(N)) and a first homology group H1(X0(N)3 Z) of a modular curve X0(N). Explicitly such an isomorphism was first constructed in [1] in the case when N = p is prime. In the thesis we review the structure of an isomorphism in the case when N = p2 and p ≥ 3 is prime.

This thesis consists of three parts. In the first part, we study the gaps between non-zero Fourier coefficients of cuspdial CM eigenforms in the short intervals. In the second part, we study the sign changes for the Fourier coefficients of Hilbert modular forms of half-integral weight. In the third part, we study the simultaneous behaviour of Fourier coefficients of two different Hilbert modular cusp forms of integral weight. In Chapter 1, we present the definitions and some preliminaries on classical modular forms. We shall also recall some relevant results from the literature, which are useful in the subsequent chapters. In Chapter 2, we show that for an elliptic curve E over Q of conductor N with complex multiplication (CM) by Q(i), the n-th Fourier coefficient of fE is non-zero in the short interval (X;X + cX 1 4 ) for all X � 0 and for some c > 0, where fE is the corresponding cuspidal Hecke eigenform in S2(

The aim of this thesis is to de�ne the Elliptic Curves and some of intresting properties of a special class of terms,namely,rational point of elliptic curve ,The properties of the rational pooint of curve is airthmetic ,and rational point on elliptic curve forms a �nitely generated group structure group it will be done by using the chord-tangent group law of composition.After completing the the abelian group structure,so we look some elementary properties like for a given elliptic curve we �nd the torsion subgroup of that elliptic curves.and �nally done the elliptic curve and their isomorphism. Further, we study different families of elliptic curves which depend on different parameters. Moreover, we look at the reduction modulo p of an elliptic curve and infer the meaning of good and bad reduction of an elliptic curve. Lastly, the statement and proof of Mordell-Weil theorem is given. The topics of the thesis are based on the book Elliptic Curves by Dale Husemoller. I have not added anything new, except making a few observations of my own.

In this article, we study (simultaneous) non-vanishing, (simultaneous) sign changes of Fourier coefficients of (two) Hilbert cusp forms, respectively.

In this article, we study (simultaneous) non-vanishing, (simultaneous) sign changes of Fourier coefficients of (two) Hilbert cusp forms, respectively.

In this article, we study (simultaneous) non-vanishing, (simultaneous) sign changes of Fourier coefficients of (two) Hilbert cusp forms, respectively.

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We prove an equidistribution of signs for the Fourier coefficients of Hilbert modular forms of half-integral weight. Our study focuses on certain subfamilies of coefficients that are accessible via the Shimura correspondence. This is a generalization of the result of Inam and Wiese (Arch Math (Basel) 101: 331-339, 2013) to the setting of totally real number fields.

We prove an equidistribution of signs for the Fourier coefficients of Hilbert modular forms of half-integral weight. Our study focuses on certain subfamilies of coefficients that are accessible via the Shimura correspondence. This is a generalization of the result of Inam and Wiese (Arch Math (Basel) 101: 331-339, 2013) to the setting of totally real number fields.

We prove an equidistribution of signs for the Fourier coefficients of Hilbert modular forms of half-integral weight. Our study focuses on certain subfamilies of coefficients that are accessible via the Shimura correspondence. This is a generalization of the result of Inam and Wiese (Arch Math (Basel) 101: 331-339, 2013) to the setting of totally real number fields.

We prove an equidistribution of signs for the Fourier coefficients of Hilbert modular forms of half-integral weight. Our study focuses on certain subfamilies of coefficients that are accessible via the Shimura correspondence. This is a generalization of the result of Inam and Wiese (Arch Math (Basel) 101: 331-339, 2013) to the setting of totally real number fields.

In this article, we study (simultaneous) non-vanishing, (simultaneous) sign changes of Fourier coefficients of (two) Hilbert cusp forms, respectively

In this article, we study (simultaneous) non-vanishing, (simultaneous) sign changes of Fourier coefficients of (two) Hilbert cusp forms, respectively

In this article, we study (simultaneous) non-vanishing, (simultaneous) sign changes of Fourier coefficients of (two) Hilbert cusp forms, respectively

I have greatly benefited reading a book of G. James and M. Liebeck for Representa- tion Theory and Kenneth Ireland and Michael Rosen for Classical introduction to modern number theory. I have presented a detailed version of the main results in this book.I have divided this thesis in two parts one part contains representation theory of finite groups and second part contains classical introduction to modern number theory. The main intersting thing in Representation Theory is to study a group G as a group of matrices. First we define FG -modules, group algebra , FG - Homomorphism. Then there is a big result namely Maschke’s Theorem, this tells that for a finite group G , if U be the FG -submodule of the FG-module then the FG- module can be written as the direct sum of two of its FG-submodule.And the valid- ity of Maschke’s Theorem hold is for the field of complex and real numbers.After this i discussed about the FG -homomomorphism with a big result namely Schur’s Lemma, For irreducible C G -modules , the C G -homomorphism is either a C G - isomorphism or it is the zero map and if we have a C G -homomorphism from the C G -module to itself then it is the scalar multiple of the identity endomorphism. Then i work on Characters of the C G -modules,inner product of characters and the number of the irreducible character with the proff that, all the irreducible charac- ters of a group G form a basis of the vector space of all class function on G . Then i discuss about how we can restrict a C G -module to a subgroup H of G and then how to construct a C G -module from a given C H -module.Then i discussed some topics in classical number theory including introduction to unique factorization in Z and congrunces, quadratic reciprocity. Then some results using the Legen- dre Symbol , Gauss’ Lemma. Then i worked on the Law of Quadratic Reciprocity, Quadratic Gauss sum , Jacobi sums and related results and then the law of Cubic and Biquadratic reciprocity. And then talked about the Aff

