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changed a C_p to C_p-0 and added an "I".
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The answer to the short version of your question is: yes, there is a $p$-adic theory of integration.

As to whether an analogue of Cauchy's theorem holds in such a theory, I assume you are thinking of Cauchy's integral formula, and I work with the theory of Coleman. That formula amounts to the statement that the complex logarithm is a multivalued function. But over the $p$-adics, for each choice of $\mathscr{L}\in\mathbb{C}_p$ there is a so-called branch of the $p$-adic logarithm, which is defined on all $\mathbb{C}_p$$\mathbb{C}_p^\times$, uniquely determined by the fact that it behaves as expected for the product and it extends the convergent power series

$$\log(1+x)=\sum_{n=1}^\infty\frac{(-x)^n}{n}$$

of $x\in\mathbb{C}_p$ with $v_p(x)>0$. Each of these provides a primitive of the function $f(x)=\frac{1}{x}$ on $\mathbb{C}_p$, and the ``Fundamental Theorem of Calculus´´ satisfied by Coleman's theory of $p$-adic line integrals, will tell you that the integral of $f(x)$ along a closed loop around $0\in\mathbb{C}_p$ is zero.

A good reference:

MR0782557 (86j:14014) Coleman, Robert F. Torsion points on curves and $p$-adic abelian integrals. Ann. of Math. (2) 121 (1985), no. 1, 111–168.

The answer to the short version of your question is: yes, there is a $p$-adic theory of integration.

As to whether an analogue of Cauchy's theorem holds in such a theory, I assume you are thinking of Cauchy's integral formula, and work with the theory of Coleman. That formula amounts to the statement that the complex logarithm is a multivalued function. But over the $p$-adics, for each choice of $\mathscr{L}\in\mathbb{C}_p$ there is a so-called branch of the $p$-adic logarithm, which is defined on all $\mathbb{C}_p$, uniquely determined by the fact that it behaves as expected for the product and it extends the convergent power series

$$\log(1+x)=\sum_{n=1}^\infty\frac{(-x)^n}{n}$$

of $x\in\mathbb{C}_p$ with $v_p(x)>0$. Each of these provides a primitive of the function $f(x)=\frac{1}{x}$ on $\mathbb{C}_p$, and the ``Fundamental Theorem of Calculus´´ satisfied by Coleman's theory of $p$-adic line integrals, will tell you that the integral of $f(x)$ along a closed loop around $0\in\mathbb{C}_p$ is zero.

A good reference:

MR0782557 (86j:14014) Coleman, Robert F. Torsion points on curves and $p$-adic abelian integrals. Ann. of Math. (2) 121 (1985), no. 1, 111–168.

The answer to the short version of your question is: yes, there is a $p$-adic theory of integration.

As to whether an analogue of Cauchy's theorem holds in such a theory, I assume you are thinking of Cauchy's integral formula, and I work with the theory of Coleman. That formula amounts to the statement that the complex logarithm is a multivalued function. But over the $p$-adics, for each choice of $\mathscr{L}\in\mathbb{C}_p$ there is a so-called branch of the $p$-adic logarithm, which is defined on all $\mathbb{C}_p^\times$, uniquely determined by the fact that it behaves as expected for the product and it extends the convergent power series

$$\log(1+x)=\sum_{n=1}^\infty\frac{(-x)^n}{n}$$

of $x\in\mathbb{C}_p$ with $v_p(x)>0$. Each of these provides a primitive of the function $f(x)=\frac{1}{x}$ on $\mathbb{C}_p$, and the ``Fundamental Theorem of Calculus´´ satisfied by Coleman's theory of $p$-adic line integrals, will tell you that the integral of $f(x)$ along a closed loop around $0\in\mathbb{C}_p$ is zero.

A good reference:

MR0782557 (86j:14014) Coleman, Robert F. Torsion points on curves and $p$-adic abelian integrals. Ann. of Math. (2) 121 (1985), no. 1, 111–168.

Source Link
monodromy
monodromy
  • 786
  • 7
  • 13

The answer to the short version of your question is: yes, there is a $p$-adic theory of integration.

As to whether an analogue of Cauchy's theorem holds in such a theory, I assume you are thinking of Cauchy's integral formula, and work with the theory of Coleman. That formula amounts to the statement that the complex logarithm is a multivalued function. But over the $p$-adics, for each choice of $\mathscr{L}\in\mathbb{C}_p$ there is a so-called branch of the $p$-adic logarithm, which is defined on all $\mathbb{C}_p$, uniquely determined by the fact that it behaves as expected for the product and it extends the convergent power series

$$\log(1+x)=\sum_{n=1}^\infty\frac{(-x)^n}{n}$$

of $x\in\mathbb{C}_p$ with $v_p(x)>0$. Each of these provides a primitive of the function $f(x)=\frac{1}{x}$ on $\mathbb{C}_p$, and the ``Fundamental Theorem of Calculus´´ satisfied by Coleman's theory of $p$-adic line integrals, will tell you that the integral of $f(x)$ along a closed loop around $0\in\mathbb{C}_p$ is zero.

A good reference:

MR0782557 (86j:14014) Coleman, Robert F. Torsion points on curves and $p$-adic abelian integrals. Ann. of Math. (2) 121 (1985), no. 1, 111–168.