The Achievements of C. S. Rajan

Description: C. S. Rajan was an Indian mathematician who made significant contributions to the field of number theory. He was particularly known for his work on modular forms and automorphic forms. This quiz will test your knowledge of the achievements of C. S. Rajan.
Number of Questions: 15
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What was C. S. Rajan's most significant contribution to number theory?

  1. His work on modular forms

  2. His work on automorphic forms

  3. His work on the Riemann hypothesis

  4. His work on the Goldbach conjecture


Correct Option: A
Explanation:

C. S. Rajan's most significant contribution to number theory was his work on modular forms. He developed a new method for constructing modular forms, which led to a number of important results in the field.

What is a modular form?

  1. A function that is invariant under the action of a group of matrices

  2. A function that is periodic in two variables

  3. A function that has a Fourier expansion

  4. A function that is holomorphic on the upper half-plane


Correct Option: A
Explanation:

A modular form is a function that is invariant under the action of a group of matrices. This means that if you apply any matrix in the group to the function, the function will remain unchanged.

What is an automorphic form?

  1. A function that is invariant under the action of a group of matrices

  2. A function that is periodic in two variables

  3. A function that has a Fourier expansion

  4. A function that is holomorphic on the upper half-plane


Correct Option: A
Explanation:

An automorphic form is a function that is invariant under the action of a group of matrices. This means that if you apply any matrix in the group to the function, the function will remain unchanged.

What was C. S. Rajan's work on modular forms used for?

  1. To prove the Riemann hypothesis

  2. To prove the Goldbach conjecture

  3. To construct new modular forms

  4. To study the distribution of prime numbers


Correct Option: C
Explanation:

C. S. Rajan's work on modular forms was used to construct new modular forms. This led to a number of important results in the field, including the discovery of new types of modular forms and the development of new methods for studying modular forms.

What was C. S. Rajan's work on automorphic forms used for?

  1. To prove the Riemann hypothesis

  2. To prove the Goldbach conjecture

  3. To construct new automorphic forms

  4. To study the distribution of prime numbers


Correct Option: C
Explanation:

C. S. Rajan's work on automorphic forms was used to construct new automorphic forms. This led to a number of important results in the field, including the discovery of new types of automorphic forms and the development of new methods for studying automorphic forms.

What was C. S. Rajan's most famous result?

  1. The Ramanujan-Petersson conjecture

  2. The Selberg trace formula

  3. The Langlands program

  4. The modularity theorem


Correct Option: A
Explanation:

C. S. Rajan's most famous result was the Ramanujan-Petersson conjecture. This conjecture was first proposed by Srinivasa Ramanujan in 1916, and it was eventually proved by C. S. Rajan in 1973.

What is the Ramanujan-Petersson conjecture?

  1. A conjecture about the distribution of prime numbers

  2. A conjecture about the modularity of elliptic curves

  3. A conjecture about the existence of new types of modular forms

  4. A conjecture about the trace formula for automorphic forms


Correct Option: C
Explanation:

The Ramanujan-Petersson conjecture is a conjecture about the existence of new types of modular forms. It was first proposed by Srinivasa Ramanujan in 1916, and it was eventually proved by C. S. Rajan in 1973.

What was C. S. Rajan's role in the development of the Langlands program?

  1. He was one of the main architects of the program

  2. He was a major contributor to the program

  3. He was a minor contributor to the program

  4. He had no role in the development of the program


Correct Option: B
Explanation:

C. S. Rajan was a major contributor to the development of the Langlands program. He made a number of important contributions to the program, including the development of new methods for studying automorphic forms and the discovery of new types of modular forms.

What is the Langlands program?

  1. A program to unify number theory and representation theory

  2. A program to study the distribution of prime numbers

  3. A program to construct new modular forms

  4. A program to study the trace formula for automorphic forms


Correct Option: A
Explanation:

The Langlands program is a program to unify number theory and representation theory. It was first proposed by Robert Langlands in 1967, and it has since become one of the most important and influential programs in mathematics.

What was C. S. Rajan's role in the development of the modularity theorem?

  1. He was one of the main architects of the theorem

  2. He was a major contributor to the theorem

  3. He was a minor contributor to the theorem

  4. He had no role in the development of the theorem


Correct Option: B
Explanation:

C. S. Rajan was a major contributor to the development of the modularity theorem. He made a number of important contributions to the theorem, including the development of new methods for studying modular forms and the discovery of new types of modular forms.

What is the modularity theorem?

  1. A theorem that relates modular forms to elliptic curves

  2. A theorem that relates automorphic forms to modular forms

  3. A theorem that relates number theory to representation theory

  4. A theorem that relates the distribution of prime numbers to modular forms


Correct Option: A
Explanation:

The modularity theorem is a theorem that relates modular forms to elliptic curves. It was first proposed by Andrew Wiles in 1993, and it was eventually proved by Wiles and Richard Taylor in 1995.

What was C. S. Rajan's role in the development of the trace formula for automorphic forms?

  1. He was one of the main architects of the formula

  2. He was a major contributor to the formula

  3. He was a minor contributor to the formula

  4. He had no role in the development of the formula


Correct Option: B
Explanation:

C. S. Rajan was a major contributor to the development of the trace formula for automorphic forms. He made a number of important contributions to the formula, including the development of new methods for studying automorphic forms and the discovery of new types of modular forms.

What is the trace formula for automorphic forms?

  1. A formula that relates the trace of an automorphic form to its Fourier coefficients

  2. A formula that relates the trace of an automorphic form to its eigenvalues

  3. A formula that relates the trace of an automorphic form to its poles

  4. A formula that relates the trace of an automorphic form to its zeros


Correct Option: A
Explanation:

The trace formula for automorphic forms is a formula that relates the trace of an automorphic form to its Fourier coefficients. It was first proposed by Atle Selberg in 1956, and it has since become one of the most important and influential formulas in mathematics.

What was C. S. Rajan's role in the development of the theory of automorphic forms?

  1. He was one of the main architects of the theory

  2. He was a major contributor to the theory

  3. He was a minor contributor to the theory

  4. He had no role in the development of the theory


Correct Option: B
Explanation:

C. S. Rajan was a major contributor to the development of the theory of automorphic forms. He made a number of important contributions to the theory, including the development of new methods for studying automorphic forms and the discovery of new types of modular forms.

What is the theory of automorphic forms?

  1. A theory that studies the properties of automorphic forms

  2. A theory that studies the relationship between automorphic forms and modular forms

  3. A theory that studies the relationship between automorphic forms and elliptic curves

  4. A theory that studies the relationship between automorphic forms and number theory


Correct Option: A
Explanation:

The theory of automorphic forms is a theory that studies the properties of automorphic forms. It is a vast and complex theory, and it has applications in a wide variety of areas of mathematics, including number theory, representation theory, and algebraic geometry.

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