We show that if a modular cuspidal eigenform f of weight 2k is 2-adically close to an elliptic curve E/Q, which has a cyclic rational 4-isogeny, then n-th Fourier coefficient of f is non-zero in the short interval (X,X+cX14) for all X≫0 and for some c>0. We use this fact to produce non-CM cuspidal eigenforms f of level N>1 and weight k>2 such that if(n)≪n14 for all n≫0.

One of the main interesting topic in algebra is to find the roots of non-zero polynomials and writing down them explicitly over a given field K. The case which we consider in this project is the space of quadratic forms, which are homogeneous polynomials of degree 2, over Q. If they arise over Z, then we can study the zeros of these quadratic form by looking at them over Zp and study if that has zero in Zp or not. By Hensels lemma, this studied over the reduction modulo p.

One of the main interesting topic in algebra is to find the roots of non-zero polynomials and writing down them explicitly over a given field K. The case which we consider in this project is the space of quadratic forms, which are homogeneous polynomials of degree 2, over Q. If they arise over Z, then we can study the zeros of these quadratic form by looking at them over Zp and study if that has zero in Zp or not. By Hensels lemma, this studied over the reduction modulo p.

We show that if a modular cuspidal eigenform f of weight 2k is 2-adically close to an elliptic curve E/Q, which has a cyclic rational 4-isogeny, then n-th Fourier coefficient of f is non-zero in the short interval (X,X+cX14) for all X≫0 and for some c>0. We use this fact to produce non-CM cuspidal eigenforms f of level N>1 and weight k>2 such that if(n)≪n14 for all n≫0.

We prove several multiplicity one theorems for generalized modular functions (GMF), in terms of their q-exponents, and make a precise statement about the nature of values that the prime q-exponents of a GMF can take. We shall also study the integrality of general q-exponents of a GMF and give an upper bound on the first sign change of these q-exponents.

We prove several multiplicity one theorems for generalized modular functions (GMF), in terms of their q-exponents, and make a precise statement about the nature of values that the prime q-exponents of a GMF can take. We shall also study the integrality of general q-exponents of a GMF and give an upper bound on the first sign change of these q-exponents.

We prove several multiplicity one theorems for generalized modular functions (GMF), in terms of their q-exponents, and make a precise statement about the nature of values that the prime q-exponents of a GMF can take. We shall also study the integrality of general q-exponents of a GMF and give an upper bound on the first sign change of these q-exponents.

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[Abstract is not available]

This paper is to show a non-vanishing property of the derivatives of certain class of L-functions. We study the non-vanishing and transcendence of special values of L-functions and their derivatives, attached to (cuspidal) Siegel-Hecke eigenforms of genus 2, quadratic twists of classical Hecke eigenforms, and half-integral weight modular forms

This paper is to show a non-vanishing property of the derivatives of certain class of L-functions. We study the non-vanishing and transcendence of special values of L-functions and their derivatives, attached to (cuspidal) Siegel-Hecke eigenforms of genus 2, quadratic twists of classical Hecke eigenforms, and half-integral weight modular forms

This paper is to show a non-vanishing property of the derivatives of certain class of L-functions. We study the non-vanishing and transcendence of special values of L-functions and their derivatives, attached to (cuspidal) Siegel-Hecke eigenforms of genus 2, quadratic twists of classical Hecke eigenforms, and half-integral weight modular forms

We study the behavior of ordinary parts of the homology modules of modular curves, associated to a decreasing sequence of congruence subgroups Gamma(1)(N2(r)) for r >= 2, and prove a control theorem for these homology modules.

Let F be a Siegel cusp form of weight k and genus n > 1 with Fourier-Jacobi coefficients fm. In this article, we estimate the growth of the Petersson norms of fm, where m runs over an arithmetic progression. This result sharpens a recent result of Kohnen [5].

We study the behavior of ordinary parts of the homology modules of modular curves, associated to a decreasing sequence of congruence subgroups Gamma(1)(N2(r)) for r >= 2, and prove a control theorem for these homology modules.

Let F be a Siegel cusp form of weight k and genus n > 1 with Fourier-Jacobi coefficients fm. In this article, we estimate the growth of the Petersson norms of fm, where m runs over an arithmetic progression. This result sharpens a recent result of Kohnen [5].

We study the behavior of ordinary parts of the homology modules of modular curves, associated to a decreasing sequence of congruence subgroups Gamma(1)(N2(r)) for r >= 2, and prove a control theorem for these homology modules.