P-adic valuation

Definition

The p-adic valuation is a way of measuring the 'size' or 'divisibility' of a number with respect to a prime number p. It assigns to each rational number a non-negative integer or infinity, indicating how many times p divides that number, which is crucial in the study of p-adic numbers and their applications, including modular forms and number theory.

5 Must Know Facts For Your Next Test

1. The p-adic valuation of a rational number can be expressed as v_p(a/b) = v_p(a) - v_p(b), where a and b are integers and v_p denotes the p-adic valuation.
2. If a number is not divisible by p, its p-adic valuation is defined to be zero.
3. The p-adic valuation has an important role in defining the topology of the p-adic numbers, making them complete with respect to this valuation.
4. In the context of modular forms, the p-adic valuation helps understand the properties of coefficients in power series expansions related to these forms.
5. The concept of p-adic valuation can also be extended to include infinite values, allowing for comparisons between different valuations across various primes.

Review Questions

• How does the p-adic valuation define the structure of p-adic numbers and their properties?
  • The p-adic valuation fundamentally shapes the structure of p-adic numbers by establishing a way to measure divisibility by prime p. This valuation allows for a unique topology on the set of p-adic numbers, making them complete in this sense. As such, it leads to interesting properties such as the convergence of series and the behavior of sequences in the realm of number theory.
• Discuss how the concept of p-adic valuation interacts with modular forms and its significance in number theory.
  • The p-adic valuation plays a significant role in understanding modular forms by providing insight into their coefficients and functional behavior. For instance, the evaluation of these coefficients under the lens of p-adic numbers can reveal deep connections between modular forms and other areas like Galois representations. This interaction is vital for applications in areas such as arithmetic geometry and cryptography.
• Evaluate the implications of using p-adic valuations in modern mathematical research and its contributions to advanced theories.
  • Using p-adic valuations has profound implications in modern mathematical research, especially within number theory and algebraic geometry. By allowing researchers to analyze problems through different lenses—such as local fields and completions—these valuations enable new approaches to classical problems. This perspective aids in proving results like the Langlands program, which connects number theory with representation theory, showcasing the depth and versatility of p-adic analysis in contemporary mathematics.

Related terms

p-adic numbers: The p-adic numbers are a system of numbers that extend the ordinary arithmetic of rational numbers, allowing for convergence with respect to the p-adic valuation.

Valuation ring: A valuation ring is a special type of integral domain that corresponds to a specific valuation, like the p-adic valuation, and includes all elements that have a non-negative valuation.

Modular forms: Modular forms are complex functions that are invariant under certain transformations and have applications in number theory, particularly in connection with the p-adic valuation and its properties.