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Modulus (algebraic number theory)

In mathematics, in the field of algebraic number theory, a modulus (or an extended ideal) is a formal product of places of an algebraic number field. It is used to encode ramification data for abelian extensions of a number field.

Definition

Let K be an algebraic number field with ring of integers R. A modulus is a formal product

\[\mathbf{m} = \prod_{\mathbf{p}} \mathbf{p}^{\nu(\mathbf{p})} \]

where p runs over all places of K, finite or infinite, the exponents ν are zero except for finitely many p, for real places r we have ν(r)=0 or 1 and for complex places ν=0.

We extend the notion of congruence to this setting. Let x and y be elements of K. For a finite place p, that is, a prime ideal of the ring of integers, we define x and y to be congruent modulo pⁿ if x/y is in the valuation ring R_p of p and congruent to 1 modulo pⁿ in R_p in the usual sense of ring theory. For a real place r we define x and y to be congruent modulo r if x/y is positive in the real embedding of K associated to r. Finally, we define x and y to be congruent modulo m if they are congruent modulo p^ν(p) whenever ν(p) > 0.

Ray class group

We split the modulus m into m_fin and m_inf, the product over the finite and infinite places respectively. Define

\[ K_{\mathbf{m}} = \left\lbrace a/b \in K \mid a,b \in R,~ ab ~\mbox{coprime to}~ \mathbf{m}_\mbox{fin} \right\rbrace ,\] \[ K_{\mathbf{m},1} = \left\lbrace x \in K_{\mathbf{m}} \mid x \equiv 1 \pmod \mathbf{m} \right\rbrace .\]

We call the group K_m,1 the ray modulo m.

Further define the subgroup of the ideal group I^m to be the subgroup generated by ideals coprime to m_fin. The ray class group modulo m is the quotient I^m / i(K_m,1), where i is the map from K to principal ideals in the ideal group. A coset of i(K_m,1) is a ray class.

Hecke's original definition of Hecke characters may be interpreted in terms of characters of the ray class group with respect to some modulus m.

Properties

• When m = 1, the ray class group is just the ideal class group.
• The ray class group is finite. Its order is the ray class number.
• The ray class number divides the class number of K.

References

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Continuant

An algebraic function of a sequence of variables which has applications in generalized continued fractions and as the determinant of a tridiagonal matrix.

The $n$-th continuant, $K(n)$, of a sequence $\mathbf{a} = a_1,\ldots,a_n,\ldots$ defined recursively by $$ K(0) = 1 ; $$ $$ K(1) = a_1 ; $$ $$ K(n) = a_n K(n-1) + K(n-2) \ . $$ It may also be obtained by taking the sum of all possible products of $a_1,\ldots,a_n$ in which any pairs of consecutive terms are deleted.

An extended definition takes the continuant with respect to three sequences $\mathbf a$, $\mathbf b$, $\mathbf c$, so that $K(n)$ is a function of $a_1,\ldots,a_n$, $b_1,\ldots,b_{n-1}$, $c_1,\ldots,c_{n-1}$. In this case the recurrence relation becomes $$ K(0) = 1 ; $$ $$ K(1) = a_1 ; $$ $$ K(n) = a_n K(n-1) - b_{n-1}c_{n-1} K(n-2) \ . $$ Since $b_r$ and $c_r$ enter into $K$ only as the product $b_r c_r$ there is no loss of generality in assuming that the $b_r$ are all equal to 1.

The simple continuant gives the value of a continued fraction of the form $[a_0;a_1,a_2,\ldots]$. The $n$-th convergent is $$ \frac{K(n+1,(a_0,\ldots,a_n))}{K(n,(a_1,\ldots,a_n))} \ . $$

The extended continuant is the determinant of the tridiagonal matrix $$ \begin{pmatrix} a_1 & b_1 & 0 & 0 & \ldots & 0 & 0 \\ c_1 & a_2 & b_2 & 0 & \ldots & 0 & 0 \\ 0 & c_2 & a_3 & b_3 & \ldots & 0 & 0 \\ \vdots & \ddots & \ddots & \ddots & & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \ldots & c_{n-1} & a_n \end{pmatrix} \ . $$

References

• Thomas Muir. A treatise on the theory of determinants. (Dover Publications, 1960 [1933]), pp. 516-525.

ABC conjecture

In mathematics, the ABC conjecture relates the prime factors of two integers to those of their sum. It was proposed by David Masser and Joseph Oesterlé in 1985. It is connected with other problems of number theory: for example, the truth of the ABC conjecture would provide a new proof of Fermat's Last Theorem.

Statement

Define the radical of an integer to be the product of its distinct prime factors

\[ r(n) = \prod_{p|n} p \ . \]

Suppose now that the equation \(A + B + C = 0\) holds for coprime integers \(A,B,C\). The conjecture asserts that for every \(\epsilon > 0\) there exists \(\kappa(\epsilon) > 0\) such that

\[ |A|, |B|, |C| < \kappa(\epsilon) r(ABC)^{1+\epsilon} \ . \]

A weaker form of the conjecture states that

\[ (|A| \cdot |B| \cdot |C|)^{1/3} < \kappa(\epsilon) r(ABC)^{1+\epsilon} \ . \]

If we define

\[ \kappa(\epsilon) = \inf_{A+B+C=0,\ (A,B)=1} \frac{\max\{|A|,|B|,|C|\}}{N^{1+\epsilon}} \ , \]

then it is known that \(\kappa \rightarrow \infty\) as \(\epsilon \rightarrow 0\).

Baker introduced a more refined version of the conjecture in 1998. Assume as before that \(A + B + C = 0\) holds for coprime integers \(A,B,C\). Let \(N\) be the radical of \(ABC\) and \(\omega\) the number of distinct prime factors of \(ABC\). Then there is an absolute constant \(c\) such that

\[ |A|, |B|, |C| < c (\epsilon^{-\omega} N)^{1+\epsilon} \ . \]

This form of the conjecture would give very strong bounds in the method of linear forms in logarithms.

Results

It is known that there is an effectively computable \(\kappa(\epsilon)\) such that

\[ |A|, |B|, |C| < \exp\left({ \kappa(\epsilon) N^{1/3} (\log N)^3 }\right) \ . \]

References

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Szpiro's conjecture

A conjectural relationship between the conductor and the discriminant of an elliptic curve. In a general form, it is equivalent to the well-known ABC conjecture. It is named for Lucien Szpiro who formulated it in the 1980s.

The conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with minimal discriminant Δ and conductor f, we have

\[ \vert\Delta\vert \leq C(\varepsilon ) \cdot f^{6+\varepsilon }. \, \]

The modified Szpiro conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with invariants c₄, c₆ and conductor f, we have

\[ \max\{\vert c_4\vert^3,\vert c_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon }. \, \]

References

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