## Some congruences for binomial coefficients. II.(English)Zbl 0971.11046

Let $$p= tn+r$$ be a prime where $$t$$, $$n$$, $$r$$ are positive integers $$(1\leq r\leq t-1)$$ and either (1) $$t= k>3$$ for a prime $$k\equiv 3\pmod 4$$ or (2) $$t=4k$$ for a prime $$k\equiv 1\pmod 4$$. Let $$H$$ be the subgroup with the operator $$r$$ of $$(\mathbb{Z}/t\mathbb{Z})^\times$$ isomorphic to $$\text{Gal} (\mathbb{Q}(\xi_t)/ \mathbb{Q}(\sqrt{-t}))$$, $$h$$ be the class number of $$\mathbb{Q}(\sqrt{-t})$$, $$s= \frac{\varphi(t)} {2}$$ and $$d= ((p^s-1)/t) (t-1)= \sum_{j=0}^{s-1} d_j p^j$$ $$(0\leq d_j< p)$$. The existence of integers $$a$$, $$b$$ was shown by the authors [Proc. Class Field Theory – its Centenary and Prospect (1998), Adv. Stud. Pure Math. 30, 445-461 (2001)] such that $$4p^h= a^2+ tb^2$$ and $a\equiv\pm \prod_{j=0}^{s-1} (d_j)!\pmod p.$ If we consider $$k=1$$ and $$r=1$$, then we get Gauss’ statement $$a\equiv \binom {2n}{n}\pmod p$$ $$(4p= a^2+ 4b^2$$, $$\frac{a}{2}\equiv 1\pmod 4)$$.
The main result of this paper determines the sign of $$a$$ in the following sense: In case (1) $$a\equiv 2p^\beta\pmod t$$ and $$a\equiv (-1)^\beta \prod_{j=0}^{s-1} (d_j)!\pmod p$$ and in case (2) $$a\equiv -p^\beta\pmod k$$ and $$2a\equiv (-1)^\alpha \prod_{j=0}^{k-2} (d_j)!\pmod p$$.
The integer $$\beta$$ is defined by $\beta= \Bigl( \sum_{1\leq i\leq t, -i\in H} i\Bigr)/t$ and $$\alpha= s-\beta$$. To prove these results some equalities on Eisenstein sums are presented.

### MSC:

 11L05 Gauss and Kloosterman sums; generalizations 11T24 Other character sums and Gauss sums 11B65 Binomial coefficients; factorials; $$q$$-identities 11A07 Congruences; primitive roots; residue systems

### Keywords:

Gauss sum; Eisenstein sum; binomial coefficients; congruences
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### References:

 [1] Berndt, B. C., Evans, R. J., and Williams, K. S.: Gauss and Jacobi Sums. Canad. Math. Soc. Ser. Monogr. Adv. Texts, vol. 21, Wiley, New York (1998). · Zbl 0906.11001 [2] Clarke, F.: Closed formulas for representing primes by binary quadratic forms (1998) (preprint, announced in ICM, Berlin). [3] Cohen, H.: A Course in Computational Algebraic Number Theory. Grad. Texts in Math., vol. 138, Springer, Berlin-Heidelberg-New York (1995). [4] Eisenstein, G.: Zur Theorie der quadratischen Zerfällung der Primzahlen $$8n+3$$, $$7n+2$$ und $$7n+\nobreak4$$. J. Reine Angew. Math., 37 , 97-126 (1848). [5] Hahn, S., and Lee, D. H.: Some congruences for binomial coefficients. Proc. for Class Field Theory - its Centenary and Prospect (1998) (to appear). · Zbl 1070.11007 [6] Lang, S.: Cyclotomic Fields I and II. 2nd ed., Grad. Texts in Math., vol. 121, Springer, Berlin-Heidelberg-New York (1990). · Zbl 0704.11038 [7] Lee, D. H., and Hahn, S.: Gauss sums and binomial coefficients. J. Number Theory (2000) (submitted). · Zbl 0994.11029 · doi:10.1006/jnth.2001.2688 [8] Washington, L. C.: Introduction to Cyclotomic Fields. 2nd ed., Grad. Texts in Math., vol. 83, Springer, Berlin-Heidelberg-New York (1997). · Zbl 0966.11047
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